TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 9, September 1997, Pages 3469–3516 S 0002-9947(97)01793-5

ON PERFECT ISOMETRIES AND ISOTYPIES IN ALTERNATING GROUPS

PAUL FONG AND MORTON E. HARRIS

Abstract. Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broué conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized Fitting subgroup isomorphic to an alternating group.

Let G be a finite group and let O be a complete discrete valuation ring with field of quotients K of characteristic 0 and residue class field k of characteristic p > 0. We suppose that K contains a primitive |G|-th root of unity. In [4, (6.1)] Michel Broué posed the following

Isotypy Conjecture. Let e be a block of OG with abelian defect group D and let f be the Brauer correspondent of e in ON_{G}(D). Then e and f are isotypic blocks.

If G has an abelian Sylow p-subgroup, then the conjecture can be posed for the principal block of OG. In this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian Sylow p-subgroup (see [10, (5E)]). In this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups. In addition, the extended conjecture is proved for the principal p-block of almost simple groups with abelian Sylow p-subgroups and generalized Fitting subgroup isomorphic to an alternating group.

We recall the basic definitions. Let ¯ : O ⟶ k be the canonical quotient mapping and let ¯ : OG ⟶ kG be the induced O-algebra homomorphism of the group algebras. In particular, ¯ : e ⟶ ē induces a bijection between central idempotents of OG and kG. If e is a block idempotent of OG, let _{KGe}Mod be the category of left KGe-modules of finite type and let R_{K}(G, e) be the Grothendieck group of _{KGe}Mod. Let G^{∨} be the set of irreducible characters of G over K. We identify R_{K}(G, e) with the free abelian group on

(G, e)^{∨} = {χ ∈ G^{∨} : χ(ge) = χ(g) for all g ∈ G}.

Let CF(G, K) be the K-space of K-valued class functions on G, and let CF(G, e, K) be the K-subspace of class functions α in CF(G, K) such that α(ge) = α(g). The

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors March 26, 1996.

1991 Mathematics Subject Classification. Primary 20C15, 20C20; Secondary 20C30.

The first author was supported in part by NSF grant DMS 9100310.

The second author was supported in part by NSA grant MDA 904 92-H-3027.

K-subspaces of CF(G, K) and CF(G, e, K) of class functions vanishing on the psingular conjugacy classes of G will be denoted by CF_{p′}(G, K) and CF_{p′}(G, e, K).

Similar notation will be used for a second finite group H and a block idempotent f of OH. We denote R_{K}(G × H, e ⊗ f) also as R_{K}((G, e), (H, f)). Each μ in R_{K}((G, e), (H, f)) determines adjoint linear maps

R_{μ}: CF(G, e, K) ⟶ CF(H, f, K), I_{μ}: CF(H, f, K) ⟶ CF(G, e, K),

where

R_{μ}(α)(h) = \frac{1}{|G|} ∑_{g∈G} μ(g, h^{−1})α(g^{−1}), I_{μ}(β)(g) = \frac{1}{|H|} ∑_{h∈H} μ(g, h^{−1})β(h)

for α ∈ CF(G, e, K) and β ∈ CF(H, f, K). Moreover, R_{μ}: R_{K}(G, e) ⟶ R_{K}(H, f) and I_{μ}: R_{K}(H, f) ⟶ R_{K}(G, e).

Definition. A virtual character μ of G × H is perfect if the following separation and integrality conditions hold for all g ∈ G, h ∈ H:

1. μ(g, h) = 0 if {g, h} consists of a p-regular element and a p-singular element.

2. μ(g, h) is divisible in O by |C_{G}(g)|_{p} and by |C_{H}(h)|_{p}. If μ is perfect, then R_{μ} and I_{μ} induce K-linear mappings

R_{μ,p′}: CF_{p′}(G, e, K) ⟶ CF_{p′}(H, f, K),
I_{μ,p′}: CF_{p′}(H, f, K) ⟶ CF_{p′}(G, e, K)

by restriction. We say μ is an isometry if R_{μ} is an isometry of R_{K}(G, e) onto R_{K}(H, f). An isometry μ is thus a pair (J_{μ}, ϵ_{μ}) consisting of a bijection and a sign function

J_{μ}: (G, e)^{∨} ⟶ (H, f)^{∨}, ϵ_{μ}: (G, e)^{∨} ⟶ {±1}

such that R_{μ}(χ) = ϵ_{μ}(χ) J_{μ}(χ) for all χ ∈ (G, e)^{∨}. In particular,

μ(g, h) = ∑_{χ∈(G,e)^{∨}} ϵ_{μ}(χ)(χ(g) × J_{μ}(χ)(h^{−1})).

It will be convenient to set G_{P} = C_{G}(P) and G_{x} = C_{G}(x) when P is a p-subgroup of G and x is a p-element of G. The Brauer morphism Br_{P}^{G}: OG ⟶ kG_{P} is the O-linear mapping defined by

Br_{P}^{G}(g) = \begin{cases}g & if g ∈ G_{P} , \\ 0 & if g ∉ G_{P} . \end{cases}

Br_{P}^{G} preserves the N_{G}(P)-conjugation action. Let e and ē be corresponding blocks of OG and kG. An ē-subpair is a pair (P, ū), where P is a p-subgroup of G, ū is a block of kG_{P}, and (1, ē) ≤ (P, ū) is containment in the sense of [1, Definition 3.3]. An e-subpair is a pair (P, u), where P is a p-subgroup of G, u is a block of OG_{P}, and (P, ū) is an ē-subpair. Containment of such pairs is defined in the obvious way: (Q, v) ≤ (P, u) if (Q, v̄) ≤ (P, ū). Let Sp(G, e) denote the set of e-subpairs of G. An ē-element is a pair (x, ū), where x is a p-element of G, ū is a block of kG_{x}, and (⟨x⟩, ū) is an ē-subpair. An e-element is a pair (x, u), where x is a p-element of G, u is a block of OG_{x}, and (x, ū) is an ē-element. The decomposition mapping

d_{G}^{(x,u)} : CF(G, K) ⟶ CF_{p′}(G_{x}, u, K)

is defined by d_{G}^{(x,u)}(α)(y) = α(xyu) for α ∈ CF(G, K), y ∈ G_{x,p′}. If (x, u) is an e-element, then α(xyu) = α(xyeu) by Brauer’s Second Main Theorem (see [2, §A2, Théorème A2.1]).

The objects and morphisms of the Brauer category Br_{e}(G) are the e-subpairs (P, u) of G and the mappings (P, u) ⟶ (Q, v) in Hom(P, Q) induced by conjugation by elements g of G such that (P, u)^{g} ≤ (Q, v). Let (D, e_{D}) be a maximal e-subpair of G. Each subgroup P of D determines a unique block e_{P} of OG_{P} such that (P, e_{P}) ≤ (D, e_{D}). If P = ⟨z⟩, we also write e_{z} for e_{P}. Let Br_{e,D}(G) be the full subcategory of Br_{e}(G) whose objects are such subpairs (P, e_{P}). Since any two maximal e-subpairs are conjugate under G, Br_{e,D}(G) is independent of the choice of (D, e_{D}) up to a categorical isomorphism. Moreover, the inclusion functor of Br_{e,D}(G) into Br_{e}(G) is a categorical equivalence, since each e-subpair of G is conjugate in G to an e-subpair (P, e_{P}) contained in (D, e_{D}). Similar notation will be used for a block idempotent f of OH with defect group D.

Definition. ([4, Definition 4.3]) Let e and f be block idempotents of OG and OH with a common defect group D. Let (D, e_{D}) and (D, f_{D}) be a maximal e-subpair and a maximal f-subpair of G and H. Then e and f are isotypic if the following conditions hold:

(I1) The inclusion of D in G and H induces an equivalence of the categories

Br_{e,D}(G) and Br_{f,D}(H). (I2) There exists a perfect isometry μ_{P} in R_{K}((G_{P} , e_{P}), (H_{P} , f_{P})) for each

cyclic subgroup P of D such that the diagram

CF(G, e, K) ⟶^{R_{μ_{⟨1⟩}}} CF(H, f, K)
↓_{d_{G}^{(x,e_{P} )}}  ↓_{d_{H}^{(x,f_{P} )}}
CF_{p′}(G_{P} , e_{P} , K) ⟶^{R_{μ_{P} ,p′}} CF_{p′}(H_{P} , f_{P} , K)

(0.1)

is commutative for every generator x of P.

We note that the horizontal arrows in the diagram depend only on P, whereas the vertical arrows depend on the generator x of P. The set of perfect isometries {μ_{P} } is called the local system of the isotypy.

Remark. The actual statement of the Isotypy Conjecture in [4, (6.1)] asserts the following: Let e be a block of OG with abelian defect group D, let (D, e_{D}) be a maximal e-subpair of G, and let T = N_{G}(D, e_{D}). Then e and e_{D} are isotypic blocks of OG and OT. This statement is equivalent to the one given in the introduction of this paper since f and e_{D} are isotypic blocks of OH and OT by the following general fact: Suppose C is a normal subgroup of a finite group H, b is a block of OC, and T = N_{H}(b). Let u and v be blocks of OH and OT covering b such that u = Tr_{T}^{H}(v). Then u and v are isotypic blocks of OH and OT. Indeed, u and v have a common defect group D. If (P, ϕ) is a v-subpair of T, then (P, Tr_{T_{P}}^{H_{P}} (ϕ)) is a u-subpair of H. The mapping Sp(T, v) ⟶ Sp(H, u) defined by (P, ϕ) ↦ (P, Tr_{T_{P}}^{H_{P}} (ϕ)) implies condition (I1) by [13, Theorem 1] and [4, 4.2, Remark 2]. Let (D, u_{D}) and (D, v_{D}) be a maximal u-subpair of H and a maximal v-subpair of T such that u_{D} = Tr_{T_{D}}^{H_{D}}(v_{D}). For each cyclic subgroup P of D, the categorical equivalences

OT_{P} v_{P} Mod ⟶^{Ind_{T_{P}}^{H_{P}}}  _{OH_{P} u_{P}} Mod,
OH_{P} u_{P} Mod ⟶^{v_{P} Res_{T_{P}}^{H_{P}}} _{OT_{P} v_{P}} Mod

induce perfect isometries μ_{P} in R_{K}((H_{P} , u_{P}), (T_{P} , v_{P})) satisfying (I2).

The isotypies for the alternating group A_{n} will be constructed from isotypies for the symmetric group S_{n}. In §1 we consider a configuration (G, G^{+}, H, H^{+}) of finite groups, where G^{+} is normal in G, H^{+} is normal in H, G/G^{+} ≃ H/H^{+}; a configuration (e, e^{+}, f, f^{+}) of blocks of OG, OG^{+}, OH, OH^{+}, where e covers e^{+} and f covers f^{+}; and an isotypy between e and f with local system {μ_{P} }. An isotypy between e^{+} and f^{+} can be constructed under suitable conditions on {μ_{P} }. In the given context the relevant condition is that of +equivariance considered in §2. This construction will then be applied to configurations where G = S_{n}, G^{+} = A_{n}, e^{+} is a block of OG^{+} with abelian defect group D^{+}, H = N_{G}(D^{+}), H^{+} = N_{G^{+}}(D^{+}), and f^{+} is the Brauer correspondent of e^{+}. If p is odd, then e and f have D^{+} as defect group. In this case Rouquier ([14, Théorème 2.13]) has shown that e and f are isotypic blocks. However, the isotypies in [Ro] are not explicit enough to check the condition of +equivariance. We therefore give in §3 an explicit +-equivariant isotypy between e and f. §4 contains the relevant character theory of H^{+}. The main result there is an analog of Frobenius’s theorem on irreducible characters of G^{+} which do not extend to characters of G. §5 contains the construction of the isotypy between e^{+} and f^{+} when p > 2, and §6 contains the proof that this isotypy satisfies the extended conjecture of [10, (5E)] when e^{+} and f^{+} are the principal blocks. §7 treats the case p = 2.

1. Some configurations of isotypies

An isotypy between blocks e and f of OG and OH with a common defect group D requires a perfect isometry μ_{P} for each cyclic subgroup P of D such that (0.1) is commutative for every generator x of P. In practice somewhat less is required. The following reduction steps will be used in our proofs.

Step 1. Suppose e and f are blocks of principal type, that is, Br_{P}^{G}(e) and Br_{P}^{H}(f) are blocks of OG_{P} and OH_{P} for each subgroup P of D. Then the decomposition mappings d_{G}^{(x,e_{x})} and d_{H}^{(x,f_{x})} in (0.1) are simply d_{G}^{x} and d_{H}^{x} by Brauer’s Second Main Theorem.

Step 2. μ_{⟨1⟩} need only be an isometry, since μ_{⟨1⟩} is perfect provided the other conditions hold [4, Remark following 4.6].

Step 3. If (I1) holds, the commutativity of (0.1) need only be checked for (P, e_{P}) with P in a set P of representatives for the N_{G}(D, e_{D})-conjugacy classes of nontrivial cyclic subgroups of D and for all generators x of such P. This follows by transport of structure. For suppose (0.1) commutes for all generators x of such a P. Fix g ∈ N_{G}(D, e_{D}) and let Q = P^{g}, so that e_{Q} = (e_{P})^{g}. Choose h ∈ N_{H}(D, f_{D}) so that h and g induce the same automorphism of D. Then Q = P^{h} and f_{Q} = (f_{P})^{h}. Define the perfect isometry μ_{Q} in R_{K}((G_{Q}, e_{Q}), (H_{Q}, f_{Q})) by μ_{Q}(y, z) = μ_{P} (gyg^{−1}, hzh^{−1}) for y ∈ G_{Q}, z ∈ H_{Q}. Then (0.1) with P replaced by Q and x replaced by any generator of Q is commutative.

For the rest of this section σ is a fixed linear character of G. Multiplication by σ gives an action χ ↦ σχ on G^{∨}; an action e ↦ σe on central idempotents of OG, where σe = ∑_{g∈G_{p′}} σ(g^{−1})c_{g}g if e = ∑_{g∈G_{p′}} c_{g}g; and an action ē ↦ σē on central idempotents of kG, where σē = ∑_{g∈G_{p′}} σ̄(g^{−1})c̄_{g}g if ē = ∑_{g∈G_{p′}} c̄_{g}g. In particular, σē = \overline{σe}. If e is a block idempotent of OG, then χ ∈ (G, e)^{∨} if and only if σχ ∈ (G, σe)^{∨}. Moreover, e and σe have the same defect groups, and σe = e if σ has p-power order. Let e be a fixed block of OG. If P is a p-subgroup of G, set σ_{P} = Res_{G_{P}}^{G} (σ) for simplicity of notation. In particular, Br_{P}^{G}(σe) = σ_{P} Br_{P}^{G}(e). If (P, u) is an e-subpair of G, then (P, σ_{P} u) is a σe-subpair of G. More generally, let (P, u) and (Q, v) be subpairs of G. Then (P, u) ≤ (Q, v) if and only if (P, σ_{P} u) ≤ (Q, σ_{Q}v). Indeed, we need only show this for normal containment, since containment of subpairs is by definition a chain of normal containments. Let (P, u) ◁ (Q, v). Then P ◁ Q, u is stable under Q, and Br_{Q}^{G_{P} Q}(u)v = v. So σ_{P} u is stable under Q and Br_{Q}^{G_{P} Q}(Res_{G_{P} Q}^{G}(σ)u)·σ_{Q}v = σ_{Q}v. Thus (P, σ_{P} u)◁(Q, σ_{Q}v). A similar argument using σ^{−1} shows that if (P, σ_{P} u) ◁ (Q, σ_{Q}v), then (P, u) ◁ (Q, v).

Lemma (1A). Br_{e,D}(G) and Br_{σe,D}(G) are isomorphic categories under the inclusion of a common defect group D of e and σe in G.

Proof. We fix a maximal e-subpair (D, e_{D}) of G and the corresponding maximal σe-subpair (D, σ_{D}e_{D}) of G. If P ≤ D and (P, e_{P}) ≤ (D, e_{D}), then (P, σ_{P} e_{P}) ≤ (D, σ_{D}e_{D}) and σ_{P} e_{P} = (σe)_{P}. To complete the argument, we show that

Hom((P, e_{P}), (Q, e_{Q})) = Hom((P, σ_{P} e_{P}), (Q, σ_{Q}e_{Q}))

as subsets of Hom(P, Q). If (P, e_{P})^{g} = (P^{g}, (e_{P})^{g}) ≤ (Q, e_{Q}) for g ∈ G, then (P, σ_{P} e_{P})^{g} = (P^{g}, σ_{P^{g}}e_{P^{g}}) ≤ (Q, σ_{Q}e_{Q}) and g ∈ Hom((P, σ_{P} e_{P}), (Q, σ_{Q}e_{Q})). A similar argument using the linear character σ^{−1} gives the reverse inclusion.

Proposition (1B). Suppose σ is trivial on a defect group D of e and σe. Then e and σe are isotypic blocks with local system {μ_{P} }, where μ_{P} = (J_{P} , ϵ_{P}) for each cyclic subgroup P of D is the isometry in R_{K}((G_{P} , e_{P}), (G_{P} , σ_{P} e_{P})) such that J_{P} (ζ) = σ_{P} ζ and ϵ_{P} (ζ) = 1 for ζ ∈ (G_{P} , e_{P})^{∨}.

Proof. Condition (I1) for an isotypy holds by (1A). The isometry μ_{P} is perfect since

μ_{P} (x, y) = ∑_{ζ∈(G_{P} ,e_{P})^{∨}} ζ(x)σ_{P} (y^{−1})ζ(y^{−1}) = σ_{P} (y^{−1}) ∑_{ζ∈(G_{P} ,e_{P})^{∨}} ζ(x)ζ(y^{−1})

and the identity isometry of (G_{P} , e_{P})^{∨} is perfect. The diagram

CF(G, e, K) ⟶^{R_{μ_{⟨1⟩}}} CF(G, σe, K)
↓_{d_{G}^{(x,e_{P} )}}  ↓_{d_{G}^{(x,σ_{P} e_{P} )}}
CF_{p′}(G_{P} , e_{P} , K) ⟶^{R_{μ_{P} ,p′}} CF_{p′}(G_{P} , σ_{P} e_{P} , K)

is commutative for each cyclic subgroup P of D and for every generator x of P. Indeed, let χ ∈ (G, e)^{∨} and x′ ∈ G_{P,p′}. Then

d_{G}^{(x,σ_{P} e_{P})}(R_{μ_{⟨1⟩}}(χ))(x′) = d_{G}^{(x,σ_{P} e_{P})}(σχ)(x′)
= (σχ)(xx′σ_{P} e_{P}),
R_{μ_{P} ,p′}(d_{G}^{(x,e_{P})}(χ))(x′) = (σ_{P} · d_{G}^{(x,e_{P})}(χ))(x′)
= σ(x′)χ(xx′e_{P}).

Now

xx′σ_{P} e_{P} = \frac{1}{|G_{P} |} ∑_{g∈G_{P}} ∑_{ζ∈(G_{P} ,e_{P})^{∨}} ζ(1)σ(g^{−1})ζ(g^{−1})xx′g.

Thus

(σχ)(xx′σ_{P} e_{P}) = \frac{1}{|G_{P} |} ∑_{g∈G_{P}} ∑_{ζ∈(G_{P} ,e_{P})^{∨}} ζ(1)σ(g^{−1})ζ(g^{−1})σ(xx′g)χ(xx′g)

= \frac{1}{|G_{P} |} ∑_{g∈G_{P}} ∑_{ζ∈(G_{P} ,e_{P})^{∨}} σ(xx′)ζ(1)ζ(g^{−1})χ(xx′g)

= \frac{1}{|G_{P} |} ∑_{g∈G_{P}} ∑_{ζ∈(G_{P} ,e_{P})^{∨}} σ(x′)ζ(1)ζ(g^{−1})χ(xx′g) since σ(x) = 1

= σ(x′)χ(xx′e_{P})

and (1B) holds.

For the rest of this section we suppose that ⟨σ⟩ acts regularly on the orbit of e, i.e., σ^{i}e = e if and only if σ^{i} = 1. Set G^{+} = ker σ and t = |G/G^{+}|. Then G^{+} contains the defect group D of e, since t is relatively prime to p. If χ ∈ (G, e)^{∨}, then σ^{i}χ ∈ (G, σ^{i}e)^{∨} and the t characters σ^{i}χ are distinct. Thus Res_{G^{+}}^{G}(χ) is irreducible and the σ^{i}e cover a single block e^{+} of OG^{+}. In other words, e^{+} = ∑_{i=1}^{t} σ^{i}e and e^{+} is G-stable. Thus the restriction Res_{G^{+}}^{G} induces a perfect isometry ρ = (J_{ρ}, ϵ_{ρ}) in R_{K}((G, e), (G^{+}, e^{+})), where J_{ρ}(χ) = Res_{G^{+}}^{G}(χ) and ϵ_{ρ}(χ) = 1 for χ ∈ (G, e)^{∨}.

A subpair (P, u) of G covers a subpair (P, u^{+}) of G^{+} if u covers u^{+}. Suppose (P, u) is an e-subpair. Then Br_{P}^{G}(e)u = u, so that Br_{P}^{G}(σ^{i}e) · σ_{P}^{i} u = σ_{P}^{i} u. This implies ⟨σ_{P} ⟩ acts regularly on the orbit of u. Moreover, G_{P}^{+} = G_{P} ∩ G^{+} is the kernel of σ_{P}, G = G^{+}G_{P}, and G/G^{+} ≃ G_{P} /G_{P}^{+}. In particular, Res_{G_{P}^{+}}^{G_{P}} induces a

perfect isometry ρ_{P} = (J_{ρ_{P}} , ϵ_{ρ_{P}}) in R_{K}((G_{P} , e_{P}), (G_{P}^{+}, e_{P}^{+})).

Lemma (1C). Suppose ⟨σ⟩ acts regularly on the orbit of e. Then the following hold:

(i) Each e^{+}-subpair (P, u^{+}) of G^{+} is covered by a unique e-subpair (P, u) of

G.

(ii) Each e-subpair (P, u) of G covers a unique e^{+}-subpair (P, u^{+}) of G^{+}.

Proof. We have Br_{P}^{G^{+}}(e^{+}) = Br_{P}^{G}(e^{+}) = ∑_{i=1}^{t} Br_{P}^{G}(σ^{i}e) since e^{+} = ∑_{i=1}^{t} σ^{i}e. If (P, u^{+}) is an e^{+}-subpair of G^{+}, then ∑_{i} Br_{P}^{G}(σ^{i}e)u^{+} = u^{+} and Br_{P}^{G}(σ^{i}e)u^{+} ≠ 0 for some i. So there exists a block u of OG_{P} such that σ_{P}^{i} u covers u^{+} and Br_{P}^{G}(σ^{i}e) · σ_{P}^{i} u = σ_{P}^{i} u. This implies Br_{P}^{G}(σ^{j}e) · σ_{P}^{j} u = σ_{P}^{j} u for all j. In particular, ⟨σ_{P} ⟩ acts regularly on the orbit of u, u^{+} = ∑_{i=1}^{t} Res_{G_{P}^{+}}^{G_{P}}(σ_{P}^{i})u, and (P, u) is the unique

e-subpair of G covering (P, u^{+}). Conversely, suppose (P, u) is an e-subpair of G. Then ⟨σ_{P} ⟩ acts regularly on the orbit of u, u covers a unique block u^{+} of OG_{P}^{+}, and u^{+} = ∑_{i=1}^{t} σ_{P}^{i} u. Since Br_{P}^{G^{+}}(e^{+}) = ∑_{i=1}^{t} Br_{P}^{G}(σ^{i}e), it follows that Br_{P}^{G^{+}}(e^{+})u^{+} = u^{+}. Thus (P, u^{+}) is the unique e^{+}-subpair of G^{+} covered by (P, u).

Proposition (1D). Suppose ⟨σ⟩ acts regularly on the orbit of e and D is a defect group of e. Then e and e^{+} are isotypic blocks with local system {ρ_{P} : P ≤ D, P cyclic}.

Proof. By (1C), u covers u^{+} if and only if there is a bijection γ: Sp(G^{+}, e^{+}) ⟶ Sp(G, e) such that γ((P, u^{+})) = (P, u). To show condition (I1) for an isotypy it suffices to show that γ satisfies the following conditions (see [4, 4.2, Remark 2]):

(C1) γ is injective and γ((P, u^{+})^{h}) = γ((P, u^{+}))^{h} for all h ∈ G^{+}.

(C2) Every element in Sp(G, e) is G-conjugate to γ((P, u^{+})) for some element

(P, u^{+}) in Sp(G^{+}, e^{+}). (C3) If γ((Q, v^{+})) = γ((P, u^{+}))^{g} for (P, u^{+}) and (Q, v^{+}) in Sp(G^{+}, e^{+}) and g

in G, then there exist z ∈ G_{P} and h ∈ G^{+} such that g = zh. (C4) γ((P, u^{+})) ≤ γ((Q, v^{+})) if and only if (P, u^{+}) ≤ (Q, v^{+}). Conditions (C1), (C2), (C3) follow easily from (1C) and the factorization G =

G_{P} G^{+}. Suppose γ((P, u^{+})) = (P, u), γ((Q, v^{+})) = (Q, v), and (P, u^{+}) ≤ (Q, v^{+}). We claim (P, u) ≤ (Q, v). It suffices to show this for normal containment. If

(P, u^{+}) ◁ (Q, v^{+}), then P ◁ Q, u^{+} is stable under Q, and Br_{Q}^{G_{P}^{+}Q}(u^{+})v^{+} = v^{+}. Thus u is stable under Q by (1C). Moreover, u covers u^{+} viewed as blocks of OG_{P} Q and OG_{P}^{+}Q, and u^{+} = ∑_{i} Res_{G_{P} Q}^{G}(σ^{i})u. Since Br_{Q}^{G_{P} Q}(Res_{G_{P} Q}^{G}(σ^{i})u) · σ_{Q}^{j}v = 0 for i ≠ j and since v^{+} = ∑_{j} σ_{Q}^{j}v, it follows that

v^{+} = Br_{Q}^{G_{P}^{+}Q}(u^{+})v^{+} = ∑_{i} Br_{Q}^{G_{P} Q}(Res_{G_{P} Q}^{G}(σ^{i})u)σ_{Q}^{i}v.

Thus Br_{Q}^{G_{P} Q}(u)v = v and (P, u) ◁ (Q, v). The argument is reversible, and so (C4) holds.

Let P be a cyclic subgroup of D. The diagram

CF(G, e, K) ⟶^{R_{ρ}} CF(G^{+}, e^{+}, K)
↓_{d_{G}^{(x,e_{P} )}}  ↓_{d_{G^{+}}^{(x,e{+ \atop P})}}
CF_{p′}(G_{P} , e_{P} , K) ⟶^{R_{ρ_{P} ,p′}} CF_{p′}(G_{P}^{+}, e_{P}^{+}, K)

is commutative for all generators x of P. Indeed, let χ ∈ (G, e)^{∨} and x′ ∈ G_{P,p′}^{+}. Then

\begin{aligned}d_{G^{+}}^{(x,e_{P}^{+})}(Res_{G^{+}}^{G}(χ))(x′) & = χ(xx′e_{P}^{+}), \\ Res_{G_{P}^{+}}^{G_{P}}(d_{G}^{(x,e_{P})}(χ))(x′) & = χ(xx′e_{P}). \end{aligned}

The commutativity now follows since χ(xx′e_{P}^{+}) = ∑_{i} χ(xx′σ_{P}^{i} e_{P}) = χ(xx′e_{P}).

Corollary (1E). Suppose ⟨σ⟩ acts regularly on the orbit of e and D is a defect group of e. Let H = N_{G}(D) and H^{+} = N_{G^{+}}(D), and let f and f^{+} be the Brauer correspondents of e and e^{+} in OH and OH^{+}. Then e and f are isotypic blocks of OG and OH if and only if e^{+} and f^{+} are isotypic blocks of OG^{+} and OH^{+}.

Proof. We have Br_{D}^{G}(σ^{i}e) = σ_{D}^{i} Br_{D}^{G}(e) = σ_{D}^{i}f for 1 ≤ i ≤ t. In particular, Res_{H}^{G}(σ) acts regularly on the orbit of f, σ_{D}^{i}f is the Brauer correspondent of σ^{i}e, and f^{+} = ∑_{i=1}^{t} σ_{D}^{i}f. Now e and e^{+} are isotypic blocks by (1D); f and f^{+} are also isotypic blocks by (1D). Thus (1E) holds.

2. +Equivariance

Let G be a finite group, let G^{+} be a subgroup of index 2, and let sgn_{G} be the sign character of G with kernel G^{+}. We write χ ↦ χ^{∗} and e ↦ e^{∗} for the action of sgn_{G} on the irreducible characters and blocks of G. Let e and e^{+} be blocks of OG and OG^{+}. The configuration (G, G^{+}, e, e^{+}) is a +configuration if e^{+} is the unique block of OG^{+} covered by e and if e is the unique block of OG covering e^{+}. This is equivalent to the condition e = e^{+} or to the conditions e^{+} is stable under G and e^{∗} = e. Let (H, H^{+}, f, f^{+}) be a second +configuration with corresponding sign character sgn_{H}, and write ψ ↦ ψ^{∗} and f ↦ f^{∗} for the action of sgn_{H} on the irreducible characters and blocks of H. An isometry μ = (J, ϵ) in R_{K}((G, e), (H, f)) is +equivariant with respect to (G^{+}, H^{+}) if

J(χ^{∗}) = J(χ)^{∗}, ϵ(χ^{∗}) = ϵ(χ)

for all χ ∈ (G, e)^{∨}.

Lemma (2A). Let (G, G^{+}, e, e^{+}) and (H, H^{+}, f, f^{+}) be +configurations, and let μ be a +equivariant isometry in R_{K}((G, e), (H, f)).

(i) χ ∈ (G, e)^{∨} is reducible on G^{+} if and only if J(χ) ∈ (H, f)^{∨} is reducible

on H^{+}.
(ii) μ(g, h) = 0 if sgn_{G}(g) ≠ sgn_{H}(h).

Proof. (i) holds since Res_{G^{+}}^{G}(χ) and Res_{H^{+}}^{H}(J(χ)) are reducible if and only if χ^{∗} = χ and J(χ)^{∗} = J(χ). In (ii) either g ∉ G^{+} or h ∉ H^{+}. Now μ(g, h) is a sum of terms ϵ(χ)χ(g) J(χ)(h^{−1}). If Res_{G^{+}}^{G}(χ) and Res_{H^{+}}^{H}(J(χ)) are reducible, then one of χ(g) or J(χ)(h^{−1}) is zero and ϵ(χ)χ(g) J(χ)(h^{−1}) = 0. If Res_{G^{+}}^{G}(χ) and Res_{H^{+}}^{H}(J(χ)) are irreducible, then χ ≠ χ^{∗}, J(χ) ≠ J(χ)^{∗}, and

ϵ(χ)χ(g) J(χ)(h^{−1}) + ϵ(χ^{∗})χ^{∗}(g) J(χ^{∗})(h^{−1})
= ϵ(χ)χ(g) J(χ)(h^{−1})(1 + sgn_{G}(g) sgn_{H}(h))

is zero. Thus μ(g, h) is a sum of zero subsums and μ(g, h) = 0.

Let μ = (J, ϵ) be a +equivariant isometry in R_{K}((G, e), (H, f)). An isometry μ^{+} = (J^{+}, ϵ^{+}) in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})) is μ-induced if one and hence both of the following diagrams are commutative:

R_{K}(G, e) ⟶^{R_{μ}} R_{K}(H, f)
↓^{Res}_{G^{+}}^{G}  ↓^{Res}_{H^{+}}^{H}
R_{K}(G^{+}, e^{+}) ⟶^{R_{μ^{+}}} R_{K}(H^{+}, f^{+})

(2.1)

R_{K}(G^{+}, e^{+}) ⟶^{R_{μ^{+}}} R_{K}(H^{+}, f^{+})
↓^{Ind}_{G^{+}}^{G}  ↓^{Ind}_{H^{+}}^{H}
R_{K}(G, e) ⟶^{R_{μ}} R_{K}(H, f)

(2.2)

All μ-induced isometries μ^{+} = (J^{+}, ϵ^{+}) arise as follows: Let

(G^{+}, e^{+})^{∨} = (G^{+}, e^{+})_{e}^{∨} ∪ (G^{+}, e^{+})_{n}^{∨},

where the characters in (G^{+}, e^{+})_{e}^{∨} extend to G and the characters in (G^{+}, e^{+})_{n}^{∨} do not extend to G. If χ^{+} ∈ (G^{+}, e^{+})_{e}^{∨} and χ^{+} = Res_{G^{+}}^{G}(χ) for χ ∈ (G, e)^{∨}, let

J^{+}(χ^{+}) = Res_{H^{+}}^{H}(J(χ)), ϵ^{+}(χ^{+}) = ϵ(χ).

J^{+}(χ^{+}) and ϵ^{+}(χ^{+}) are well-defined since

Res_{H^{+}}^{H}(J(χ^{∗})) = Res_{H^{+}}^{H}(J(χ)^{∗}) = Res_{H^{+}}^{H}(J(χ))

and ϵ(χ^{∗}) = ϵ(χ). If χ_{+} and χ_{−} are characters in (G^{+}, e^{+})_{n}^{∨} such that χ_{+} + χ_{−} = Res_{G^{+}}^{G}(χ) for χ ∈ (G, e)^{∨}, let Res_{H^{+}}^{H}(J(χ)) = J(χ)_{+} + J(χ)_{−} and set

J^{+}(χ_{±}) = J(χ)_{±}, ϵ^{+}(χ_{±}) = ϵ(χ).

J^{+}(χ_{±}) depends on the labeling of the constituents χ_{±} and J(χ)_{±}. In all cases χ covers χ^{+} if and only if J(χ) covers J^{+}(χ^{+}). Moreover, J^{+}((χ^{+})^{g}) = J^{+}(χ^{+})^{h} and ϵ^{+}((χ^{+})^{g}) = ϵ^{+}(χ^{+}) for g ∈ G − G^{+} and h ∈ H − H^{+}.

Lemma (2B). Let (G, G^{+}, e, e^{+}) and (H, H^{+}, f, f^{+}) be +configurations, let μ be a +equivariant isometry in R_{K}((G, e), (H, f)), and let μ^{+} = (J^{+}, ϵ^{+}) be a μ-induced isometry in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})). Suppose p is odd and μ is perfect. Then μ^{+} is perfect if and only if the generalized character of G^{+} × H^{+} defined by

Δ(μ^{+})(g, h) = ∑_{{χ_{+},χ_{−}}⊆(G^{+},e^{+})_{n}^{∨}}ϵ^{+}(χ_{±}) (χ_{+} − χ_{−})(g) (J^{+}(χ_{+}) − J^{+}(χ_{−}))(h^{−1})

is perfect.

Proof. If (g, h) ∈ G^{+} × H^{+}, then

μ(g, h) = ∑_{χ∈(G,e)^{∨}} ϵ(χ) χ(g) J(χ)(h^{−1})

= ∑_{χ^{+}∈(G^{+},e^{+})_{e}^{∨}} 2ϵ^{+}(χ^{+}) χ^{+}(g) J^{+}(χ^{+})(h^{−1})

+ ∑_{{χ_{+},χ_{−}}⊆(G^{+},e^{+})_{n}^{∨}}ϵ^{+}(χ_{±}) (χ_{+} + χ_{−})(g) (J^{+}(χ_{+}) + J^{+}(χ_{−}))(h^{−1})

= 2μ^{+}(g, h) − ∑_{{χ_{+},χ_{−}}⊆(G^{+},e^{+})_{n}^{∨}}ϵ^{+}(χ_{±}) (χ_{+} − χ_{−})(g) (J^{+}(χ_{+}) − J^{+}(χ_{−}))(h^{−1}).

Thus

Res_{G^{+}×H^{+}}^{G×H}(μ) = 2μ^{+} − Δ(μ^{+}), (2.3)

and (2B) follows.

Remark (2C). If (χ_{+}−χ_{−})(g) (J^{+}(χ_{+})−J^{+}(χ_{−}))(h^{−1}) is perfect for all {χ_{+}, χ_{−}} ⊆ (G^{+}, e^{+})_{n}^{∨}, then all μ-induced isometries μ^{+} are perfect. We note Δ(μ^{+})(g, h) ≠ 0 only if g and h lie in conjugacy classes of G and H which split in G^{+} and H^{+}.

Suppose G_{0} and G_{1} are finite groups, G_{0}^{+} and G_{1}^{+} are normal subgroups of G_{0} and G_{1} of index 2, and sgn_{G_{0}} and sgn_{G_{1}} are the corresponding sign characters. Let G = G_{0} × G_{1}, and let sgn_{G} = sgn_{G_{0}} × sgn_{G_{1}} be the sign character of G with kernel

G^{+} = (G_{0} × G_{1})^{+} = {(g_{0}, g_{1}) ∈ G_{0} × G_{1} : sgn_{G_{0}}(g_{0}) = sgn_{G_{1}}(g_{1})}.

In addition, let G^{++} = G_{0}^{+} × G_{1}^{+} and let sgn_{G^{+}} be the sign character of G^{+} with respect to G^{++}, that is,

sgn_{G^{+}} = Res_{G^{+}}^{G}(sgn_{G_{0}} ×1_{G_{1}}) = Res_{G^{+}}^{G}(1_{G_{0}} × sgn_{G_{1}}).

A character χ in G^{∨} with components χ_{i} ∈ G_{i}^{∨} is reducible on G^{+} if and only if each χ_{i} is reducible on G_{i}^{+}. In particular, if χ is reducible on G^{+} and Res_{G^{+}}^{G}(χ) = χ_{+} + χ_{−}, then Res_{G^{++}}^{G^{+}}(χ_{+}) and Res_{G^{++}}^{G^{+}}(χ_{−}) are also reducible and all three characters χ, χ_{+}, χ_{−} vanish off G^{++}.

Proposition (2D). Let (G_{0}, G_{0}^{+}, e_{0}, e_{0}^{+}) and (G_{1}, G_{1}^{+}, e_{1}, e_{1}^{+}) be +configurations, let e be the block e_{0}e_{1} of OG, and let e^{++} be the block e_{0}^{+}e_{1}^{+} of OG^{++}. Then there exists a unique block e^{+} of OG^{+} covered by e. Moreover, (G, G^{+}, e, e^{+}) and (G^{+}, G^{++}, e^{+}, e^{++}) are +configurations.

Proof. Every block of G^{+} covered by e covers e^{++}. The hypotheses imply that sgn_{G} e = e and e^{++} is stable under G. Suppose first that χ_{0}^{+}χ_{1}^{+} is a character in (G^{++}, e^{++})^{∨} stable under G. Each factor χ_{i}^{+} extends to two characters χ_{i} and χ_{i}^{∗} in (G_{i}, e_{i})^{∨}, and the four characters χ_{0}χ_{1}, χ_{0}^{∗}χ_{1}^{∗}, χ_{0}^{∗}χ_{1}, χ_{0}χ_{1}^{∗} in (G, e)^{∨} are distinct. The restrictions of these four characters to G^{+} are irreducible and so belong to a unique block e^{+} of OG^{+}. Thus e^{+} is the unique block of OG^{+} covered by e, and (G, G^{+}, e, e^{+}) is a +configuration. The two distinct characters of G^{+} extending χ_{0}^{+}χ_{1}^{+} are

χ^{+} = Res_{G^{+}}^{G}(χ_{0}χ_{1}) = Res_{G^{+}}^{G}(χ_{0}^{∗}χ_{1}^{∗}),
χ^{−} = Res_{G^{+}}^{G}(χ_{0}χ_{1}^{∗}) = Res_{G^{+}}^{G}(χ_{0}^{∗}χ_{1}),

and χ^{−} = sgn_{G^{+}} χ^{+}. Thus (G^{+}, G^{++}, e^{+}, e^{++}) is a +configuration and sgn_{G^{+}} e^{+} = e^{+}. Suppose next that χ_{0}^{+}χ_{1}^{+} is a character in (G^{++}, e^{++})^{∨} not stable under G. Then χ_{0}^{+}χ_{1}^{+} is not stable under G^{+} and Ind_{G^{++}}^{G^{+}}(χ_{0}^{+}χ_{1}^{+}) is irreducible. The block e^{+} of OG^{+} containing Ind_{G^{++}}^{G^{+}}(χ_{0}^{+}χ_{1}^{+}) is the unique block of OG^{+} covering e^{++} and sgn_{G^{+}} e^{+} = e^{+}. Thus (2D) also holds in this case.

Let (H_{0}, H_{0}^{+}, f_{0}, f_{0}^{+}) and (H_{1}, H_{1}^{+}, f_{1}, f_{1}^{+}) be a second pair of +configurations, and let H, H^{+}, H^{++}, sgn_{H}, sgn_{H^{+}}, f^{+}, f^{++} have the analogous meanings for this pair. In particular, (H, H^{+}, f, f^{+}) and (H^{+}, H^{++}, f^{+}, f^{++}) are +configurations.

Proposition (2E). Let (G_{i}, G_{i}^{+}, e_{i}, e_{i}^{+}) and (H_{i}, H_{i}^{+}, f_{i}, f_{i}^{+}) be +configurations and let μ_{i} = (J_{i}, ϵ_{i}) be a +equivariant isometry in R_{K}((G_{i}, e_{i}), (H_{i}, f_{i})) for i = 0,

1. Then the following hold:

(i) μ = μ_{0} × μ_{1} is a +equivariant isometry in R_{K}((G, e), (H, f)) with respect to (G^{+}, H^{+}).

(ii) All μ-induced isometries μ^{+} in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})) are +equivariant with respect to (G^{++}, H^{++}).

Proof. Let μ = (J, ϵ), where J = J_{0} × J_{1} and ϵ = ϵ_{0} × ϵ_{1}. Let χ ∈ (G, e)^{∨}, where χ = χ_{0}χ_{1} and χ_{i} ∈ (G_{i}, e_{i})^{∨}. Then

\begin{aligned}ϵ(χ^{∗}) & = ϵ_{0}(χ_{0}^{∗})ϵ_{1}(χ_{1}^{∗}) = ϵ_{0}(χ_{0})ϵ_{1}(χ_{1}) = ϵ(χ), \\ J(χ^{∗}) & = J_{0}(χ_{0}^{∗}) J_{1}(χ_{1}^{∗}) = J_{0}(χ_{0})^{∗} J_{1}(χ_{1})^{∗} = J(χ)^{∗}, \end{aligned}

and (i) holds. We set (χ^{+})^{#} = sgn_{G^{+}} χ^{+} and (ψ^{+})^{#} = sgn_{H^{+}} ψ^{+} for χ^{+} in (G^{+})^{∨} and for ψ^{+} in (H^{+})^{∨}. Let μ^{+} = (J^{+}, ϵ^{+}) be a μ-induced isometry in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})). Let χ^{+} ∈ (G^{+}, e^{+})^{∨} and choose χ in (G, e)^{∨} covering χ^{+}. Let χ = χ_{0}χ_{1}, where χ_{i} ∈ (G_{i}, e_{i})^{∨}. Then J(χ) = J_{0}(χ_{0}) J_{1}(χ_{1}) is a character in (H, f)^{∨} covering J^{+}(χ^{+}). If Res_{G^{++}}^{G^{+}}(χ^{+}) is reducible, then (χ^{+})^{#} = χ^{+} and Res_{G_{i}^{+}}^{G_{i}}(χ_{i}) is reducible for some i. So Res_{H_{i}^{+}}^{H_{i}}(J_{i}(χ_{i})) is reducible for the same i

and Res_{H^{++}}^{H^{+}}(J^{+}(χ^{+})) is reducible. Thus ϵ^{+}((χ^{+})^{#}) = ϵ^{+}(χ^{+}) and J^{+}((χ^{+})^{#}) = (J^{+}(χ^{+}))^{#}. If Res_{G^{++}}^{G^{+}}(χ^{+}) is irreducible, then χ^{+} = Res_{G^{+}}^{G}(χ) and

χ^{+} = Res_{G^{+}}^{G}(χ_{0}χ_{1}) = Res_{G^{+}}^{G}(χ_{0}^{∗}χ_{1}^{∗}),
(χ^{+})^{#} = Res_{G^{+}}^{G}(χ_{0}χ_{1}^{∗}) = Res_{G^{+}}^{G}(χ_{0}^{∗}χ_{1}).

Since μ_{0} and μ_{1} are +equivariant, ϵ^{+}((χ^{+})^{#}) = ϵ^{+}(χ^{+}) and

J^{+}((χ^{+})^{#}) = Res_{H^{+}}^{H}(J_{0}(χ_{0}) J_{1}(χ_{1}^{∗}))
= Res_{H^{+}}^{H}(J_{0}(χ_{0}) J_{1}(χ_{1})^{∗})
= (Res_{H^{+}}^{H}(J_{0}(χ_{0}) J_{1}(χ_{1})))^{#}
= J^{+}(χ^{+})^{#}.

This completes the proof of (ii).

We continue with the situation of (2E). Suppose μ_{i}^{+} = (J_{i}^{+}, ϵ_{i}^{+}) is a μ_{i}-induced isometry in R_{K}((G_{i}^{+}, e_{i}^{+}), (H_{i}^{+}, f_{i}^{+})) for i = 0, 1, and μ^{++} is the isometry μ_{0}^{+}×μ_{1}^{+} in R_{K}((G^{++}, e^{++}), (H^{++}, f^{++})). We may choose a μ-induced isometry μ^{+} = (J^{+}, ϵ^{+}) such that μ^{++} is μ^{+}-induced. Indeed, fix notation so that J_{i}^{+}(χ_{i,±}) = ψ_{i,±} whenever J_{i}(χ_{i}) = ψ_{i}, Res_{G_{i}^{+}}^{G_{i}}(χ_{i}) = χ_{i,+} + χ_{i,−}, and Res_{H_{i}^{+}}^{H_{i}}(ψ_{i}) = ψ_{i,+} + ψ_{i,−}. Suppose

J(χ) = ψ, where Res_{G^{+}}^{G}(χ) = χ_{+} + χ_{−} and Res_{H^{+}}^{H}(ψ) = ψ_{+} + ψ_{−} are reducible. Then χ = χ_{0}χ_{1} and ψ = ψ_{0}ψ_{1}, where Res_{G_{i}^{+}}^{G_{i}}(χ_{i}) = χ_{i,+} + χ_{i,−} and Res_{H_{i}^{+}}^{H_{i}}(ψ_{i}) = ψ_{i,+} + ψ_{i,−} are reducible for i = 0, 1. We label χ_{±} and ψ_{±} so that

Res_{G^{++}}^{G^{+}}(χ_{+}) = χ_{0,+}χ_{1,+} + χ_{0,−}χ_{1,−},
Res_{G^{++}}^{G^{+}}(χ_{−}) = χ_{0,+}χ_{1,−} + χ_{0,−}χ_{1,+},
Res_{H^{++}}^{H^{+}}(ψ_{+}) = ψ_{0,+}ψ_{1,+} + ψ_{0,−}ψ_{1,−},
Res_{H^{++}}^{H^{+}}(ψ_{−}) = ψ_{0,+}ψ_{1,−} + ψ_{0,−}ψ_{1,+},

(2.4) and choose J^{+} so that J^{+}(χ_{±}) = ψ_{±} for all such χ and ψ. Then

R_{K}(G^{+}, e^{+}) ⟶^{R_{μ^{+}}} R_{K}(H^{+}, f^{+})
↓_{Res}_{G^{++}}^{G^{+}}  ↓_{Res}_{H^{++}}^{H^{+}}
R_{K}(G^{++}, e^{++}) ⟶^{R_{μ^{++}}} R_{K}(H^{++}, f^{++})

is a commutative diagram. Thus μ^{+} is μ-induced and μ^{++} is μ^{+}-induced.

Proposition (2F). Suppose the hypotheses of (2E) hold and μ_{i}^{+} is a μ_{i}-induced isometry in R_{K}((G_{i}^{+}, e_{i}^{+}), (H_{i}^{+}, f_{i}^{+})) for i = 0, 1.

(i) There exists a μ-induced isometry μ^{+} in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})) such that μ^{++} = μ_{0}^{+} × μ_{1}^{+} is μ^{+}-induced.

(ii) If p is odd, μ_{i}^{+} is perfect for i = 0, 1, and μ^{+} is as in (i), then μ^{+} is perfect.

Proof. (i) follows from the preceding paragraph. For (ii) it suffices by (2B) to prove that

Δ(μ^{+})(g, h) = ∑_{{χ_{+},χ_{−}}⊆(G^{+},e^{+})_{n}^{∨}} ϵ^{+}(χ_{±})(χ_{+} − χ_{−})(g)(ψ_{+} − ψ_{−})(h^{−1})

is perfect on its support G^{++} × H^{++}. We may suppose μ^{+} satisfies (2.4). Let g_{0} ∈ G_{0}^{+}, g_{1} ∈ G_{1}^{+}, h_{0} ∈ H_{0}^{+}, h_{1} ∈ H_{1}^{+}. Then

\begin{aligned}Δ(μ^{+})(g_{0}g_{1}, h_{0}h_{1}) & =[ ∑_{\begin{matrix}\{χ_{0,+},χ_{0,−}\} \\ ⊆(G_{0},e_{0})_{n}^{∨} \end{matrix}}ϵ_{0}^{+}(χ_{0,±})(χ_{0,+} − χ_{0,−})(g_{0})(ψ_{0,+} − ψ_{0,−})(h_{0}^{−1})] \\ & ·[ ∑_{\begin{matrix}\{χ_{1,+},χ_{1,−}\} \\ ⊆(G_{1},e_{1})_{n}^{∨} \end{matrix}}ϵ_{1}^{+}(χ_{1,±})(χ_{1,+} − χ_{1,−})(g_{1})(ψ_{1,+} − ψ_{1,−})(h_{1}^{−1})]. \end{aligned}

The bracketed sums are perfect generalized characters of G_{0}^{+} × H_{0}^{+} and G_{1}^{+} × H_{1}^{+} since μ_{0}^{+} and μ_{1}^{+} are perfect. Thus Δ(μ^{+}) is perfect.

3. The isotypy conjecture in symmetric groups

Let G be the symmetric group S(Π) on a set Π of cardinality n. By the Nakayama conjecture a p-block e of OG is parametrized by a p-core λ_{0}, and irreducible characters χ_{λ} in (G, e)^{∨} are parametrized by partitions λ of n with p-core λ_{0}. In particular, n = n_{0} + pw, where n_{0} = |λ_{0}| and w ≥ 0 is the so-called weight of e. We suppose that e has an abelian defect group D, or equivalently, that w < p. Thus Π = Π_{0} ∪ Π_{1} is the disjoint union of sets Π_{0} and Π_{1} of cardinality n_{0} and pw. We may suppose Π_{1} = Γ × Ω, where Γ = {1, 2, …, p} and Ω is a set of w elements. Let X = S(Γ) and Y = N_{X}(P), where P is a fixed Sylow p-subgroup of X. In particular, Y is a Frobenius group with kernel P of order p and a cyclic complement of order p − 1.

We take D as the Sylow p-subgroup P^{Ω} of S(Π_{1}) and set H = N_{G}(D), so that H = H_{0} × H_{1} with H_{0} = S(Π_{0}) and H_{1} = Y ≀ S(Ω). We view Y ≀ S(Ω) as the product set Y^{Ω} × S(Ω), where Y^{Ω} is the set of functions from Ω to Y, S(Ω) acts on Y^{Ω} on the left by (^{σ}f)(i) = f(σ^{−1}(i)) or (^{σ}f)_{i} = f_{σ^{−1}(i)}, and multiplication in Y ≀ S(Ω) is defined by

(f, σ)(g, τ) = (f(^{σ}g), στ).

The Brauer correspondent f of e in OH has the form f = f_{0} × f_{1}, where f_{0} is the block of defect 0 of OH_{0} parametrized by λ_{0} and f_{1} is the principal block of OH_{1}. We note that f_{1} is the unique block of OH_{1}, since H_{1}/D is a p′-group and C_{H_{1}}(D) = D.

Let λ be a partition of n with p-core λ_{0} and p-quotient Λ = (λ_{1}, λ_{2}, …, λ_{p}). Then λ_{0} can be gotten by removing a sequence (ν_{1}, ν_{2}, …, ν_{w}) of p-hooks ν_{i} from λ. The sign v(λ) = (−1)^{∑_{1}^{w} ℓ(ν_{i})}, where ℓ(ν_{i}) is the leg length of ν_{i}, is independent of the choice of (ν_{1}, ν_{2}, …, ν_{w}). The residue of the (i, j)-node in the Young diagram of λ is the integer r such that 0 ≤ r < p and r ≡ j − i(mod p). A hook of λ has p-singular length if and only if the hand and foot node residues of the hook are respectively k, k + 1 for some k modulo p. We normalize Λ so that λ_{i} corresponds to hooks of λ of p-singular length and hand node residue i − 1. Let λ^{∗} be the dual partition of λ, let λ_{0}^{∗} be the p-core of λ^{∗}, and let Λ^{∗} = (λ_{1}^{∗}, λ_{2}^{∗}, …, λ_{p}^{∗}) be the p-quotient of λ^{∗}. Then λ_{0}^{∗} is the dual (λ_{0})^{∗} of λ_{0} and λ_{γ}^{∗} = (λ_{p+1−γ})^{∗} for 1 ≤ γ ≤ p. In particular, λ = λ^{∗} if and only if λ_{0} = λ_{0}^{∗} and λ_{γ} = (λ_{p+1−γ})^{∗} for 1 ≤ γ ≤ p. Indeed, the (i, j)-hook of λ and the (j, i)-hook of λ^{∗} have the same length. So if this length is p-singular, then the hand node residues of these hooks are of the form k and −k − 1 modulo p.

Let Y^{∨} = {ξ_{γ} : 1 ≤ γ ≤ p}. The characters in (H, f)^{∨} have the form χ_{λ_{0}} × ψ_{Λ}, where (H_{0}, f_{0})^{∨} = {χ_{λ_{0}}}, ψ_{Λ} ∈ H_{1}^{∨}, and Λ is a mapping

Y^{∨} ⟶ {partitions}, ξ_{γ} ↦ λ_{γ},

such that ∑_{γ} |λ_{γ}| = w. We also represent Λ by the p-tuple (λ_{1}, λ_{2}, …, λ_{p}). We recall the definition of ψ_{Λ}. Let Ω = ∐_{γ=1}^{p} Ω_{γ}^{Λ} be a partition where |Ω_{γ}^{Λ}| = |λ_{γ}|, and let ξ_{Λ} ∈ (Y^{Ω})^{∨} be such that (ξ_{Λ})_{i} = ξ_{γ} for i ∈ Ω_{γ}^{Λ}. The stabilizer H_{1,Λ} of ξ_{Λ} in H_{1} is Y^{Ω}S(Ω^{Λ}), where

S(Ω^{Λ}) = S(Ω_{1}^{Λ}) × S(Ω_{2}^{Λ}) × ⋯ × S(Ω_{p}^{Λ}),

and ξ_{Λ} extends to a character E(ξ_{Λ}) of H_{1,Λ}. Let ω_{Λ} = ω_{λ_{1}} × ω_{λ_{2}} × ⋯ × ω_{λ_{p}} be the irreducible character of S(Ω^{Λ}), where ω_{λ_{γ}} is the character of S(Ω_{γ}^{Λ}) parametrized by λ_{γ}. We may view ω_{Λ} as a character of H_{1,Λ} by inflation. Then

ψ_{Λ} = Ind_{H_{1,Λ}}^{H_{1}}(E(ξ_{Λ})ω_{Λ}).

We give a formula for E(ξ_{Λ})(f, σ) for f ∈ Y^{Ω} and σ ∈ S(Ω^{Λ}). Write σ = σ_{1}σ_{2} ⋯ σ_{d} as a product of disjoint cycles in S(Ω) and write

(f, σ) = (f_{1}, σ_{1})(f_{2}, σ_{2}) ⋯ (f_{d}, σ_{d}),

where f_{δ} ∈ Y^{Ω_{δ}^{σ}} and Ω_{δ}^{σ} is the support of σ_{δ}. Let n_{δ} = |Ω_{δ}^{σ}|, let i_{δ} ∈ Ω_{δ}^{σ}, and let

p_{i_{δ}}(f, σ) = f(i_{δ})f(σ^{−1}(i_{δ})) ⋯ f(σ^{−(n_{δ}−1)}(i_{δ})).

In particular, (f_{δ}, σ_{δ})^{n_{δ}} = (p_{i_{δ}}(f, σ), 1). Moreover

\begin{aligned}p_{i_{δ}}(^{h}(f, σ)) & = h_{i_{δ}}p_{i_{δ}}(f, σ)h_{i_{δ}}^{−1}, \\ p_{g(i_{δ})}(^{g}(f, σ)) & = p_{i_{δ}}(f, σ) \end{aligned} (3.1)

for h ∈ Y^{Ω} and g ∈ S(Ω). Since (f, σ) ∈ H_{1,Λ}, σ stabilizes each Ω_{γ}^{Λ}. Let Δ_{γ} be index sets such that Ω_{γ}^{Λ} = ∐_{δ∈Δ_{γ}} Ω_{δ}^{σ} is the orbit decomposition of σ on Ω_{γ}^{Λ}. Then

E(ξ_{Λ})(f, σ) = ∏_{γ} ∏_{δ∈Δ_{γ}} ξ_{γ}(p_{i_{δ}}(f, σ)) (3.2)

(see [12, §4.3]). We call E(ξ_{Λ}) the James-Kerber extension of ξ_{Λ}.

We give a Murnaghan-Nakayama type formula for ψ_{Λ}(f, σ) for the preceding element (f, σ). We fix s with 1 ≤ s ≤ d and consider pairs (\underline{ζ}, \underline{γ}) of sequences \underline{ζ} = (ζ_{1}, ζ_{2}, …, ζ_{s}) and \underline{γ} = (γ_{1}, γ_{2}, …, γ_{s}) with the following properties: ζ_{1} is a hook of length |σ_{1}| of the γ_{1}-component of Λ; ζ_{2} is a hook of length |σ_{2}| of the γ_{2}-component of (ζ_{1}, γ_{1}) ∗ Λ, the p-tuple gotten from Λ by removing ζ_{1} from the γ_{1}-component of Λ; ζ_{3} is a hook of length |σ_{3}| of the γ_{3}-component of (ζ_{2}, γ_{2})∗(ζ_{1}, γ_{1})∗Λ, the p-tuple gotten from (ζ_{1}, γ_{1}) ∗ Λ by removing ζ_{2} from the γ_{2}-component of (ζ_{1}, γ_{1}) ∗ Λ; and so on. Set

(\underline{ζ}, \underline{γ}) ∗ Λ = (ζ_{s}, γ_{s}) ∗ ⋯ ∗ (ζ_{2}, γ_{2}) ∗ (ζ_{1}, γ_{1}) ∗ Λ

and v(\underline{ζ}) = (−1)^{∑_{1}^{s} ℓ(ζ_{k})}.

Proposition (3A). With the above notation

ψ_{Λ}(f, σ) = ∑_{(\underline{ζ},\underline{γ})} v(\underline{ζ}) ∏_{k=1}^{s} ξ_{γ_{k}}(p_{i_{k}}(f, σ))ψ_{(\underline{ζ},\underline{γ})∗Λ}(f′, σ′), (3.3)

where (f′, σ′) = (f_{s+1}, σ_{s+1}) ⋯ (f_{d}, σ_{d}) and (\underline{ζ}, \underline{γ}) runs over all pairs of the above type.

Proof. We consider first the case s = 1. By definition

ψ_{Λ}(f, σ) = Ind_{H_{1,Λ}}^{H_{1}}(E(ξ_{Λ})ω_{Λ})(f, σ)

= ∑_{g∈S(Ω^{Λ})\S(Ω)}(E(ξ_{Λ})ω_{Λ})(^{g}(f, σ)),

since H_{1} = Y^{Ω}S(Ω) and H_{1,Λ} = Y^{Ω}S(Ω^{Λ}). We identify S(Ω′) for Ω′ ⊆ Ω with the subgroup of S(Ω) fixing Ω − Ω′ elementwise. Let Ω′ = Ω − Ω_{1}^{σ} and decompose

S(Ω) = ⋃_{g∈Γ} S(Ω^{Λ})gS(Ω′)

as a disjoint union of double cosets modulo (S(Ω^{Λ}), S(Ω′)). We say that a representative g in Γ is good and that the double coset S(Ω^{Λ})gS(Ω′) is good if g(Ω_{1}^{σ}) ⊆ Ω_{γ}^{Λ} for some γ. This definition is independent of the choice of coset representative g, since S(Ω′) fixes Ω_{1}^{σ} elementwise and S(Ω^{Λ}) stabilizes each Ω_{γ}^{Λ}. Moreover, the integer γ such that g(Ω_{1}^{σ}) ⊆ Ω_{γ}^{Λ} depends only on the double coset. Let Γ′ be the subset of good representatives g in Γ. If g and g′ are in Γ′ and g(Ω_{1}^{σ}) and g′(Ω_{1}^{σ}) are contained in the same Ω_{γ}^{Λ}, then g = g′. Indeed, there exists x ∈ S(Ω^{Λ}) such that x(g(i)) = g′(i) for all i ∈ Ω_{1}^{σ}, so g^{−1}x^{−1}g′ ∈ S(Ω′) and g′ ∈ S(Ω^{Λ})gS(Ω′). Thus there is a bijection between good double cosets and components λ_{γ} of Λ such that |λ_{γ}| ≥ |σ_{1}|.

Given g in Γ, let S(Ω^{Λ})gS(Ω′) = ⋃_{t∈T_{g}} S(Ω^{Λ})gt, where T_{g} is a set of representatives t for the cosets (g^{−1}S(Ω^{Λ})g ∩ S(Ω′))\S(Ω′). If g ∈ Γ′ and g(Ω_{1}^{σ}) ⊆ Ω_{γ_{1}}^{Λ}, then

g^{−1}S(Ω^{Λ})g ∩ S(Ω′) = S(g^{−1}(Ω^{Λ})) ∩ S(Ω′)

= S(g^{−1}(Ω_{γ_{1}}^{Λ}) − Ω_{1}^{σ}) × ∏_{γ≠γ_{1}} S(g^{−1}(Ω_{γ}^{Λ})). (3.4)

For g ∈ Γ′ we say t in T_{g} and the coset S(Ω^{Λ})gt are good if ^{gt}(f, σ) ∈ H_{1,Λ}. Thus t is good if and only if for each δ = 2, 3, …, d, there is a γ such that gt(Ω_{δ}^{σ}) ⊆ Ω_{γ}^{Λ}. This definition is independent of the choice of representative t. Let T′_{g} be the subset of good representatives t in T_{g}. It follows that

ψ_{Λ}(f, σ) = ∑_{g∈Γ′} ∑_{t∈T′_{g}}(E(ξ_{Λ})ω_{Λ})(^{gt}(f, σ)) (3.5)

since E(ξ_{Λ}) and ω_{Λ} are zero by convention on arguments not in H_{1,Λ}. To prove (3.3) for the case s = 1, it suffices to show that

∑_{t∈T′_{g}}(E(ξ_{Λ})ω_{Λ})(^{gt}(f, σ)) = ∑_{ζ_{1}} v(ζ_{1})ξ_{γ_{1}}(p_{i_{1}}(f, σ))ψ_{(ζ_{1},γ_{1})∗Λ}(f′, σ′), (3.6)

where ζ_{1} runs over all hooks of length |σ_{1}| in λ_{γ_{1}}.

Given g ∈ Γ′ such that g(Ω_{1}^{σ}) ⊆ Ω_{γ_{1}}^{Λ} and given t in T′_{g}, let Δ_{gt,γ} be the index sets such that

Ω_{γ}^{Λ} = ⋃_{δ∈Δ_{gt,γ}} gt(Ω_{δ}^{σ}).

Let

Δ′_{gt,γ} = \begin{cases}Δ_{gt,γ} & if γ ≠ γ_{1}, \\ Δ_{gt,γ} − \{1\} & if γ = γ_{1}. \end{cases}

Then

E(ξ_{Λ})(^{gt}(f, σ)) = ∏_{γ} ∏_{δ∈Δ_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f, σ)))

= ξ_{γ_{1}}(p_{gt(i_{1})}(^{gt}(f_{1}, σ_{1})))[∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f′, σ′)))]

= ξ_{γ_{1}}(p_{i_{1}}(f_{1}, σ_{1}))[∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f′, σ′)))].

The usual Murnaghan-Nakayama formula (see [12, 21.1]) gives

ω_{Λ}(^{gt}σ) = ∏_{γ} ω_{λ_{γ}} ( ∏_{δ∈Δ_{gt,γ}} ^{gt}(σ_{δ}))

= ω_{λ_{γ_{1}}} ( ∏_{δ∈Δ_{gt,γ_{1}}} ^{gt}(σ_{δ})) ∏_{γ≠γ_{1}} ω_{λ_{γ}} ( ∏_{δ∈Δ_{gt,γ}} ^{gt}(σ_{δ}))

= ∑_{ζ_{1}} v(ζ_{1})ω_{ζ_{1}∗λ_{γ_{1}}} ( ∏_{δ∈Δ′_{gt,γ_{1}}} ^{gt}(σ_{δ})) ∏_{γ≠γ_{1}} ω_{λ_{γ}} ( ∏_{δ∈Δ_{gt,γ}} ^{gt}(σ_{δ}))

= ∑_{ζ_{1}} v(ζ_{1})ω_{(ζ_{1},γ_{1})∗Λ}(^{gt}σ′),

where ζ_{1} runs over all hooks of length |σ_{1}| in λ_{γ_{1}}. Thus

\begin{matrix}∑_{t∈T′_{g}}(E(ξ_{Λ})ω_{Λ}) & (^{gt}(f, σ)) = ∑_{ζ_{1}} v(ζ_{1})ξ_{γ_{1}}(p_{i_{1}}(f_{1}, σ_{1})) \\ × ∑_{t∈T′_{g}}[∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f′, σ′)))]ω_{(ζ_{1},γ_{1})∗Λ}(^{gt}σ′). \end{matrix}

It suffices to show that if ζ_{1} is a fixed hook in λ_{γ_{1}} of length |σ_{1}|, then

ψ_{(ζ_{1},γ_{1})∗Λ}(f′, σ′) = ∑_{t∈T′_{g}}[∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f′, σ′)))]ω_{(ζ_{1},γ_{1})∗Λ}(^{gt}σ′). (3.7)

Now (ζ_{1}, γ_{1}) ∗ Λ corresponds to the p-tuple Λ′ = (λ′_{1}, λ′_{2}, …, λ′_{p}), where

λ′_{γ} = \begin{cases}λ_{γ} & if γ ≠ γ_{1}, \\ ζ_{1} ∗ λ_{γ} & if γ = γ_{1}. \end{cases}

Let ψ_{Λ′} be the irreducible character of Y ≀ S(Ω′) parametrized by Λ′. We use notation analogous to that for ψ_{Λ}. Namely, let Ω′ = ⋃_{γ} Ω′_{γ}^{Λ′}, where

Ω′_{γ}^{Λ′} = \begin{cases}g^{−1}(Ω_{γ}^{Λ}) & if γ ≠ γ_{1}, \\ g^{−1}(Ω_{γ_{1}}^{Λ}) − Ω_{1}^{σ} & if γ = γ_{1}. \end{cases}

So λ′_{γ} is a partition of |Ω′_{γ}^{Λ′}|. Let ξ_{Λ′} ∈ (Y^{Ω′})^{∨} be such that (ξ_{Λ′})_{i} = ξ_{γ} for i ∈ Ω′_{γ}^{Λ′}. Then the stabilizer of ξ_{Λ′} in Y^{Ω′}S(Ω′) is Y^{Ω′}S(Ω′^{Λ′}), where

S(Ω′^{Λ′}) = S(Ω′_{1}^{Λ′}) × S(Ω′_{2}^{Λ′}) × ⋯ × S(Ω′_{p}^{Λ′}).

In particular, T′_{g} is a set of coset representatives of S(Ω′^{Λ′}) in S(Ω′) by (3.4). Let ω_{Λ′} = ω_{λ′_{1}} × ω_{λ′_{2}} × ⋯ × ω_{λ′_{p}} be the irreducible character of S(Ω′^{Λ′}), where ω_{λ′_{γ}} is the irreducible character of S(Ω′_{γ}^{Λ′}) parametrized by λ′_{γ}. Let E(ξ_{Λ′}) be the James-Kerber extension of ξ_{Λ′} to Y^{Ω′}S(Ω′^{Λ′}). Then

ψ_{(ζ_{1},γ_{1})∗Λ}(f′, σ′) = ψ_{Λ′}(f′, σ′)

= Ind_{Y^{Ω′} S(Ω′^{Λ′})}^{Y^{Ω′} S(Ω′)}(E(ξ_{Λ′})ω_{Λ′})(f′, σ′)

= ∑_{t∈T′_{g}}(E(ξ_{Λ′})ω_{Λ′})(^{t}(f′, σ′)).

We can now show (3.7). Indeed, ^{t}(f′, σ′) ∈ S(Ω′^{Λ′}) if and only if for each δ ≥ 2, there exists a γ such that t(Ω_{δ}^{σ}) ⊆ Ω′_{γ}^{Λ′}, or equivalently, such that gt(Ω_{δ}^{σ}) ⊆ Ω_{γ}^{Λ}. Moreover, Ω′_{γ}^{Λ′} = ∐_{δ∈Δ′_{gt,γ}} t(Ω_{δ}^{σ}), since Ω_{γ}^{Λ} = ∐_{δ∈Δ_{gt,γ}} gt(Ω_{δ}^{σ}). Thus

\begin{aligned}E(ξ_{Λ′})(^{t}(f′, σ′)) & = ∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{t(i_{δ})}(^{t}(f′, σ′))) \\ & = ∏_{γ} ∏_{δ∈Δ′_{gt,γ}} ξ_{γ}(p_{gt(i_{δ})}(^{gt}(f′, σ′))) \end{aligned}

and

ω_{Λ′}(^{t}(σ′)) = ∏_{γ} ω_{λ′_{γ}} ( ∏_{δ∈Δ′_{gt,γ}} ^{t}(σ_{δ}))

= ω_{λ′_{γ_{1}}} ( ∏_{δ∈Δ′_{gt,γ_{1}}} ^{gt}(σ_{δ})) ∏_{γ≠γ_{1}} ω_{λ_{γ}} ( ∏_{δ∈Δ_{gt,γ}} ^{gt}(σ_{δ}))

= ω_{Λ′}(^{gt}(σ′)).

This completes the proof of (3.3) for the case s = 1. The general case of (3.3) now follows by iteration.

Remark. Formula (3.3) does not depend on the structure of Y. Equivalent formulas can also be found in [14, Théorème 2.8] and in [15, Theorem 4.3].

Definition. Let A(Γ) be the alternating group on Γ and label ξ_{γ} in Y^{∨} so that ξ_{γ} and ξ_{p+1−γ} have the same restriction to Y^{+} = Y ∩ A(Γ) for 1 ≤ γ ≤ p. In particular, ξ_{γ}(1) = 1 for γ ≠ p^{∗} and ξ_{γ}(1) = p − 1 for γ = p^{∗}, where p^{∗} = \frac{1}{2}(p + 1). The Rouquier isometry μ = (J, ϵ) in R_{K}((G, e), (H, f)) is defined as follows: For each partition λ of n with p-core λ_{0} and p-quotient Λ = (λ_{1}, λ_{2}, …, λ_{p}), let

\begin{aligned}J(χ_{λ}) & = χ_{λ_{0}} × ψ_{Λ}, \\ J(χ_{λ}) & = χ_{λ_{0}} × ψΛ̂, \end{aligned} \begin{aligned}ϵ(χ_{λ}) & = 1 \\ ϵ(χ_{λ}) & = v(λ)(−1)^{|λ_{p^{∗}} |} \end{aligned} \begin{matrix}if p = 2, \\ if p > 2, \end{matrix} (3.8)

where Λ̂ is the p-tuple gotten from Λ by replacing λ_{p^{∗}} by (λ_{p^{∗}})^{∗}.

We define a Rouquier isometry in a more general context. An element x = (f, 1) of D is called standard if the non-identity values of f are all equal. For a standard element x let Ω = Ω_{x,0} ∪ Ω_{x,1}, where x acts trivially on Γ × Ω_{x,0} and fixed-point-freely on Γ × Ω_{x,1}. Thus

G_{x} = G_{x,0} × G_{x,1}, H_{x} = H_{x,0} × H_{x,1},

where G_{x,0} = S(Π_{0} ∪ (Γ × Ω_{x,0})), H_{x,0} = S(Π_{0}) × (Y ≀ S(Ω_{x,0})), and G_{x,1} = H_{x,1} = P ≀S(Ω_{x,1}). Let H_{x,0} = H_{x,00}×H_{x,01}, where H_{x,00} = S(Π_{0}) and H_{x,01} = Y ≀S(Ω_{x,0}). Let

e_{x} = e_{x,0} × e_{x,1}, f_{x} = f_{x,00} × f_{x,01} × f_{x,1},

where e_{x,0} is the block of OG_{x,0} parametrized by λ_{0}, e_{x,1} is the 1-block of OG_{x,1}, f_{x,00} is the block of OH_{x,00} parametrized by λ_{0}, f_{x,01} is the 1-block of OH_{x,01}, and f_{x,1} is the 1-block of OH_{x,1}. In particular, e_{x,1}, f_{x,01}, and f_{x,1} are the unique blocks of OG_{x,1}, OH_{x,01}, and OH_{x,1}. Let f_{x,0} be the block f_{x,00} × f_{x,01} of OH_{x,0}. Then e_{x,0} and f_{x,0} have defect group P^{Ω_{x,0}} and f_{x,0} is the Brauer correspondent of e_{x,0}. Let μ_{x,0} = (J_{x,0}, ϵ_{x,0}) be the Rouquier isometry in R_{K}((G_{x,0}, e_{x,0}), (H_{x,0}, f_{x,0})). Let μ_{x,1} = (J_{x,1}, ϵ_{x,1}) be the identity isometry in R_{K}((G_{x,1}, 1), (H_{x,1}, 1)). Let μ_{x} = (J_{x}, ϵ_{x}) be the isometry μ_{x,0}×μ_{x,1} in R_{K}((G_{x}, e_{x}), (H_{x}, f_{x})). If μ_{x,0} is perfect, then μ_{x} is perfect We call μ_{x} the Rouquier isometry in R_{K}((G_{x}, e_{x}), (H_{x}, f_{x})). We note that μ_{x} depends only on ⟨x⟩ and not on the generator x.

For each j = 0, 1, 2, …, w let x_{j} be a standard element of D such that |Ω_{x_{j},1}| = j. The w+1 subgroups ⟨x_{j}⟩ then form a set of representatives for the N_{G}(D, e_{D})-orbits of cyclic subgroups of D, since N_{G}(D, e_{D}) ⊇ (Y ≀ S(Ω)).

Theorem (3B). e and f are isotypic blocks of OG and OH with local system given by the translates of the Rouquier isometries μ_{x} for x in {x_{0}, x_{1}, …, x_{w}}.

Proof. Condition (I1) holds by [1, Proposition 4.2] since D is abelian. The blocks e and f are of principal type by a theorem of Marichal and Puig (see [3, (2.2)]). Let x ∈ {x_{1}, x_{2}, …, x_{w}}. We prove the Rouquier isometries μ_{x} are perfect by induction on |D|. The claim is clear if |D| = 1, so we may suppose D > 1. By induction μ_{x,0} and hence μ_{x} are perfect isometries. We claim the following diagram is commutative:

\begin{matrix}R_{K}(G, e) & ⟶^{R_{μ}} & R_{K}(H, f) \\ ↓^{d_{G}^{x}} & & ↓^{d_{H}^{x}} \\ CF_{p′}(G_{x}, e_{x}) & ⟶^{R_{μx,p′}} & CF_{p′}(H_{x}, f_{x}) \end{matrix} (3.9)

Suppose p = 2. Then w = 1, G_{x} = H_{x} = S(Π_{0})×⟨x⟩, and λ is a partition with 2core λ_{0} and 2-quotient (1, 0) or (0, 1). The ordinary Murnaghan-Nakayama formula implies the commutativity of (3.9). Suppose p > 2. We consider the top-right portion of (3.9). Let χ_{λ} ∈ (G, e)^{∨}, let y ∈ H_{x,p′}, and write y = y_{00}y_{01}y_{1}, where y_{00} ∈ H_{x,00}, y_{01} ∈ H_{x,01}, and y_{1} ∈ H_{x,1}. Let xy_{1} = (f_{1}, σ_{1})(f_{2}, σ_{2}) ⋯ (f_{s}, σ_{s}), where the σ_{i} are disjoint cycles of S(Ω_{x,1}). Then

d_{H}^{x}(R_{μ}(χ_{λ}))(y) = ϵ(χ_{λ})(χ_{λ_{0}} × ψΛ̂)(xy)

= ϵ(χ_{λ})χ_{λ_{0}}(y_{00})ψΛ̂(y_{01}(xy_{1}))

= ϵ(χ_{λ})χ_{λ_{0}}(y_{00}) ∑_{(\underline{ζ},\underline{γ})} v(\underline{ζ}) ∏_{k=1}^{s} ξ_{γ_{k}}(p_{i_{k}}(f_{k}, σ_{k}))ψ_{(\underline{ζ},\underline{γ})∗Λ̂}(y_{01})

by (3A). For each pair (\underline{ζ}, \underline{γ}) in the formula, let M(\underline{γ}) = \{k : γ_{k} = p^{∗}\}. Then

d_{H}^{x}(R_{μ}(χ_{λ}))(y) = ϵ(χ_{λ})χ_{λ_{0}}(y_{00}) ∑_{(\underline{ζ},\underline{γ})} v(\underline{ζ})(−1)^{|M(\underline{γ})|}ψ_{(\underline{ζ},\underline{γ})∗Λ̂}(y_{01}) (3.10)

since p_{i_{k}}(f_{k}, σ_{k}) ∈ P^{#}, so that ξ_{γ_{k}}(p_{i_{k}}(f_{k}, σ_{k})) = 1 or −1 according as γ_{k} ≠ p^{∗} or γ_{k} = p^{∗}.

We consider the left-bottom portion of (3.9). Let g ∈ G_{x,p′} and write g = g_{0}g_{1}, where g_{0} ∈ G_{x,0,p′}, g_{1} ∈ G_{x,1,p′}. Each p′-conjugacy class of G_{x,1} contains a unique conjugacy class of S(Ω_{x,1}), since S(Ω_{x,1}) is a Hall p-complement in G_{x,1}. We may suppose without loss of generality that g_{1} = (1, σ) for some σ in S(Ω_{x,1}). Let σ = σ_{1}σ_{2} ⋯ σ_{s} be the cycle decomposition of σ in S(Ω_{x,1}). The Murnaghan-Nakayama formula implies that

d_{G}^{x}(χ_{λ})(g) = χ_{λ}(xg)

= χ_{λ}(g_{0}(xg_{1}))

= ∑_{\underline{ν}} v(\underline{ν})χ_{\underline{ν}∗λ}(g_{0}),

where \underline{ν} runs over sequences (ν_{1}, ν_{2}, …, ν_{s}) of hooks ν_{k} of length p|σ_{k}| of λ. Let τ run over representatives of the p′-conjugacy classes of S(Ω_{x,1}), let η_{τ} be the characteristic function of the G_{x,1}-conjugacy class of τ, and set

α_{λ} = ∑_{\underline{ν},τ} v(\underline{ν})(χ_{\underline{ν}∗λ} × η_{τ}).

ON PERFECT ISOMETRIES AND ISOTYPIES IN ALTERNATING GROUPS 3487

Then

α_{λ}(g_{0}g_{1}) = \begin{cases}∑\underline{ν} v(\underline{ν})χ_{\underline{ν}∗λ}(g_{0}) & if g_{1} ∈ G_{x,1,p′}, \\ 0 & if g_{1} ∉ G_{x,1,p′}. \end{cases}

for g_{0} ∈ G_{x,0} and g_{1} ∈ G_{x,1}. Thus d_{G}^{x}(χ_{λ}) = d_{G_{x}}^{1}(α_{λ}) and

\begin{aligned}R_{μ_{x},p′}(d_{G}^{x}(χ_{λ})) & = R_{μ_{x},p′}(d_{G_{x}}^{1}(α_{λ})) \\ & = d_{H_{x}}^{1}(R_{μ_{x}}(α_{λ})), \end{aligned}

where the second equality holds since μ_{x} is perfect by induction. Now

\begin{aligned}R_{μ_{x}}(α_{λ}) & = ∑_{\underline{ν},τ} v(\underline{ν})(R_{μ_{x,0}}(χ_{\underline{ν}∗λ}) × R_{μ_{x,1}}(η_{τ})) \\ & = ∑_{\underline{ν},τ} v(\underline{ν})ϵ_{x,0}(χ_{\underline{ν}∗λ})(χ_{λ_{0}} × ψ_{(\underline{ζ},\underline{γ})∗Λ̂} × η_{τ} ), \end{aligned}

where (\underline{ζ}, \underline{γ}) is obtained from \underline{ν} as follows: First \underline{ν} determines a pair (\underline{ζ̃}, \underline{γ}) of sequences (ζ̃_{1}, ζ̃_{2}, …, ζ̃_{s}) and (γ_{1}, γ_{2}, …, γ_{s}), where ζ̃_{k} is a hook of length |σ_{k}| of the γ_{k}-component of Λ. So \underline{ν} determines a pair (\underline{ζ}, \underline{γ}) of sequences (ζ_{1}, ζ_{2}, …, ζ_{s}) and (γ_{1}, γ_{2}, …, γ_{s}), where ζ_{k} is a hook of length |σ_{k}| of the γ_{k}-component of Λ̂, and ζ_{k} = ζ̃_{k} or (ζ̃_{k})^{∗} according as γ_{k} ≠ p^{∗} or γ_{k} = p^{∗}. Thus

R_{μ_{x},p′}(d_{G}^{x}(χ_{λ})) = ∑_{\underline{ν},τ} v(\underline{ν})ϵ_{x,0}(χ_{\underline{ν}∗λ}) d_{H_{x}}^{1}(χ_{λ_{0}} × ψ_{(\underline{ζ},\underline{γ})∗Λ̂} × η_{τ}).

If y ∈ H_{x,p′} and y = y_{00}y_{01}y_{1}, where y_{00} ∈ H_{x,00}, y_{01} ∈ H_{x,01}, and y_{1} ∈ H_{x,1}, then

R_{μ_{x},p′}(d_{G}^{x}(χ_{λ}))(y) = ∑_{\underline{ν}} v(\underline{ν})ϵ_{x,0}(χ_{\underline{ν}∗λ})χ_{λ_{0}}(y_{00})ψ_{(\underline{ζ},\underline{γ})∗Λ̂}(y_{01}). (3.11)

The commutativity of the diagram (3.9) holds by (3.10) and (3.11) if ϵ(χ_{λ})v(\underline{ζ})(−1)^{|M(\underline{γ})|} = v(\underline{ν})ϵ_{x,0}(χ_{\underline{ν}∗λ}) (3.12) for corresponding (\underline{ζ}, \underline{γ}) and \underline{ν}. Now

\begin{aligned}ϵ(χ_{λ}) & = v(λ)(−1)^{|λ_{p^{∗}} |}, \\ ϵ_{x,0}(χ_{\underline{ν}∗λ}) & = v(\underline{ν} ∗ λ)(−1)^{|(\underline{ν}∗λ)_{p^{∗}} |}. \end{aligned}

So (3.12) becomes

v(λ)(−1)^{|λ_{p^{∗}} |}v(\underline{ζ})(−1)^{|M(\underline{γ})|} = v(\underline{ν})v(\underline{ν} ∗ λ)(−1)^{|(\underline{ν}∗λ)_{p^{∗}} |}.

Since |(\underline{ν} ∗ λ)_{p^{∗}}| = |λ_{p^{∗}}| − ∑_{k∈M(\underline{γ})} |ζ_{k}|, (3.12) is equivalent to

\begin{aligned}v(λ)v(\underline{ν} ∗ λ) & = v(\underline{ν})v(\underline{ζ})(−1)^{|λ_{p^{∗}} |}(−1)^{|M(\underline{γ})|}(−1)^{|(\underline{ν}∗λ)_{p^{∗}} |} \\ & = v(\underline{ν})v(\underline{ζ}) ∏_{k∈M(\underline{γ})}(−1)^{(|ζ_{k}|−1)}. \end{aligned} (3.13)

But v(λ)v(\underline{ν} ∗ λ) = v(\underline{ζ̃})v(\underline{ν}) by [8, (2.15)] or [9, Théorème 1.4, b)]. Since

v(\underline{ζ})v(\underline{ζ̃}) = ∏_{k∈M(\underline{γ})}(−1)^{(|ζ_{k}|−1)},

(3.13) holds and (3.9) is commutative. This argument for the commutativity of (3.9) still applies if x is replaced by any generator of ⟨x⟩, since the lower horizontal arrow in (3.9) depends only on ⟨x⟩. (3B) now follows by transport of structure, as remarked in §1.

4. An analog of Frobenius’s theorem

Let G and G^{+} be the symmetric group S(Π) and the alternating group A(Π) on Π. An irreducible character χ_{λ} of G is reducible on G^{+} if and only if λ is selfdual. For such λ let η(λ) be the partition of n with parts η_{11}(λ), η_{22}(λ), …, η_{ss}(λ), where η_{ii}(λ) is the hook length of the (i, i)-node of λ and s is the number of diagonal nodes of λ. Since the η_{ii}(λ) are distinct odd integers, the conjugacy class C_{η(λ)} of G parametrized by η(λ) lies in G^{+} and splits in G^{+}. The mapping λ ↦ η(λ) thus induces a bijection χ_{λ} ↦ C_{η(λ)} from irreducible characters χ_{λ} of G which split on G^{+} to conjugacy classes C_{η(λ)} of G which lie in G^{+} and split in G^{+}. Let Res_{G^{+}}^{G}(χ_{λ}) = χ_{λ,+} + χ_{λ,−} and C_{η(λ)} = C_{η(λ),+} ∪ C_{η(λ),−} be the corresponding decompositions. We use the convention that \sqrt{a} = \sqrt{−a}i for negative real a, where i is a fixed primitive 4-th root of unity in K. In addition, we write \sqrt{±η(λ)} for \sqrt{±η_{11}(λ)η_{22}(λ) ⋯ η_{ss}(λ)}. The following is Frobenius’s Theorem [11, §3].

Theorem (4A). Suppose χ_{λ} is an irreducible character of G which splits on G^{+}. Let g ∈ G^{+}. Then (χ_{λ,+} − χ_{λ,−})(g) ≠ 0 if and only if g is in C_{η(λ)}. Moreover, χ_{λ,±} and C_{η(λ),±} may be labeled so that

\begin{aligned}χ_{λ,±}(g) & = \frac{1}{2}(ϵ_{λ} + \sqrt{ϵ_{λ}η(λ)}) if g ∈ C_{η(λ),±}, \\ χ_{λ,±}(g) & = \frac{1}{2}(ϵ_{λ} − \sqrt{ϵ_{λ}η(λ)}) if g ∈ C_{η(λ),∓}, \end{aligned}

where ϵ_{λ} = (−1)^{\frac{1}{2} (n−s)}.

We suppose for the rest of this section that Π = Γ × Ω, where Γ = {1, 2, …, p} and |Ω| = w < p. Let X and X^{+} be the symmetric group S(Γ) and the alternating group A(Γ) on Γ; let Y = N_{X}(P) and Y^{+} = N_{X^{+}}(P), where P is a fixed Sylow p-subgroup of X; and let H = N_{G}(D) and H^{+} = N_{G^{+}}(D), where D is the abelian Sylow p-subgroup P^{Ω} of G. We denote the sign characters of G, H, X, Y, S(Ω) with respect to G^{+}, H^{+}, X^{+}, Y^{+}, A(Ω) by sgn_{G}, sgn_{H}, sgn_{X}, sgn_{Y}, sgn_{S(Ω)}. In particular, sgn_{H} = Res_{H}^{G}(sgn_{G}) and sgn_{Y} = Res_{Y}^{X}(sgn_{X}). If (f, σ) is an element of H = Y ≀ S(Ω) with f ∈ Y^{Ω} and σ ∈ S(Ω), then

(4.1) sgn_{H}(f, σ) = sgn_{S(Ω)}(σ) ∏_{i∈Ω} sgn_{Y} (f(i)).

Thus (f, σ) is in H^{+} if and only if sgn_{S(Ω)}(σ) = ∏_{i∈Ω} sgn_{Y} (f(i)). The next two results hold for any finite group Y and a normal subgroup Y^{+} of Y.

Lemma (4B). Let (f, σ) ∈ Y ≀ S(Ω) and let Ω = ∐_{δ=1}^{d} Ω_{δ}^{σ} be the orbit decomposition of σ on Ω. Then the following hold:

(i) (f, σ)^{Y ≀S(Ω)} ∩ (Y^{+} ≀ S(Ω)) ≠ ∅ if and only if p_{i_{δ}}(f, σ) ∈ Y^{+} for all δ. (ii) Let (f, σ) ∈ Y^{+} ≀ S(Ω) and g ∈ Y^{Ω}. Then ^{(g,1)}(f, σ) ∈ (Y^{+} ≀ S(Ω)) if and

only if g is constant modulo Y^{+} on Ω_{δ}^{σ} for 1 ≤ δ ≤ d.

Proof. Part (i) follows from (3.1). For (ii) we note that ^{(g,1)}(f, σ) ∈ (Y^{+} ≀ S(Ω)) if and only if g(i)f(i)(g(σ^{−1})(i))^{−1} ∈ Y^{+} for all i ∈ Ω. Since f ∈ (Y^{+})^{Ω}, this is equivalent to saying g(i)(g(σ^{−1}(i)))^{−1} ∈ Y^{+} for all i ∈ Ω.

Lemma (4C). Let ψ^{+} be a character of Y^{+}, let ψ = Ind_{Y^{+}}^{Y}(ψ^{+}), and let E((ψ^{+})^{Ω}) and E(ψ^{Ω}) be the James-Kerber extensions of (ψ^{+})^{Ω} and ψ^{Ω} to Y^{+} ≀ S(Ω) and Y ≀ S(Ω). Then

E(ψ^{Ω}) = Ind_{Y^{+}≀S(Ω)}^{Y ≀S(Ω)}(E((ψ^{+})^{Ω})).

Proof. Let (f, σ) ∈ (Y ≀ S(Ω)) and let Ω = ∐_{δ=1}^{d} Ω_{δ}^{σ} be the orbit decomposition of σ on Ω. If (f, σ)^{Y ≀S(Ω)} ∩ (Y^{+} ≀ S(Ω)) = ∅, then Ind_{Y^{+}≀S(Ω)}^{Y ≀S(Ω)}(E((ψ^{+})^{Ω}))(f, σ) = 0. On the other hand, by (4B), (i) there exists δ such that p_{i_{δ}}(f, σ) ∉ Y^{+}, so that

E(ψ^{Ω})(f, σ) = ∏_{δ} ψ(p_{i_{δ}}(f, σ)) = 0.

Thus (4C) holds in this case. We may then suppose that (f, σ) ∈ (Y^{+} ≀ S(Ω)) and that p_{i_{δ}}(f, σ) ∈ Y^{+} for all δ. Let T be a transversal for Y^{+} in Y. Then

E(ψ^{Ω})(f, σ) = ∏_{δ} ψ(p_{i_{δ}}(f, σ))

= ∏_{δ} ∑_{t∈T} ψ^{+}(tp_{i_{δ}}(f, σ)t^{−1})

= ∑_{(t_{δ})∈T^{d}} ∏_{δ} ψ^{+}(t_{δ}p_{i_{δ}}(f, σ)t_{δ}^{−1}).

T^{Ω} is a transversal for (Y^{+})^{Ω} in Y^{Ω}. Let T_{σ}^{Ω} be the subset of all g ∈ T^{Ω} such that g is constant on Ω_{δ}^{σ} for all δ. We may identify such a g with the d-tuple (t_{δ}), where t_{δ} = g(i_{δ}). This gives a bijection of T_{σ}^{Ω} with T^{d}. Then (4B), (ii) and (3.1) imply that

Ind_{Y^{+}≀S(Ω)}^{Y ≀S(Ω)}(E((ψ^{+})^{Ω}))(f, σ) = ∑_{g∈T_{σ}^{Ω}} E((ψ^{+})^{Ω})(g(f, σ)g^{−1})

= ∑_{(t_{δ})∈T^{d}} ∏_{δ} ψ^{+}(t_{δ}p_{i_{δ}}(f, σ)t_{δ}^{−1}),

and (4C) holds.

Y is a Frobenius group with kernel P = ⟨a⟩ of order p and cyclic complement ⟨b⟩ of order p − 1. The set R = {a, b^{i} : 1 ≤ i ≤ p − 1} is a transversal for the conjugacy classes of Y; the classes contained in Y^{+} are represented by a and b^{2i} for 1 ≤ i ≤ \frac{1}{2}(p − 1). Of these only a^{Y} splits in Y^{+}, say a^{Y} = a_{+}^{Y^{+}} ∪ a_{−}^{Y^{+}}. We choose ξ_{1} = sgn_{Y}, ξ_{p} = 1_{Y}, and label the characters ξ_{γ} in Y^{∨} so that ξ_{p+1−γ} = ξ_{γ} sgn_{Y} for 1 ≤ γ ≤ p. We also write ξ_{p+1−γ} as ξ_{γ}^{∗}. In particular, Res_{Y^{+}}^{Y}(ξ_{p^{∗}}) = ξ_{p^{∗},+} + ξ_{p^{∗},−}, where p^{∗} = \frac{1}{2}(p + 1). We label ξ_{p^{∗},±} and a_{±} so that

ξ_{p^{∗},+}(x) − ξ_{p^{∗},−}(x) = ±\sqrt{ϵ_{p}p} for x ∈ (a_{±})^{Y^{+}},

where ϵ_{p} = (−1)^{\frac{p−1}{2}} . As in §3 we parametrize the irreducible characters ψ_{Λ} of H by mappings

Λ: Y^{∨} ⟶ {partitions}, ξ_{γ} ↦ λ_{γ},

where ∑_{γ} |λ_{γ}| = w. We also represent Λ by the p-tuple (λ_{1}, λ_{2}, …, λ_{p}). In particular, sgn_{H} = ψ_{Λ_{sgn}}, where Λ_{sgn} = ({1^{w}}, ∅, ∅, …, ∅). Indeed, the stabilizer H_{Λ_{sgn}} of Λ_{sgn} is H, and

E(ξ_{Λ_{sgn}})(f, σ) = ∏_{δ∈Δ_{1}} ξ_{1}(p_{i_{δ}}(f, σ))

= ∏_{δ∈Δ_{1}} ∏_{i∈Ω_{δ}^{σ}} sgn_{Y} (f(i))

= ∏_{i∈Ω} sgn_{Y} (f(i))

by (3.2). Thus ψ_{Λ_{sgn}} = sgn_{H}. Let ^{∗} be the duality Λ ↦ Λ^{∗}, where Λ^{∗}: ξ_{γ} ↦ (λ_{p+1−γ})^{∗}. We say Λ is a splitting mapping of H if ψ_{Λ} is a splitting character of H, i.e., Res_{H^{+}}^{H}(ψ_{Λ}) = ψ_{Λ,+} + ψ_{Λ,−} is reducible on H^{+}. The main result of this section is an analog of Frobenius’s theorem for splitting characters ψ_{Λ}.

Proposition (4D). Let ψ_{Λ} ∈ H^{∨}. Then sgn_{H} ψ_{Λ} = ψ_{Λ^{∗}}. In particular, ψ_{Λ} is a splitting character if and only if Λ = Λ^{∗}.

Proof. Clearly

ψ_{Λ} sgn_{H} = (Ind_{H_{Λ}}^{H}(E(ξ_{Λ})ω_{Λ})) sgn_{H}
= Ind_{H_{Λ}}^{H}(E(ξ_{Λ})ω_{Λ} sgn_{H_{Λ}})

where sgn_{H_{Λ}} = Res_{H_{Λ}}^{H}(sgn_{H}). Let (f, σ) ∈ H_{Λ}, and write (f, σ) = ∏_{γ}(f_{γ}, σ_{γ}), where (f_{γ}, σ_{γ}) in Y ≀ S(Ω_{γ}^{Λ}). Let Ω_{γ}^{Λ} = ∐_{δ∈Δ_{γ}} Ω_{δ}^{σ} be the orbit decomposition of σ_{γ} on Ω_{γ}^{Λ}. Since

sgn_{H_{Λ}}(f, σ) = ∏_{γ}[ ∏_{δ∈Δ_{γ}} sgn_{Y} (p_{i_{δ}}(f_{γ}, σ_{γ}))] sgn_{S(Ω_{γ}^{Λ})}(σ_{γ}),

we have that (E(ξ_{Λ})ω_{Λ} sgn_{H_{Λ}})(f, σ)

= ∏_{γ}[ ∏_{δ∈Δ_{γ}} ξ_{γ}(p_{i_{δ}}(f_{γ}, σ_{γ})) sgn_{Y} (p_{i_{δ}}(f_{γ}, σ_{γ}))]ω_{λ_{γ}} (σ_{γ}) sgn_{S(Ω_{γ}^{Λ})}(σ_{γ})

= ∏_{γ}[ ∏_{δ∈Δ_{γ}} ξ_{p+1−γ}(p_{i_{δ}}(f_{γ}, σ_{γ}))]ω_{λ_{γ}^{∗}} (σ_{γ})

= (E(ξ_{Λ^{∗}})ω_{Λ^{∗}})(f, σ).

Thus ψ_{Λ} sgn_{H} = ψ_{Λ^{∗}}, and Res_{H^{+}}^{H}(ψ_{Λ}) is reducible if and only if Λ = Λ^{∗}.

Each element of H is Y^{Ω}-conjugate to an element (f, σ) of the following type: Let Ω = ∐_{δ}^{d} Ω_{δ}^{σ} be the orbit decomposition of σ, let (f, σ) = ∏_{δ}(f_{δ}, σ_{δ}), where f_{δ} ∈ Y^{Ω_{δ}^{σ}} and σ_{δ} is the restriction of σ to Ω_{δ}^{σ}, and let n_{δ} = |Ω_{δ}^{σ}|. Then f_{δ} takes a value r_{δ} in R on a chosen element i_{δ} in Ω_{δ}^{σ} and value 1 on Ω_{δ}^{σ} − {i_{δ}}. In particular, p_{i_{δ}}(f, σ) = r_{δ}. The data ∏_{δ}[r_{δ}, n_{δ}] is called the cycle type of (f, σ) and characterizes the conjugacy class C of (f, σ) in H. The cycle type of (f, σ) in H and the cycle type of (f, σ) in S(Γ × Ω) are related as follows: Let Γ = ∐_{κ=1}^{k_{δ}} Γ_{δ,κ} be the orbit decomposition of ⟨p_{i_{δ}}(f, σ)⟩. The subsets Γ_{δ,κ} × Ω_{δ} for 1 ≤ κ ≤ k_{δ} are the orbits of (f_{δ}, σ_{δ}) on Γ × Ω_{δ}^{σ}. Thus the cycle type of (f, σ) in S(Γ × Ω) is the partition with parts |Γ_{δ,κ}||Ω_{δ}| for 1 ≤ δ ≤ d and 1 ≤ κ ≤ k_{δ}. A conjugacy class C of H is a splitting class if C ⊆ H^{+} and C = C_{+} ∪ C_{−} is the union of two conjugacy classes of H^{+}. The following result is [6, §4.2].

Lemma (4E). A conjugacy class C of H is a splitting class if and only if its cycle type satisfies the following conditions:

(i) Cycles [a, k] occur only for odd k, and such cycles have distinct lengths k. (ii) Cycles [b^{i}, j] occur only for odd i and even j.

A splitting class C thus has cycle type

∏_{δ=1}^{d}[a, k_{δ}] ∏_{i=1}^{p^{∗}−1} ∏_{j}[b^{2i−1}, 2j]^{w_{ij}}, (4.2)

where the k_{δ} are distinct odd integers. The cycle type of a splitting C is p-regular if no cycle of type [a, k] occurs, p-singular if no cycle of type [b^{2i−1}, 2j] occurs. A splitting class C consists of p-singular elements if and only if d > 0. If (f, σ) ∈ C, then

C_{Y^{Ω}}(f, σ) ≃ ⟨a⟩^{d} × ∏_{i,j}⟨b⟩^{w_{ij}}.

In particular,

|C_{H}(f, σ)|_{p} = p^{d}. (4.3)

We define a bijection between splitting mappings Λ and splitting classes C of H. If Λ is a splitting mapping, let C_{Λ} be the splitting class of the following cycle type (4.2): d is the number of diagonal nodes of λ_{p^{∗}}, k_{δ} is the hook length η_{δδ}(λ_{p^{∗}}), and w_{ij} is the number of parts of length j in λ_{i}. Conversely, if C is a splitting class of cycle type (4.2), let Λ_{C} be the following p-tuple (λ_{1}, λ_{2}, …, λ_{p}): λ_{i} for i < p^{∗} is the partition having a part j for each cycle of type [b^{2i−1}, 2j], λ_{p^{∗}} is the unique self-dual partition such that η_{δδ}(λ_{p^{∗}}) = k_{δ} for 1 ≤ δ ≤ d, and λ_{i} for i > p^{∗} is the partition λ_{p+1−i}^{∗}. Then (Λ_{C})^{∗} = Λ_{C} and Λ_{C} is a splitting mapping.

If Λ is a splitting mapping of H, let Λ′ and Λ′′ be the p-tuples gotten from Λ by replacing λ_{p^{∗}} by ∅ in the one case and by replacing λ_{γ} for γ ≠ p^{∗} by ∅ in the other case. We call Λ′ and Λ′′ the p-regular and the p-singular parts of Λ. The p-tuple with all components ∅ will be called the trivial p-tuple. Let Ω = Ω′ ∪ Ω′′, where

Ω′ = ∐_{γ≠p^{∗}} Ω_{γ}^{Λ}, Ω′′ = Ω_{p^{∗}}^{Λ},

and |Ω_{γ}^{Λ}| = |λ_{γ}| for all γ. Let w′ = |Ω′|, w′′ = |Ω′′|, H′ = Y ≀S(Ω′), H′′ = Y ≀S(Ω′′). We may regard a non-trivial Λ′ or Λ′′ as a splitting mapping parametrizing a splitting character ψ_{Λ′} or ψ_{Λ′′} of H′ or H′′, since (Λ′)^{∗} = Λ′ and (Λ′′)^{∗} = Λ′′. In that case let (H′)^{+} = H′ ∩ A(Γ × Ω′) and (H′′)^{+} = H′′ ∩ A(Γ × Ω′′), and let

Res_{(H′)^{+}}^{H′}(ψ_{Λ′}) = ψ_{Λ′,+} + ψ_{Λ′,−} and Res_{(H′′)^{+}}^{H′′}(ψ_{Λ′′}) = ψ_{Λ′′,+} + ψ_{Λ′′,−}.

Suppose C_{Λ} has cycle type (4.2). If Λ′ is trivial, then H′ = (H′)^{+} = 1 and we define C_{Λ′} as the identity class of H′. If Λ′ is non-trivial, then C_{Λ′} = C_{Λ′,+} ∪ C_{Λ′,−} is a splitting class of H′ of cycle type ∏_{i,j}[b^{2i−1}, 2j]^{w_{ij}}. If Λ′′ is trivial, then H′′ = (H′′)^{+} = 1 and we define C_{Λ′′} as the identity class of H′′. If Λ′′ is non-trivial, then C_{Λ′′} = C_{Λ′′,+}∪C_{Λ′′,−} is a splitting class of H′′ of cycle type ∏_{δ}[a, k_{δ}]. Moreover, the irreducible character ω_{λ_{p^{∗}}} of S(Ω′′) splits as ω_{λ_{p^{∗},+}} + ω_{λ_{p^{∗},−}} on A(Ω′′).

Proposition (4F). Suppose Λ is a splitting mapping of H such that Λ equals its p-singular part Λ′′. Let (f, σ) ∈ H^{+}. Then (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0 if and only if (f, σ) ∈ C_{Λ}. Moreover, ψ_{Λ,±} and C_{Λ,±} may be labeled so that

(ψ_{Λ,+} − ψ_{Λ,−})(f, σ) = ±(\sqrt{ϵ_{p}p})^{d}\sqrt{ϵ_{λ_{p^{∗}}} η(λ_{p^{∗}})} for (f, σ) ∈ C_{Λ,±},

where d is the number of diagonal nodes in λ_{p^{∗}}.

Proof. For simplicity of notation we write ξ for ξ_{p^{∗}} and λ for λ_{p^{∗}}. By hypothesis H_{Λ} = H and ψ_{Λ} = E(ξ_{Λ})ω_{Λ}. Suppose w = 1. Then H = Y, H^{+} = Y^{+}, ψ_{Λ} = ξ, and C_{Λ} = a^{Y}. So (4F) holds if we choose ψ_{Λ,±} = ξ_{±} and C_{Λ,±} = (a_{±})^{Y^{+}}. We may suppose w > 1. The irreducible character θ_{Λ} = (ξ_{+})^{Ω} of (Y^{+})^{Ω} is stable under Y^{+} ≀ S(Ω). Let E(θ_{Λ}) be its James-Kerber extension to Y^{+} ≀ S(Ω). Since ξ = Ind_{Y^{+}}^{Y}(ξ_{+}), it follows by (4C) that

E(ξ_{Λ}) = Ind_{Y^{+}≀S(Ω)}^{H}(E(θ_{Λ})).

Frobenius reciprocity and Mackey decomposition give

Res_{H^{+}}^{H}(ψ_{Λ}) = Res_{H^{+}}^{H}(E(ξ_{Λ})ω_{Λ})
= Res_{H^{+}}^{H}(Ind_{Y^{+}≀S(Ω)}^{H}(E(θ_{Λ}))) Res_{H^{+}}^{H}(ω_{Λ})
= Ind_{Y^{+}≀A(Ω)}^{H^{+}}(Res_{Y^{+}≀A(Ω)}^{Y^{+}≀S(Ω)}(E(θ_{Λ}))) Res_{H^{+}}^{H}(ω_{Λ}).

Let (Y^{Ω})^{+} = Y^{Ω} ∩ G^{+} and let Θ_{Λ,+} = Ind_{Y^{+}≀A(Ω)}^{(Y^{Ω})^{+}A(Ω)}(Res_{Y^{+}≀A(Ω)}^{Y^{+}≀S(Ω)}(E(θ_{Λ}))). Then

Res_{H^{+}}^{H}(ψ_{Λ}) = Ind_{(Y^{Ω})^{+}A(Ω)}^{H^{+}}(Θ_{Λ,+}) Res_{H^{+}}^{H}(ω_{Λ})
= Ind_{(Y^{Ω})^{+}A(Ω)}^{H^{+}}(Θ_{Λ,+} Res_{(Y^{Ω})^{+}A(Ω)}^{H}(ω_{Λ}))
= Ind_{(Y^{Ω})^{+}A(Ω)}^{H^{+}}(Θ_{Λ,+}(ω_{Λ,+} + ω_{Λ,−})).

Since Res_{H^{+}}^{H}(ψ_{Λ}) = ψ_{Λ,+} + ψ_{Λ,−}, it follows that Θ_{Λ,+} is an irreducible character of (Y^{Ω})^{+}A(Ω). We label ψ_{Λ,±} so that

ψ_{Λ,+} = Ind_{(Y^{Ω})^{+}A(Ω)}^{H^{+}}(Θ_{Λ,+}ω_{Λ,+}),
ψ_{Λ,−} = Ind_{(Y^{Ω})^{+}A(Ω)}^{H^{+}}(Θ_{Λ,+}ω_{Λ,−}).

(4.5)

Let H^{+} = ⟨(Y^{Ω})^{+}A(Ω), s⟩, where s = (g, τ) with g ∉ (Y^{Ω})^{+} and τ ∉ A(Ω), and set Θ_{Λ,−} = (Θ_{Λ,+})^{s}. Then

Res_{(Y^{+})^{Ω}}^{(Y^{Ω})^{+}A(Ω)}(Θ_{Λ,+}) = ∑_{\begin{matrix}(ϵ_{1},ϵ_{2},... ,ϵ_{w}) \\ ϵ_{1}ϵ_{2}···ϵ_{w}=+ \end{matrix}} ξ_{ϵ_{1}} × ξ_{ϵ_{2}} × ⋯ × ξ_{ϵ_{w}},

Res_{(Y^{+})^{Ω}}^{(Y^{Ω})^{+}A(Ω)}(Θ_{Λ,−}) = ∑_{\begin{matrix}(ϵ_{1},ϵ_{2},... ,ϵ_{w}) \\ ϵ_{1}ϵ_{2}···ϵ_{w}=− \end{matrix}} ξ_{ϵ_{1}} × ξ_{ϵ_{2}} × ⋯ × ξ_{ϵ_{w}}.

In particular, Θ_{Λ,−} ≠ Θ_{Λ,+}. By (4.5)

(4.6) ψ_{Λ,+} − ψ_{Λ,−} = \begin{cases}(Θ_{Λ,+} − Θ_{Λ,−})(ω_{Λ,+} − ω_{Λ,−}) & on (Y^{Ω})^{+}A(Ω), \\ 0 & off (Y^{Ω})^{+}A(Ω). \end{cases}

Suppose (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0. Then (f, σ) ∈ (Y^{Ω})^{+}A(Ω) by (4.6), and

(Θ_{Λ,+} − Θ_{Λ,−})(f, σ) ≠ 0, (ω_{Λ,+} − ω_{Λ,−})(σ) ≠ 0.

(4A) implies that σ is in the class C_{η(λ)} of S(Ω) and

(ω_{Λ,+} − ω_{Λ,−})(σ) = ±\sqrt{ϵ_{λ}η(λ)}.

Since (f, σ) is in a splitting class of H, (4E) then implies that (f, σ) ∈ C_{Λ}. We may suppose p_{i_{δ}}((f, σ)) = a for all δ. The proof of (4C) then gives

Θ_{Λ,+}((f, σ)) = ∑_{h∈(Y^{Ω})^{+}/(Y^{+})^{Ω}}E(θ_{Λ})(^{h}(f, σ)) = ∑_{\begin{matrix}(ϵ_{1},ϵ_{2},... ,ϵ_{d}) \\ ϵ_{1}ϵ_{2}···ϵ_{d}=+ \end{matrix}} ∏_{δ=1}^{d} ξ_{ϵ_{δ}}(a).

Since Θ_{Λ,+} is (Y^{Ω})^{+}S(Ω)-invariant and Θ_{Λ,−} = (Θ_{Λ,+})^{s},

Θ_{Λ,−}((f, σ)) = ∑_{\begin{matrix}(ϵ_{1},ϵ_{2},... ,ϵ_{d}) \\ ϵ_{1}ϵ_{2}···ϵ_{d}=− \end{matrix}} ∏_{δ=1}^{d} ξ_{ϵ_{δ}}(a).

Thus

(Θ_{Λ,+} − Θ_{Λ,−})((f, σ)) = (ξ_{+}(a) − ξ_{−}(a))^{d} = (\sqrt{ϵ_{p}p})^{d}.

We then label ψ_{Λ,±} and C_{Λ,±} so that (4F) holds.

Proposition (4G). Suppose Λ is a splitting mapping of H such that Λ equals its p-regular part Λ′. Let (f, σ) ∈ H^{+}. Then (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0 only if (f, σ) is in a p-regular splitting class of H.

Proof. Let H_{Λ}^{+} = H_{Λ} ∩ H^{+} and Θ_{Λ} = Res_{H_{Λ}^{+}}^{H_{Λ}}(E(ξ_{Λ})ω_{Λ}). Then Θ_{Λ} is irreducible

since (E(ξ_{Λ})ω_{Λ})(h, 1) ≠ 0 for any element h in Y^{Ω} − (Y^{Ω})^{+} taking value b on some i ∈ Ω and value 1 on Ω − {i}. We fix such an h. Let τ be an involution in S(Ω) interchanging Ω_{γ}^{Λ} and Ω_{p+1−γ}^{Λ} for 1 ≤ γ < p^{∗}. Let t = (g, τ), where g = 1 if τ ∈ A(Ω) and g = h if τ ∉ A(Ω). Then t ∈ N_{H^{+}}(H_{Λ}^{+}) and T_{Λ} = ⟨H_{Λ}^{+}, t⟩ is an extension of H_{Λ}^{+} of order 2.

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We claim T_{Λ} stabilizes Θ_{Λ}. Indeed, let (f, σ) ∈ H_{Λ}^{+}, and write (f, σ) = ∏_{γ}^{p}(f_{γ}, σ_{γ}),

where f_{γ} ∈ Y^{Ω_{γ}^{Λ}}, σ_{γ} ∈ S(Ω_{γ}^{Λ}). Then

ω_{Λ}^{t}(σ) = ∏_{γ=1}^{p} ω_{λ_{γ}} (^{t}σ_{γ})

= ∏_{γ=1}^{p} ω_{λ_{γ}^{∗}} (σ_{γ})

= ∏_{γ=1}^{p} ω_{λ_{γ}} (σ_{γ}) sgn_{S(Ω_{γ}^{Λ})}(σ_{γ})

= (ω_{Λ} sgn_{S(Ω)})(σ).

(4.7)

On the other hand, if (f_{γ}, σ_{γ}) decomposes into cycles ∏_{δ∈Δ_{γ}} (f_{δ}, σ_{δ}), then

E(ξ_{Λ})^{t}(f, σ) = E(ξ_{Λ})^{(g,1)}(^{(1,τ)}(f, σ))

= E(ξ_{Λ})(^{(1,τ)}(f, σ))

= ∏_{γ} ∏_{δ∈Δ_{γ}} ξ_{p+1−γ}(p_{τ(i_{δ})}(^{(1,τ)}(f, σ)))

= ∏_{γ} ∏_{δ∈Δ_{γ}} ξ_{γ}^{∗}(p_{i_{δ}}(f, σ))

= ∏_{γ} ∏_{δ∈Δ_{γ}}(ξ_{γ}(p_{i_{δ}}(f, σ)) sgn_{Y} (p_{i_{δ}}(f, σ)))

= E(ξ_{Λ})(f, σ) sgn_{S(Ω)}(σ), (4.8)

since ∏_{γ} ∏_{i_{δ}∈Δ_{γ}} sgn_{Y} (p_{i_{δ}}(f, σ) = ∏_{i∈Ω} sgn_{Y} (f(i)) = sgn_{S(Ω)}(σ). So T_{Λ} stabilizes Θ_{Λ} by (4.7) and (4.8).

Let Θ_{Λ}^{+} and Θ_{Λ}^{−} be the two extensions of Θ_{Λ} to T_{Λ}. By Mackey decomposition we may label ψ_{Λ,±} so that

(4.9)  ψ_{Λ,+} = Ind_{T_{Λ}}^{H^{+}}(Θ_{Λ}^{+}),  ψ_{Λ,−} = Ind_{T_{Λ}}^{H^{+}}(Θ_{Λ}^{−}).

We note for later use that any odd element s of Y^{Ω} normalizes T_{Λ} and interchanges Θ_{Λ}^{+} and Θ_{Λ}^{−}, since ψ_{Λ,+}^{s} = ψ_{Λ,−}.

Suppose (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0. Replacing (f, σ) by a conjugate, we may suppose by (4.9) that (f, σ) ∈ T_{Λ} and (Θ_{Λ}^{+} − Θ_{Λ}^{−})(f, σ) ≠ 0. Thus (f, σ) ∉ H_{Λ}^{+}. Elements of T_{Λ} − H_{Λ}^{+} have the form (h, ρ), where ρ interchanges ∐_{γ<p^{∗}} Ω_{γ}^{Λ} and ∐_{γ>p^{∗}} Ω_{γ}^{Λ}. In particular, ρ contains no cycles of odd length. Thus (f, σ) is contained in a p-regular splitting class of H. This completes the proof of (4G).

Proposition (4H). Suppose Λ is a splitting mapping of H, Λ′ and Λ′′ are the pregular and p-singular parts of Λ, and H′ and H′′ are the subgroups of H associated to Λ′ and Λ′′.

(i) If (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0, then (f, σ) is H^{+}-conjugate to (f′, σ′)(f′′, σ′′), where (f′, σ′) is in a p-regular splitting class of H′ and (f′′, σ′′) is in the p-singular splitting class C_{Λ′′} of H′′.

(ii) Given labelings of ψ_{Λ′,±} and ψ_{Λ′′,±}, there exists a labeling of ψ_{Λ,±} such that

\begin{aligned}(ψ_{Λ,+} − ψ_{Λ,−})(f, σ) & = (ψ_{Λ′,+} − ψ_{Λ′,−})(f′, σ′) \\ & × (ψ_{Λ′′,+} − ψ_{Λ′′,−})(f′′, σ′′) \end{aligned} (4.10)

for all (f, σ) = (f′, σ′)(f′′, σ′′) with (f′, σ′) a p-regular element of (H′)^{+}

and (f′′, σ′′) an arbitrary element of (H′′)^{+}.

Proof. (4H) follows trivially from (4F) or (4G) if Λ′ or Λ′′ is trivial. So we may suppose Λ′ and Λ′′ are non-trivial Let ξ_{Λ} be the irreducible character ∏_{γ}(ξ_{γ})^{Ω_{γ}^{Λ}} of Y^{Ω}. Let ξ_{Λ′} and ξ_{Λ′′} be the irreducible characters of Y^{Ω′} and Y^{Ω′′} defined by

ξ_{Λ′} = ∏_{γ≠p^{∗}}(ξ_{γ})^{Ω_{γ}^{Λ}}, ξ_{Λ′′} = (ξ_{p^{∗}})^{Ω_{p^{∗}}^{Λ}} .

The stabilizers of ξ_{Λ′} in H′ and ξ_{Λ′′} in H′′ are H_{Λ′} = Y^{Ω′}S(Ω^{Λ′}) and H_{Λ′′} = Y^{Ω′′}S(Ω^{Λ′′}), where S(Ω^{Λ′}) and S(Ω^{Λ′′}) are the components of S(Ω^{Λ}) on Ω′ and Ω′′. Let E(ξ_{Λ′}) and E(ξ_{Λ′′}) be the James-Kerber extensions of ξ_{Λ′} and ξ_{Λ′′} to H_{Λ′} and H_{Λ′′}, and let ω_{Λ′} and ω_{Λ′′} be the irreducible characters of S(Ω^{Λ′}) and S(Ω^{Λ′′}) parametrized by Λ′ and Λ′′. Then

ψ_{Λ′} = Ind_{H_{Λ′}}^{H′} (E(ξ_{Λ′})ω_{Λ′}), ψ_{Λ′′} = E(ξ_{Λ′′})ω_{Λ′′},

and

E(ξ_{Λ})ω_{Λ} = E(ξ_{Λ′})ω_{Λ′} × E(ξ_{Λ′′})ω_{Λ′′}.

Let

Θ_{Λ} = Res_{H_{Λ}^{+}}^{H_{Λ}}(E(ξ_{Λ})ω_{Λ}), Θ_{Λ′} = Res_{H_{Λ′}^{+}}^{H_{Λ′}} (E(ξ_{Λ′})ω_{Λ′}).

Then Θ_{Λ} and Θ_{Λ′} are irreducible since (E(ξ_{Λ})ω_{Λ})(f′, 1) and (E(ξ_{Λ′})ω_{Λ′})(f′, 1) are non-zero for any element f′ in Y^{Ω′} − (Y^{Ω′})^{+} taking value b on some i ∈ Ω′ and value 1 on Ω′ − {i}. We fix such an f′. On the other hand,

Res_{(H′′)^{+}}^{H′′}(ψ_{Λ′′}) = ψ_{Λ′′,+} + ψ_{Λ′′,−}, Res_{H_{Λ′}^{+} ×H_{Λ′′}^{+}}^{H_{Λ}^{+}} (Θ_{Λ}) = Θ_{Λ,+} + Θ_{Λ,−}

are reducible. Given a labeling of ψ_{Λ′′,±}, we label Θ_{Λ,±} so that Θ_{Λ,+} = Θ_{Λ′} ×ψ_{Λ′′,+} and Θ_{Λ,−} = Θ_{Λ′} × ψ_{Λ′′,−}.

Let s = (h, 1) in Y^{Ω} be such that h takes the value b once on Ω′ and once on Ω′′, and the value 1 everywhere else. Then H_{Λ}^{+} = ⟨H_{Λ′}^{+} × H_{Λ′′}^{+}, s⟩. Let τ ∈ S(Ω′) be an involution interchanging Ω_{γ}^{Λ} and Ω_{p+1−γ}^{Λ} for 1 ≤ γ < p^{∗}. Let t = (g, τ), where g is the identity of Y^{Ω′} if τ ∈ A(Ω′), and g = f′ if τ ∉ A(Ω′). Then t ∈ N_{(H′)^{+}}(H_{Λ′}^{+}), T_{Λ′} = ⟨H_{Λ′}^{+}, t⟩ is an extension of H_{Λ′}^{+} of order 2, and [s, t] ∈ H_{Λ′}^{+} × H_{Λ′′}^{+}. In particular, if T_{Λ} = ⟨H_{Λ}^{+}, t⟩, then T_{Λ} = ⟨H_{Λ′}^{+} × H_{Λ′′}^{+}, s, t⟩ = ⟨T_{Λ′} × H_{Λ′′}^{+}, s⟩.

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 • • • • • • •  • •  H H^{+} T_{Λ′} × H_{Λ′′}^{+} H_{Λ′}^{+} × H_{Λ′′}^{+}  H_{Λ′}^{+} × H_{Λ′′} H_{Λ′} × H_{Λ′′} H_{Λ′} × H_{Λ′′}^{+} T_{Λ} H_{Λ}^{+}

T_{Λ′} stabilizes Θ_{Λ′} as in the proof of (4G). In particular, Θ_{Λ′} has two extensions Θ_{Λ′}^{+} and Θ_{Λ′}^{−} to T_{Λ′}, Ind_{H_{Λ′}^{+}}^{T_{Λ′}} (Θ_{Λ′}) = Θ_{Λ′}^{+} +Θ_{Λ′}^{−}, and Ind_{T_{Λ′}}^{(H′)^{+}}(Θ_{Λ′}^{+} +Θ_{Λ′}^{−}) = ψ_{Λ′,+}+ψ_{Λ′,−}. Given a labeling of ψ_{Λ′,±}, we label Θ_{Λ′}^{±} so that

ψ_{Λ′,+} = Ind_{T_{Λ′}}^{(H′)^{+}}(Θ_{Λ′}^{+}), ψ_{Λ′,−} = Ind_{T_{Λ′}}^{(H′)^{+}}(Θ_{Λ′}^{−}).

Then T_{Λ′} ×H_{Λ′′}^{+} stabilizes Θ_{Λ} since Θ_{Λ} = Ind_{H_{Λ′}^{+} ×H_{Λ′′}^{+}}^{H_{Λ}^{+}} (Θ_{Λ,+}), and T_{Λ} stabilizes Θ_{Λ} since s ∈ H_{Λ}^{+}. Now

(Θ_{Λ′}^{+} × ψ_{Λ′′,+})^{s} = Θ_{Λ′}^{−} × ψ_{Λ′′,−},
(Θ_{Λ′}^{+} × ψ_{Λ′′,−})^{s} = Θ_{Λ′}^{−} × ψ_{Λ′′,+},

since any odd element of Y^{Ω′} normalizes T_{Λ′} and interchanges Θ_{Λ′}^{+} and Θ_{Λ′}^{−}, and any odd element of Y^{Ω′′} interchanges ψ_{Λ′′,+} and ψ_{Λ′′,−}. Thus

Ξ_{Λ,+} = Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{T_{Λ}} (Θ_{Λ′}^{+} × ψ_{Λ′′,+}),
Ξ_{Λ,−} = Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{T_{Λ}} (Θ_{Λ′}^{+} × ψ_{Λ′′,−})

are distinct irreducible characters, and

(4.11) Ξ_{Λ,+} − Ξ_{Λ,−} = \begin{cases}(Θ_{Λ′}^{+} − Θ_{Λ′}^{−}) × (ψ_{Λ′′,+} − ψ_{Λ′′,−}) & on T_{Λ′} × H_{Λ′′}^{+}, \\ 0 & on T_{Λ} − (T_{Λ′} × H_{Λ′′}^{+}). \end{cases}

We may label ψ_{Λ,±} so that

ψ_{Λ,+} = Ind_{T_{Λ}}^{H^{+}}(Ξ_{Λ,+}) = Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{H^{+}} (Θ_{Λ′}^{+} × ψ_{Λ′′,+}),
ψ_{Λ,−} = Ind_{T_{Λ}}^{H^{+}}(Ξ_{Λ,−}) = Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{H^{+}} (Θ_{Λ′}^{+} × ψ_{Λ′′,−}).

(4.12)

Indeed, Mackey decomposition implies that Res_{H^{+}}^{H}(ψ_{Λ}) = Ind_{H_{Λ}^{+}}^{H^{+}}(Θ_{Λ}) = Ind_{T_{Λ}}^{H^{+}}(Ind_{H_{Λ}^{+}}^{T_{Λ}}(Θ_{Λ})), since ψ_{Λ} = Ind_{H_{Λ}}^{H}(E(ξ_{Λ})ω_{Λ}). But

Ind_{H_{Λ}^{+}}^{T_{Λ}}(Θ_{Λ}) = Ind_{H_{Λ}^{+}}^{T_{Λ}}(Ind_{H_{Λ′}^{+} ×H_{Λ′′}^{+}}^{H_{Λ}^{+}} (Θ_{Λ′} × ψ_{Λ′′,+}))
= Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{T_{Λ}} (Ind_{H_{Λ′}^{+} ×H_{Λ′′}^{+}}^{T_{Λ′} ×H_{Λ′′}^{+}} (Θ_{Λ′} × ψ_{Λ′′,+}))
= Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{T_{Λ}} ((Θ_{Λ′}^{+} + Θ_{Λ′}^{−}) × ψ_{Λ′′,+})
= Ξ_{Λ,+} + Ξ_{Λ,−}.

Thus (4.12) holds since Res_{H^{+}}^{H}(ψ_{Λ}) = ψ_{Λ,+} + ψ_{Λ,−}.

Suppose (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0. By (4.11) and (4.12) there exist (f′, σ′) in T_{Λ′} − H_{Λ′}^{+} and (f′′, σ′′) in H_{Λ′′}^{+} such that (f, σ) is H^{+}-conjugate to (f′, σ′)(f′′, σ′′) and

(Θ_{Λ′}^{+} − Θ_{Λ′}^{−})(f′, σ′) ≠ 0, (ψ_{Λ′′,+} − ψ_{Λ′′,−})(f′′, σ′′) ≠ 0. (4.13)

(4H),(i) now follows by (4F) and the last paragraph of the proof of (4G).

Let (f, σ) = (f′, σ′)(f′′, σ′′) ∈ H with (f′, σ′) ∈ (H′)_{p′}^{+} and (f′′, σ′′) ∈ (H′′)^{+}. If (f′, σ′) or (f′′, σ′′) is not in a splitting class of H′ or H′′, then (f, σ) is not in a splitting class of H. Thus both sides of (4.10) are zero and (4.10) holds. So we may suppose (f′, σ′) and (f′′, σ′′) are in splitting classes of H′ and H′′. Suppose (f′′, σ′′) is not in C_{Λ′′}. Then the right-hand side of (4.10) is zero by (4F). Suppose the left-hand side of (4.10) is non-zero. By (4H) (i) there exist u′ in a p-regular splitting class of H′, u′′ in a p-singular splitting class of H′′, and h ∈ H^{+} such that ((f′, σ′)(f′′, σ′′))^{h} = u′u′′. Moreover, u′′ ∈ C_{Λ′′}. In particular, u′′ is a product of cycles of p-singular length as an element of S(Γ × Ω′′). A comparison of the pregular and p-singular cycle types of these elements as elements of S(Γ × Ω) shows that (f′, σ′)^{h} = u′, (f′′, σ′′)^{h} = u′′. Since u′′ has no fixed-points on Ω′′, neither does (f′′, σ′′). So h stabilizes Ω′ and Ω′′, and h ∈ H′H′′. Thus (f′, σ′) is conjugate to u′ in H′, (f′′, σ′′) is conjugate to u′′ in H′′, and (f′′, σ′′) ∈ C_{Λ′′}. This is a contradiction. So we may suppose (f′, σ′) is in a p-regular splitting class of H′ and (f′′, σ′′) is in C_{Λ′′}.

Suppose there exists (g, τ) in H^{+} such that ^{(g,τ)}(f, σ) = (h′, ρ′)(h′′, ρ′′), where (h′, ρ′) ∈ T_{Λ′}, (h′′, ρ′′) ∈ H_{Λ′′}^{+}, and

(Θ_{Λ′}^{+} − Θ_{Λ′}^{−})(h′, ρ′) ≠ 0, (ψ_{Λ′′,+} − ψ_{Λ′′,−})(h′′, ρ′′) ≠ 0. (4.14)

If τ(i_{δ}) ∈ Ω′′ for some i_{δ} ∈ Ω′, then p_{τ(i_{δ})}(h′′, ρ′′) ∈ a^{Y} by (4F). But

p_{τ(i_{δ})}(h′′, ρ′′) = p_{τ(i_{δ})}(^{(g,τ)}(f, σ))
= g(τ(i_{δ}))p_{i_{δ}}(f, σ)g(τ(i_{δ}))^{−1}
= g(τ(i_{δ}))p_{i_{δ}}(f′, σ′)g(τ(i_{δ}))^{−1}

and p_{i_{δ}}(f′, σ′) ∈ (b^{i})^{Y} for odd i by hypothesis. This is impossible. Thus τ stabilizes Ω′ and Ω′′ and (g, τ) = (g′, τ′)(g′′, τ′′), where (g′, τ′) ∈ H′ and (g′′, τ′′) ∈ H′′. Then

(ψ_{Λ,+} − ψ_{Λ,−})(f, σ)
= Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{H^{+}} ((Θ_{Λ′}^{+} × ψ_{Λ′′,+}) − (Θ_{Λ′}^{+} × ψ_{Λ′′,−}))(f, σ)
= Ind_{T_{Λ′} ×H_{Λ′′}^{+}}^{(H′×H′′)^{+}}((Θ_{Λ′}^{+} × ψ_{Λ′′,+}) − (Θ_{Λ′}^{+} × ψ_{Λ′′,−}))(f, σ).

Let D be a transversal for T_{Λ′} in (H′)^{+}. Then D ∪Ds is a transversal for T_{Λ′} ×H_{Λ′′}^{+} in (H′ × H′′)^{+}, where s = (h, 1) is the element introduced earlier in the proof. It follows that

(ψ_{Λ,+} − ψ_{Λ,−})(f, σ)
= Ind_{T_{Λ′}}^{(H′)^{+}}(Θ_{Λ′}^{+})(f′, σ′) × (ψ_{Λ′′,+} − ψ_{Λ′′,−})(f′′, σ′′)
+ Ind_{T_{Λ′}}^{(H′)^{+}}(Θ_{Λ′}^{−})(f′, σ′) × (ψ_{Λ′′,−} − ψ_{Λ′′,+})(f′′, σ′′)
= (ψ_{Λ′,+} − ψ_{Λ′,−})(f′, σ′) × (ψ_{Λ′′,+} − ψ_{Λ′′,−})(f′′, σ′′),

and (4.10) holds. Suppose no (g, τ) in H^{+} satisfies (4.14). Then (ψ_{Λ,+}−ψ_{Λ,−})(f, σ) is 0 by (4.12). Thus (4.10) holds unless

(ψ_{Λ′,+} − ψ_{Λ′,−})(f′, σ′) ≠ 0, (ψ_{Λ′′,+} − ψ_{Λ′′,−})(f′′, σ′′) ≠ 0.

In this case there exists (g′, τ′) in (H′)^{+} such that ^{(g′,τ′)}(f′, σ′) = (h′, ρ′) is in T_{Λ′} and (Θ_{Λ′}^{+} − Θ_{Λ′}^{−})(h′, ρ′) ≠ 0. This is impossible since ^{(g′,τ′)}(f′′, σ′′) = (f′′, σ′′), so that (4.14) holds. This completes the proof of (4H), (ii).

Corollary (4I). Let Λ be a splitting mapping of H. If (ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≠ 0, then

(ψ_{Λ,+} − ψ_{Λ,−})(f, σ) ≡ 0 (mod \sqrt{|C_{H}(f, σ)|_{p}})

and |C_{H}(f, σ)|_{p} = p^{d}, where d is the number of diagonal nodes in λ_{p^{∗}}.

Proof. This follows from (4H), (4F), and (4.3).

5. The isotypy conjecture for alternating groups for p > 2

We suppose p is odd in this section. Let G and G^{+} be the symmetric and alternating groups on the set Π of n elements. Let e and e^{+} be blocks of OG and OG^{+} such that e covers e^{+}. In particular, e and e^{+} have the same defect group D. If D is non-trivial, then there exists χ_{λ} in (G, e)^{∨} such that Res_{G^{+}}^{G}(χ_{λ}) is irreducible. Indeed, let λ_{0} be the p-core parametrizing e, and choose a partition λ of n with p-core λ_{0} and a p-quotient Λ such that Λ^{∗} ≠ Λ. In particular, e^{+} is the unique block of OG^{+} covered by e.

Let H = N_{G}(D) and H^{+} = N_{G^{+}}(D). Let f = Br_{D}^{G}(e) and f^{+} = Br_{D}^{G^{+}}(e^{+}) be the Brauer correspondents of e and e^{+} in OH and OH^{+}. In particular, f^{+} is H-stable since e^{+} is G-stable. Moreover, ff^{+} = f, since ee^{+} = e and Br_{D}^{G}(e^{+}) = Br_{D}^{G^{+}}(e^{+}). Thus f covers f^{+}, and f^{+} is the unique block of OH^{+} covered by f. We suppose from now on that D is abelian and that |D| = p^{w} with 1 ≤ w < p. Then e and f are isotypic blocks by (3B). If λ_{0}^{∗} ≠ λ_{0}, then e^{+} and f^{+} are isotypic blocks by (1E). So we also suppose from now on that λ_{0}^{∗} = λ_{0}. Thus e^{∗} = e = e^{+}, f^{∗} = f = f^{+}, and (G, G^{+}, e, e^{+}) and (H, H^{+}, f, f^{+}) are +configurations.

We use the notation of §3: Π = Π_{0} ∪ Π_{1}, where Π_{0} and Π_{1} are disjoint sets of n_{0} and pw elements; Π_{1} = Γ × Ω, where Γ = {1, 2, …, p} and |Ω| = w; X = S(Γ) and Y = N_{X}(P), where P is a Sylow p-subgroup of X; and D is the Sylow p-subgroup P^{Ω} of S(Π_{1}). Let

\begin{aligned}H_{0} & = S(Π_{0}), H_{1} = Y ≀ S(Ω), \\ H_{0}^{+} & = A(Π_{0}), H_{1}^{+} = (Y ≀ S(Ω))^{+}, \end{aligned} (5.1)

and set H^{++} = H_{0}^{+} × H_{1}^{+}. Then H = H_{0} × H_{1}, H > H^{+} ≥ H^{++}, and H^{+}/H^{++} has order 1 or 2 according as |Π_{0}| ≤ 1 or |Π_{0}| > 1. We have

f^{+} = f = e_{λ_{0}} × 1_{H_{1}}, (5.2)

where e_{λ_{0}} is the block of defect 0 of OH_{0} parametrized by λ_{0}. Let χ_{λ_{0}} denote the character in (H_{0}, e_{λ_{0}})^{∨}. Since λ_{0} is self-dual, either |λ_{0}| ≤ 1 or |λ_{0}| ≥ 3. It will be convenient to interpret H_{0} and H_{0}^{+} as the identity group when |λ_{0}| ≤ 1 and to omit identity factors in expressions such as (5.2). We fix a maximal e^{+}-subpair (D, e_{D}^{+}) and a maximal f^{+}-subpair (D, f_{D}^{+}) as follows: C_{G^{+}}(D) = C_{H^{+}}(D) = H_{0}^{+} × D. If |Π_{0}| ≤ 1, let e_{D}^{+} = f_{D}^{+} be the 1-block; if |Π_{0}| ≥ 3, let e_{D}^{+} = f_{D}^{+} = e_{λ_{0},+}, where e_{λ_{0},+} is the block of OH_{0}^{+} containing an irreducible constituent χ_{λ_{0},+} of Res_{H_{0}^{+}}^{H_{0}}(χ_{λ_{0}}). In both cases

N_{G^{+}}(D, e_{D}^{+}) = N_{H^{+}}(D, f_{D}^{+}) = H^{++}. (5.3)

We describe certain subpairs in Br_{e^{+}}(G^{+}) and Br_{f^{+}}(H^{+}). Every e^{+}-subpair (Q, u^{+}) is covered by an e-subpair (Q, u) in the sense that u covers u^{+}. Conversely, any subpair (Q, u^{+}) of G^{+} covered by an e-subpair (Q, u) is an e^{+}-subpair. Indeed, these claims follow from the relation Br_{Q}^{G}(e) = Br_{Q}^{G^{+}}(e^{+}). If |Π_{0}| ≤ 1, then e^{+} and f^{+} are the 1-blocks of OG^{+} and OH^{+}, and Brauer’s Third Main Theorem implies the subpairs in Br_{e^{+}}(G^{+}) and Br_{f^{+}}(H^{+}) are (Q, e_{Q}^{+}) and (Q, f_{Q}^{+}), where e_{Q}^{+} and f_{Q}^{+} are the 1-blocks of OG_{Q}^{+} and OH_{Q}^{+}. So we suppose |Π_{0}| ≥ 3. Let x be a standard non-identity element of D in the sense of §3, and recall that

G_{x} = G_{x,0} × G_{x,1}, H_{x} = H_{x,0} × H_{x,1},

where G_{x,0} = S(Π_{0} ∪ (Γ × Ω_{x,0})), H_{x,0} = S(Π_{0}) × (Y ≀ S(Ω_{x,0})), and G_{x,1} = H_{x,1} = P ≀S(Ω_{x,1}). Let H_{x,0} = H_{x,00}×H_{x,01}, where H_{x,00} = S(Π_{0}) and H_{x,01} = Y ≀S(Ω_{x,0}). Let

e_{x} = e_{x,0} × e_{x,1}, f_{x} = f_{x,00} × f_{x,01} × f_{x,1},

where e_{x,0} is the block of OG_{x,0} parametrized by λ_{0}, e_{x,1} is the 1-block of OG_{x,1}, f_{x,00} is the block of OH_{x,00} parametrized by λ_{0}, f_{x,01} is the 1-block of OH_{x,01}, and f_{x,1} is the 1-block of OH_{x,1}. Let f_{x,0} be the block f_{x,00} × f_{x,01} of OH_{x,0}. Let G_{x}^{+} and H_{x}^{+} be the intersections of G_{x} and H_{x} with A(Π). Similarly, let G_{x,0}^{+} and H_{x,0}^{+} be the intersections of G_{x,0} and H_{x,0} with A(Π_{0} ∪ (Γ × Ω_{x,0})), and let G_{x,1}^{+} and H_{x,1}^{+} be the intersections of G_{x,1} and H_{x,1} with A(Γ × Ω_{x,1}). In particular, G_{x} > G_{x}^{+} if |Π_{0}| > 1 or w > 1. Let G_{x}^{++} = G_{x,0}^{+} × G_{x,1}^{+} and H_{x}^{++} = H_{x,0}^{+} × H_{x,1}^{+}. If w > 1, then e_{x} and f_{x} cover unique blocks e_{x}^{+} and f_{x}^{+} of OG_{x}^{+} and OH_{x}^{+}. Indeed, let χ_{x} = χ_{x,0} × χ_{x,1}, where χ_{x,0} ∈ (G_{x,0}, e_{x,0})^{∨}, χ_{x,1} ∈ (G_{x,1}, e_{x,1})^{∨}, and χ_{x,0} or χ_{x,1} is irreducible on G_{x,0}^{+} or G_{x,1}^{+}. Such a χ_{x,0} or χ_{x,1} exists since |Ω_{x,0}| > 0 or |Ω_{x,1}| ≥ 2. So χ_{x} ∈ (G_{x}, e_{x})^{∨}, χ_{x} is irreducible on G_{x}^{+}, and e_{x} covers a unique block e_{x}^{+} of OG_{x}^{+}. A similar argument shows that f_{x} covers a unique block f_{x}^{+} of OH_{x}^{+}.

Proposition (5A). Let x be a non-identity standard element in D. If |Π_{0}| ≤ 1 or w > 1, then Br_{⟨x⟩}^{G^{+}}(e^{+}) and Br_{⟨x⟩}^{H^{+}}(f^{+}) are blocks of OG_{x}^{+} and OH_{x}^{+}. If |Π_{0}| ≥ 3 and w = 1, then D = ⟨x⟩, G_{x}^{+} = H_{x}^{+} = H_{0}^{+} × D, and

Br_{⟨x⟩}^{G^{+}}(e^{+}) = Br_{⟨x⟩}^{H^{+}}(f^{+}) = e_{λ_{0},+} + e_{λ_{0},−}.

Proof. If |Π_{0}| ≤ 1, then e^{+} and f^{+} are the principal blocks and (5A) is clear. If |Π_{0}| ≥ 3 and w > 1, then (5A) follows from the preceding paragraph and the fact that e and f are blocks of principal type. If |Π_{0}| ≥ 3 and w = 1, then the result follows from our discussion of the maximal e^{+}-subpair (D, e_{D}^{+}) and the maximal f^{+}-subpair (D, f_{D}^{+}).

Proposition (5B). Let μ = (J, ϵ) be the Rouquier isometry in R_{K}((G, e), (H, f)). Then the following hold:

(i) μ is +equivariant with respect to (G^{+}, H^{+}).

(ii) Every μ-induced isometry μ^{+} = (J^{+}, ϵ^{+}) in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})) is perfect.

Proof. By definition (3.8) of the Rouquier isometry

J(sgn_{G} χ_{λ}) = J(χ_{λ^{∗}}) = χ_{λ_{0}^{∗}} × ψΛ̂^{∗} = sgn_{H_{0}} χ_{λ_{0}} × sgn_{H_{1}} ψΛ̂ = sgn_{H} J(χ_{λ}).

Moreover, ϵ(sgn_{G} χ_{λ})) = ϵ(χ_{λ}), since |λ_{\frac{1}{2} (p+1)}| = |λ_{\frac{1}{2} (p+1)}^{∗}| and v(λ) = v(λ^{∗}). Here the last equality holds since the dual of a sequence of p-hooks leading from λ to λ_{0} is a sequence of p-hooks from λ^{∗} to λ_{0}^{∗}, and the leg and arm lengths of a p-hook have the same parity.

For (ii) it suffices by (2B) to show that

Φ_{λ}(g, h) = (χ_{λ,+} − χ_{λ,−})(g)(J^{+}(χ_{λ,+}) − J^{+}(χ_{λ,−}))(h^{−1})

is perfect on G^{+} × H^{+} for all self-dual partitions λ with p-core λ_{0}. Suppose Φ_{λ}(g, h) ≠ 0 for g ∈ G^{+}, h ∈ H^{+}. Then g and h are in splitting classes of G and H, and

(χ_{λ,+} − χ_{λ,−})(g) ≠ 0, (J^{+}(χ_{λ,+}) − J^{+}(χ_{λ,−}))(h^{−1}) ≠ 0. (5.4)

The first inequality and (4A) imply that g has cycle type η(λ). If d is the number of η_{ii}(λ) divisible by p, then

(χ_{λ,+} − χ_{λ,−})(g) ≡ 0 (mod \sqrt{ϵ_{λ}p^{d}})

and |C_{G}(g)|_{p} = p^{d}. Moreover, g is p-singular if and only if d > 0. The second inequality implies that h ∈ H^{++}. Indeed, if |Π_{0}| ≤ 1, then h ∈ H^{++} since H^{+} = H^{++}. If |Π_{0}| ≥ 3, then χ_{λ_{0}} and ψΛ̂ are reducible on H_{0}^{+} and H_{1}^{+}, and J^{+}(χ_{λ,±}) has support on H^{++}. Thus h = h_{0}(f_{1}, σ_{1}), where h_{0} ∈ H_{0}^{+}, (f_{1}, σ_{1}) ∈ H_{1}^{+}, and

(χ_{λ_{0},+} − χ_{λ_{0},−})(h_{0}^{−1}) ≠ 0, (ψ_{Λ̂,+} − ψ_{Λ̂,−})((f_{1}, σ_{1})^{−1}) ≠ 0.

Here (χ_{λ_{0},+} − χ_{λ_{0},−})(h_{0}^{−1}) is interpreted as 1 if |Π_{0}| ≤ 1. Since χ_{λ_{0},±} has defect 0, it follows that h_{0} is p-regular and (χ_{λ_{0},+} − χ_{λ_{0},−})(h_{0}) is divisible by |C_{H_{0}}(h_{0})|_{p}. On the other hand, (4I) implies that |C_{H_{1}}(f_{1}, σ_{1})|_{p} = p^{d} and

(ψ_{Λ̂,+} − ψ_{Λ̂,−})(f_{1}, σ_{1}) ≡ 0 (mod \sqrt{|C_{H_{1}}(f_{1}, σ_{1})|_{p}}).

Moreover, h is p-singular if and only if d > 0. It now follows that Φ_{λ} is perfect.

For each j = 1, 2, …, w, let x_{j} be a standard element of D such that |Ω_{x_{j},1}| = j. We claim the w subgroups ⟨x_{j}⟩ form a set of representatives for the N_{G^{+}}(D, e_{D}^{+})- orbits of non-trivial cyclic subgroups of D. This is trivially true when w = 1, so we may suppose w > 1. The w subgroups ⟨x_{j}⟩ form a set of representatives for the (Y ≀ S(Ω))-orbits of non-trivial cyclic subgroups of D. But the (Y ≀ S(Ω))-orbit of ⟨x_{j}⟩ is the (Y ≀ S(Ω))^{+}-orbit of ⟨x_{j}⟩, since

C_{Y ≀S(Ω)}(x_{j}) = (Y ≀ S(Ω_{x_{j},0})) × (P ≀ S(Ω_{x_{j},1}))

contains odd permutations when w > 1. Since (Y ≀ S(Ω))^{+} ≤ N_{G^{+}}(D, e_{D}^{+}), the claim holds.

We will use the isotypy (3B) between e and f to define an isometry μ^{+} and perfect isometries μ_{x}^{+} for x in {x_{1}, x_{2}, …, x_{w}} so that the diagrams

\begin{matrix}R_{K}(G^{+}, e^{+}) & ⟶^{R_{μ^{+}}} & R_{K}(H^{+}, f^{+}) \\ ↓_{d_{G^{+}}^{x,e^{+}_{x}}} & & ↓_{d_{H^{+}}^{x,f^{+}_{x}}} \\ CF_{p′}(G_{x}^{+}, e_{x}^{+}) & ⟶^{R_{μ^{+} x,p′}} & CF_{p′}(H_{x}^{+}, f_{x}^{+}) \end{matrix} (5.5)

are commutative. The definitions require a consistent labeling of splitting characters and splitting classes.

The following three configurations (5.6), (5.7), (5.8) are basic for the labeling:

G_{x,0} ⟶^{μ_{x,0}} H_{x,0} = H_{x,00} × H_{x,01} ; ↑  ↑ ; G_{x,0}^{+} ⟶^{μ_{x,0}^{+}}  H_{x,0}^{+} ; ↑ ; H_{x,00}^{+} × H_{x,01}^{+} (5.6)

where μ_{x,0} is the Rouquier isometry. If G_{x,0} = G_{x,0}^{+} or G_{x,0} = H_{x,0}, then μ_{x,0} is the identity isometry. Here G_{x,0} = G_{x,0}^{+} if |Π_{0}| ≤ 1 and j = w; G_{x,0} = H_{x,0} if j = w. In these cases, we fix μ_{x,0}^{+} as the identity isometry. If G_{x,0} > G_{x,0}^{+}, then (G_{x,0}, G_{x,0}^{+}, e_{x,0}, e_{x,0}^{+}) and (H_{x,0}, H_{x,0}^{+}, f_{x,0}, f_{x,0}^{+}) are +configurations, and μ_{x,0} is +equivariant with respect to (G_{x,0}^{+}, H_{x,0}^{+}) by (5B), (i). We will fix μ_{x,0}^{+} in what follows so that μ_{x,0}^{+} is μ_{x,0}-induced and hence perfect by (5B), (ii).

\begin{matrix}G_{x,1} & ⟶^{μ_{x,1}} & H_{x,1} \\ ↑ & & ↑ \\ G_{x,1}^{+} & ⟶^{μ_{x,1}^{+}} & H_{x,1}^{+} \end{matrix} (5.7)

where μ_{x,1} is the identity isometry. We fix μ_{x,1}^{+} as the identity isometry. If G_{x,1} > G_{x,1}^{+}, then (G_{x,1}, G_{x,1}^{+}, e_{x,1}, e_{x,1}^{+}) and (H_{x,1}, H_{x,1}^{+}, f_{x,1}, f_{x,1}^{+}) are +configurations, μ_{x,1} is trivially +equivariant with respect to (G_{x,1}^{+}, H_{x,1}^{+}), and μ_{x,1}^{+} is trivially μ_{x,1}- induced. Here G_{x,1} > G_{x,1}^{+} if and only if j > 1.

G_{x} = G_{x,0} × G_{x,1} ⟶^{μ_{x}=μ_{x,0}×μ_{x,1}} H_{x} = H_{x,0} × H_{x,1} ; ↑  ↑ ; G_{x}^{+}  ⟶^{μ_{x}^{+}}  H_{x}^{+} ; ↑  ↑ ; G_{x}^{++} = G_{x,0}^{+} × G_{x,1}^{+} ⟶^{μ_{x,0}^{+}×μ_{x,1}^{+}} H_{x}^{++} = H_{x,0}^{+} × H_{x,1}^{+} ; ↑ ; H_{x,00}^{+} × H_{x,01}^{+} × H_{x,1}^{+} (5.8)

where μ_{x,0}, μ_{x,1}, μ_{x,0}^{+}, μ_{x,1}^{+} are the perfect isometries in (5.6) and (5.7). If G_{x} = G_{x}^{+}, then G_{x,0} = G_{x,0}^{+} and G_{x,1} = G_{x,1}^{+}, that is, |Π_{0}| ≤ 1 and w = 1. Thus G_{x} = G_{x}^{+} = G_{x}^{++} and μ_{x} is the identity isometry. In this case we fix μ_{x}^{+} as the identity isometry. If G_{x} > G_{x}^{+} and G_{x}^{+} = G_{x}^{++}, then either G_{x,0} = G_{x,0}^{+} or G_{x,1} = G_{x,1}^{+}, that is, either |Π_{0}| ≤ 1 and 1 < j = w, or |Π_{0} ∪ (Γ × Ω_{x,0})| ≥ 3 and j = 1. In this case we fix μ_{x}^{+} = μ_{x,0}^{+} × μ_{x,1}^{+}. If G_{x} > G_{x}^{+} > G_{x}^{++}, then

\begin{matrix}(G_{x}, G_{x}^{+}, e_{x}, e_{x}^{+}), \\ (G_{x}^{+}, G_{x}^{++}, e_{x}^{+}, e_{x}^{++}), \end{matrix}  \begin{matrix}(H_{x}, H_{x}^{+}, f_{x}, f_{x}^{+}), \\ (H_{x}^{+}, H_{x}^{++}, f_{x}^{+}, f_{x}^{++}) \end{matrix}

are +configurations. Moreover, μ_{x} is perfect and +equivariant with respect to (G_{x}^{+}, H_{x}^{+}) by (2E), (i). We will fix μ_{x}^{+} in what follows so that μ_{x}^{+} is μ_{x}-induced and so that μ_{x,0}^{+} × μ_{x,1}^{+} is μ_{x}^{+}-induced. This is possible by (2F), (i). In particular, μ_{x}^{+} is perfect by (5B), (ii).

We fix μ_{x}^{+} for x = x_{j} in the remaining cases as follows:

Step 1. If |λ_{0}| ≥ 3, then the character χ_{λ_{0}} of H_{x,00} splits on H_{x,00}^{+}. We label the irreducible characters χ_{λ_{0},±} and the conjugacy classes C_{η(λ_{0}),±} of H_{x,00}^{+} so that (4A) holds. This labeling is fixed for all j. If |λ_{0}| ≤ 1, then H_{x,00} = H_{x,00}^{+} = 1 and we omit this step.

Step 2. If |Ω_{x,0}| ≥ 1, let Λ′ run over the p-regular splitting mappings of H_{x,01}. The splitting characters ψ_{Λ′} of H_{x,01} satisfy the hypotheses of (4G); the splitting classes C_{Λ′} of H_{x,01} have p-regular cycle type. We label arbitrarily the irreducible characters ψ_{Λ′,±} of H_{x,01}^{+} and the conjugacy classes C_{Λ′,±} of H_{x,01}^{+}. If |Ω_{x,0}| = 0, then H_{x,01} = H_{x,01}^{+} = 1 and we omit this step.

Step 3. Let Λ′′ run over the p-singular splitting mappings of Y ≀ S(Ω_{x,1}). Such mappings exist since |Ω_{x,1}| ≥ 1. The splitting characters ψ_{Λ′′} of Y ≀ S(Ω_{x,1}) satisfy the hypotheses of (4F); the splitting classes C_{Λ′′} of Y ≀ S(Ω_{x,1}) have p-singular cycle type. We label the irreducible characters ψ_{Λ′′,±} of Y ≀ S(Ω_{x,1}) and the conjugacy classes C_{Λ′′,±} of Y ≀ S(Ω_{x,1}) so that (4F) holds.

Step 4. Let λ′ run over all self-dual partitions of |Π_{0} ∪ (Γ × Ω_{x,0})| with p-core λ_{0} and p-regular p-quotient Λ′. Suppose |Ω_{x,0}| = 0. Then Λ′ is trivial, λ′ = λ_{0}, G_{x,0} = H_{x,0} = H_{x,00}, and χ_{λ′,±} and C_{η(λ′),±} have already been labeled in step 1. Suppose |Ω_{x,0}| ≥ 1. Then Λ′ is a p-regular splitting mapping of H_{x,01} of the type considered in step 2. The conjugacy class C_{η(λ′)} of G_{x,0} contains the conjugacy class C_{η(λ_{0})}C_{Λ′} of H_{x,0}. If |λ_{0}| ≥ 3, we label C_{η(λ′),±} so that

\begin{aligned}C_{η(λ′),+} & = (C_{η(λ_{0}),+}C_{Λ′,+})^{G_{x,0}^{+}} = (C_{η(λ_{0}),−}C_{Λ′,−})^{G_{x,0}^{+}}, \\ C_{η(λ′),−} & = (C_{η(λ_{0}),+}C_{Λ′,−})^{G_{x,0}^{+}} = (C_{η(λ_{0}),−}C_{Λ′,+})^{G_{x,0}^{+}}. \end{aligned}

If |λ_{0}| ≤ 1, then H_{x,00} = H_{x,00}^{+} = 1 and we label C_{η(λ′),±} so that C_{η(λ′),+} = (C_{Λ′,+})^{G_{x,0}^{+}} and C_{η(λ′),−} = (C_{Λ′,−})^{G_{x,0}^{+}}. We then label the irreducible characters χ_{λ′,±} of G_{x,0}^{+} so that (4A) holds. We label the irreducible characters (χ_{λ_{0}}ψ_{Λ′})_{±} of H_{x,0}^{+} so that

Res_{H_{x,00}^{+}H_{x,01}^{+}}^{H_{x,0}^{+}}((χ_{λ_{0}}ψ_{Λ′})_{+}) = χ_{λ_{0},+}ψ_{Λ′,+} + χ_{λ_{0},−}ψ_{Λ′,−},
Res_{H_{x,00}^{+}H_{x,01}^{+}}^{H_{x,0}^{+}}((χ_{λ_{0}}ψ_{Λ′})_{−}) = χ_{λ_{0},+}ψ_{Λ′,−} + χ_{λ_{0},−}ψ_{Λ′,+}

for |λ_{0}| ≥ 3, and so that

(χ_{λ_{0}}ψ_{Λ′})_{+} = 1_{S(Π_{0})} × ψ_{Λ′,+}, (χ_{λ_{0}}ψ_{Λ′})_{−} = 1_{S(Π_{0})} × ψ_{Λ′,−}

for |λ_{0}| ≤ 1.

Step 5. Let μ_{x,0} = (J_{x,0}, ϵ_{x,0}). We fix a μ_{x,0}-induced isometry μ_{x,0}^{+} = (J_{x,0}^{+}, ϵ_{x,0}^{+}) so that

J_{x,0}^{+}(χ_{λ′,±}) = J_{x,0}(χ_{λ′})_{±} = (χ_{λ_{0}}ψ_{Λ′})_{±}

for the λ′ in step 4. Here Λ̂′ = Λ′, since Λ^{∗} = Λ.

Step 6. We label arbitrarily the irreducible characters and the conjugacy classes of G_{x,1}^{+} = H_{x,1}^{+} occurring in splitting irreducible characters and conjugacy classes of G_{x,1} = H_{x,1}. Here μ_{x,1}^{+} was already chosen as the identity isometry. If G_{x} > G_{x}^{+} > G_{x}^{++}, we choose a μ_{x}-induced isometry μ_{x}^{+} such μ_{x,0}^{+} × μ_{x,1}^{+} is μ_{x}^{+}-induced. In the remaining cases, we choose μ_{x}^{+} as in the comments following (5.8). In all cases, μ_{x}^{+} is μ_{x}-induced, μ_{x,0}^{+} × μ_{x,1}^{+} is μ_{x}^{+}-induced, and μ_{x}^{+} is perfect.

Step 7. Let Λ run over all splitting mappings of H_{1} such that the p-regular and p′-regular parts Λ′ and Λ′′ of Λ are splitting mappings of H_{x,01} and Y ≀S(Ω_{x,1}). We label the irreducible characters ψ_{Λ,±} of H_{1}^{+} so that (4H) (ii) holds. The conjugacy class C_{Λ} of H_{1} contains the conjugacy class C_{Λ′}C_{Λ′′} of H_{x,01}H_{x,1}. If |Ω_{x,0}| ≥ 1, we label C_{Λ,±} so that

C_{Λ,+} = (C_{Λ′,+}C_{Λ′′,+})^{H_{1}^{+}} = (C_{Λ′,−}C_{Λ′′,−})^{H_{1}^{+}},
C_{Λ,−} = (C_{Λ′,+}C_{Λ′′,−})^{H_{1}^{+}} = (C_{Λ′,−}C_{Λ′′,+})^{H_{1}^{+}}.

If |Ω_{x,0}| = 0, then H_{x,01} = 1, and we label C_{Λ,±} so that C_{Λ,+} = C_{Λ′′,+} and C_{Λ,−} = C_{Λ′′,−}.

Step 8. Let λ run over all self-dual partitions of |Π| with p-core λ_{0} and p-quotient Λ as in step 7. Let λ′ and λ′′ be the self-dual partitions gotten from λ by removing respectively all (i, i)-hooks of p-singular length and all (i, i)-hooks of p-regular length. We call λ′ and λ′′ the p-regular and p′-regular parts of λ. In particular, λ′ has p-core λ_{0} and p-quotient Λ′, and λ′′ has p-core ∅ and p-quotient Λ′′. If |λ′| ≥ 3, we label the conjugacy classes C_{η(λ),±} of G^{+} so that

\begin{aligned}C_{η(λ),+} & = (C_{η(λ′),+}C_{Λ′′,+})^{G^{+}} = (C_{η(λ′),−}C_{Λ′′,−})^{G^{+}}, \\ C_{η(λ),−} & = (C_{η(λ′),+}C_{Λ′′,−})^{G^{+}} = (C_{η(λ′),−}C_{Λ′′,+})^{G^{+}}. \end{aligned}

If |λ′| ≤ 1, then G_{x,0} = 1, and we label the conjugacy classes C_{η(λ),±} of G^{+} so that C_{η(λ),+} = C_{Λ′′,+} and C_{η(λ),−} = C_{Λ′′,−}. We then label the irreducible characters χ_{λ,±} of G^{+} so that (4A) holds.

Step 9. The λ in step 8 determine splitting classes C_{η(λ_{0}),Λ} = C_{η(λ_{0})}C_{Λ} of H. If |λ_{0}| ≥ 3, we label the classes C_{η(λ_{0}),Λ,±} of H^{+} so that

\begin{aligned}C_{η(λ_{0}),Λ,+} & = (C_{η(λ_{0}),+}C_{Λ,+})^{H^{+}} = (C_{η(λ_{0}),−}C_{Λ,−})^{H^{+}}, \\ C_{η(λ_{0}),Λ,−} & = (C_{η(λ_{0}),+}C_{Λ,−})^{H^{+}} = (C_{η(λ_{0}),−}C_{Λ,+})^{H^{+}}. \end{aligned}

We label the irreducible constituents (χ_{λ_{0}}ψ_{Λ})_{±} of Res_{H^{+}}^{H}(χ_{λ_{0}}ψ_{Λ}) so that

Res_{H_{0}^{+}H_{1}^{+}}^{H^{+}}((χ_{λ_{0}}ψ_{Λ})_{+}) = χ_{λ_{0},+}ψ_{Λ,+} + χ_{λ_{0},−}ψ_{Λ,−},
Res_{H_{0}^{+}H_{1}^{+}}^{H^{+}}((χ_{λ_{0}}ψ_{Λ})_{−}) = χ_{λ_{0},+}ψ_{Λ,−} + χ_{λ_{0},−}ψ_{Λ,+}.

If |λ_{0}| ≤ 1, then H_{x,0} = 1, and we label the classes C_{η(λ_{0}),Λ,±} of H^{+} so that C_{η(λ_{0}),Λ,+} = C_{Λ,+} and C_{η(λ_{0}),Λ,−} = C_{Λ,−}. We label the irreducible constituents (χ_{λ_{0}}ψ_{Λ})_{±} of Res_{H^{+}}^{H}(χ_{λ_{0}}ψ_{Λ}) so that

(χ_{λ_{0}}ψ_{Λ})_{+} = 1_{S(Π_{0})} × ψ_{Λ,+}, (χ_{λ_{0}}ψ_{Λ})_{−} = 1_{S(Π_{0})} × ψ_{Λ,−}.

Step 10. Let λ run over all self-dual partitions of |Π| with p-core λ_{0} and a p-regular p-quotient Λ. We label arbitrarily the irreducible characters χ_{λ,±} and the conjugacy classes C_{Λ,±} of G^{+}. Similarly, we label arbitrarily the irreducible constituents (χ_{λ_{0}}ψ_{Λ})_{±} of Res_{H^{+}}^{H}(χ_{λ_{0}}ψ_{Λ}) and the conjugacy classes C_{η(λ_{0}),Λ,±} = (C_{η(λ_{0})}C_{Λ})_{±} of H^{+}.

Step 11. Let λ run over all self-dual partitions of |Π| with p-core λ_{0}. Let μ = (J, ϵ). We choose the μ-induced isometry μ^{+} = (J^{+}, ϵ^{+}) so that

J^{+}(χ_{λ,±}) = \begin{cases}J(χ_{λ})_{±} = (χ_{λ_{0}}ψ_{Λ})_{±} & if (−ϵ_{p})^{d_{λ}}σ_{λ} = 1, \\ J(χ_{λ})_{±} = (χ_{λ_{0}}ψ_{Λ})_{∓} & if (−ϵ_{p})^{d_{λ}}σ_{λ} = −1, \end{cases}

where d_{λ} is the number of diagonal nodes of λ′′ and

σ_{λ} = \frac{\sqrt{ϵ_{λ′}}(\sqrt{ϵ_{p}})^{d_{λ}} \sqrt{ϵ_{λ_{p^{∗}}}}}{\sqrt{ϵ_{λ}}} .

Note that σ_{λ} = ±1. Indeed, if η_{ii}(λ_{p^{∗}}) = 2u_{i} + 1, then

\begin{aligned}ϵ_{λ_{p^{∗}}} & = (−1)^{\frac{1}{2}[(∑(2u_{i}+1))−d_{λ}]} \\ & = (−1)^{∑ u_{i}}, \end{aligned}

and

\begin{aligned}ϵ_{λ′′} & = (−1)^{\frac{1}{2}[(∑(2u_{i}p+p))−d_{λ}]} \\ & = (−1)^{(∑ u_{i}p)+d_{λ}( \frac{p−1}{2} )}. \end{aligned}

Since ∑ u_{i}p ≡ ∑ u_{i}(mod 2) and ϵ_{p} = (−1)^{( \frac{p−1}{2} )}, it follows that ϵ_{λ′′} = (ϵ_{p})^{d_{λ}}ϵ_{λ_{p^{∗}}} and ϵ_{λ} = ϵ_{λ′}(ϵ_{p})^{d_{λ}}ϵ_{λ_{p^{∗}}}. Thus σ_{λ} = ±1.

Proposition (5C). Let λ be a self-dual partition of n with p-core λ_{0} and p-quotient Λ, and let x ∈ {x_{1}, x_{2}, …, x_{w}}. Then

R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}))(y) = 0

unless |λ_{p^{∗}}| = |Ω_{x,1}| and y ∈ (H_{x,00}^{+}H_{x,01}^{+}H_{x,1}^{+})_{p′}. Moreover, if |λ_{p^{∗}}| = |Ω_{x,1}| and y = y_{00}y_{01}y_{1} with y_{00} ∈ (H_{x,00}^{+})_{p′}, y_{01} ∈ (H_{x,01}^{+})_{p′}, y_{1} ∈ (H_{x,1}^{+})_{p′}, then

\begin{matrix}R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}))(y) = & σ_{λ}ϵ_{x,0}(χ_{λ′})(χ_{λ_{0},+} − χ_{λ_{0},−})(y_{00}) \\ × (ψ_{Λ′,+} − ψ_{Λ′,−})(y_{01})(ψ_{Λ′′,+} − ψ_{Λ′′,−})(xy_{1}). \end{matrix}

Here λ′ is the p-regular part of λ; ψ_{Λ′} and ψ_{Λ′′} are the irreducible characters of Y ≀ S(Ω_{x,0}) and Y ≀ S(Ω_{x,1}) parametrized by the p-regular and p-singular parts Λ′ and Λ′′ of Λ; and the factor (χ_{λ_{0},+} − χ_{λ_{0},−}) or (ψ_{Λ′,+} − ψ_{Λ′,−}) is interpreted as 1 when |λ_{0}| ≤ 1 or |Ω_{x,0}| = 0.

Proof. Let ρ_{Λ′′,±} = Res_{(P ≀S(Ω_{x,1}))^{+}}^{(Y ≀S(Ω_{x,1}))^{+}}(ψ_{Λ′′,±}) if |λ_{p^{∗}}| = |Ω_{x,1}|, and let ρ_{Λ′′,±} be the zero function if |λ_{p^{∗}}| ≠ |Ω_{x,1}|. We show first that

d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}) = σ_{λ} Ind_{G_{x}^{++}}^{G_{x}^{+}}(d_{G_{x,0}^{+}}^{1}(χ_{λ′,+}) × d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−})), (5.9)

where d_{G_{x,0}^{+}}^{1}(χ_{λ′,+}) is interpreted as 1 when |λ′| ≤ 1. Indeed, let z_{0}z_{1} ∈ G_{x,p′}^{+}, where

z_{0} ∈ G_{x,0} and z_{1} ∈ G_{x,1}. By (4A), (χ_{λ,+} − χ_{λ,−})(xz_{0}z_{1}) ≠ 0 only if xz_{0}z_{1} ∈ C_{η(λ)}, or equivalently, only if |λ_{p^{∗}}| = |Ω_{x,1}|, z_{0} is in the class C_{η(λ′)} of G_{x,0}, and xz_{1} is in the class C_{Λ′′} of G_{x,1}. Thus (5.9) holds if |λ_{p^{∗}}| ≠ |Ω_{x,1}|, since both sides of (5.9) are zero. Suppose |λ_{p^{∗}}| = |Ω_{x,1}|. Then

d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−})(z_{0}z_{1}) = \begin{cases}±\sqrt{ϵ_{λ}η(λ)} & if xz_{0}z_{1} ∈ C_{η(λ),±}, \\ 0 & if xz_{0}z_{1} ∉ C_{η(λ),±}. \end{cases}

On the other hand, Ind_{G_{x}^{++}}^{G_{x}^{+}}(d_{G_{x,0}^{+}}^{1}(χ_{λ′,+}) × d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−})) has support on

G_{x}^{++}, where it is equal to

d_{G_{x,0}^{+}}^{1}(χ_{λ′,+} − χ_{λ′,−}) × d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−}). (5.10)

Moreover, (4A) and (4F) imply that (χ_{λ′,+} − χ_{λ′,−})(z_{0}) and (ψ_{Λ′′,+} − ψ_{Λ′′,−})(xz_{1}) are zero except for

\begin{aligned}(χ_{λ′,+} − χ_{λ′,−})(z_{0}) & = ±\sqrt{ϵ_{λ′}η(λ′)} \\ (ψ_{Λ′′,+} − ψ_{Λ′′,−})(xz_{1}) & = ±(\sqrt{ϵ_{p}p})^{d_{λ}} \sqrt{ϵ_{λ_{p^{∗}}} η(λ_{p^{∗}})} \end{aligned} \begin{matrix}if z_{0} ∈ C_{η(λ′),±}, \\ if xz_{1} ∈ C_{Λ′′,±}. \end{matrix}

Here (χ_{λ′,+} − χ_{λ′,−})(z_{0}) and ±\sqrt{ϵ_{λ′}η(λ′)} are interpreted as 1 when |λ′| ≤ 1. Since

ϵ_{λ} = ϵ_{λ′}(ϵ_{p})^{d_{λ}}ϵ_{λ_{p^{∗}}} , \sqrt{η(λ)} = \sqrt{η(λ′)p^{d_{λ}}η(λ_{p^{∗}})},

(5.9) follows from the labeling in step 8.

Suppose |λ′| ≥ 3. Then

R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}))
= R_{μ_{x}^{+},p′}(σ_{λ} Ind_{G_{x}^{++}}^{G_{x}^{+}}(d_{G_{x,0}^{+}}^{1}(χ_{λ′,+}) × d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−})))
= σ_{λ} Ind_{H_{x}^{++}}^{H_{x}^{+}}(R_{μ_{x,0}^{+}×μ_{x,1}^{+},p′}(d_{G_{x,0}^{+}}^{1}(χ_{λ′,+}) × d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−})))
(since μ_{x}^{+} induces μ_{x,0}^{+} × μ_{x,1}^{+})
= σ_{λ} Ind_{H_{x}^{++}}^{H_{x}^{+}}(R_{μ_{x,0,p′}^{+}} (d_{G_{x,0}^{+}}^{1}(χ_{λ′,+})) × R_{μ_{x,1,p′}^{+}} (d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−})))
= σ_{λ} Ind_{H_{x}^{++}}^{H_{x}^{+}}(d_{H_{x,0}^{+}}^{1}(R_{μ_{x,0}^{+}}(χ_{λ′,+})) × d_{H_{x,1}^{+}}^{x}(R_{μ_{x,1}^{+}}(ρ_{Λ′′,+} − ρ_{Λ′′,−})))
(since μ_{x,0}^{+} is perfect by (5B) and μ_{x,1}^{+} is the identity isometry)
= σ_{λ}d_{H_{x}^{+}}^{x}(Ind_{H_{x}^{++}}^{H_{x}^{+}}(R_{μ_{x,0}^{+}×μ_{x,1}^{+}}(χ_{λ′,+} × (ρ_{Λ′′,+} − ρ_{Λ′′,−})))).

(5.11)

Thus R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−})) has support on (H_{x}^{++})_{p′}, where it is equal to

σ_{λ}d_{H_{x}^{+}}^{x}(R_{μ_{x,0}^{+}}(χ_{λ′,+} − χ_{λ′,−}) × R_{μ_{x,1}^{+}}(ρ_{Λ′′,+} − ρ_{Λ′′,−})).

In turn, R_{μ_{x,0}^{+}}(χ_{λ′,+} − χ_{λ′,−}) has support on H_{x,00}^{+}H_{x,01}^{+}, where it is equal to

ϵ_{x,0}(χ_{λ′})(χ_{λ_{0},+} − χ_{λ_{0},−}) × (ψ_{Λ′,+} − ψ_{Λ′,−}).

Thus R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−})) has support on (H_{x,00}^{+}H_{x,01}^{+}H_{x,1}^{+})_{p′}, and the first half of (5C) holds. The argument shows that

\begin{aligned}R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}))(y) & = σ_{λ}ϵ_{x,0}(χ_{λ′})(χ_{λ_{0},+} − χ_{λ_{0},−})(y_{00}) \\ & × (ψ_{Λ′,+} − ψ_{Λ′,−})(y_{01})(ψ_{Λ′′,+} − ψ_{Λ′′,−})(xy_{1}) \end{aligned}

if |λ_{p^{∗}}| = |Ω_{x,1}| and y = y_{00}y_{01}y_{1} with y_{00} ∈ (H_{x,00}^{+})_{p′}, y_{01} ∈ (H_{x,01}^{+})_{p′}, and y_{1} ∈ (H_{x,1}^{+})_{p′}. Thus (5C) holds when |λ′| ≥ 3.

Suppose |λ′| ≤ 1. Then G_{x}^{+} = G_{x}^{++}, and (5.11) reduces to

R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−})) = R_{μ_{x}^{+},p′}(σ_{λ}d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−}))
= σ_{λ}R_{μ_{x,1}^{+},p′}(d_{G_{x,1}^{+}}^{x}(ρ_{Λ′′,+} − ρ_{Λ′′,−}))
= σ_{λ}d_{H_{x,1}^{+}}^{x}(R_{μ_{x,1}^{+}}(ρ_{Λ′′,+} − ρ_{Λ′′,−})).

Then (5C) holds trivially in this case, since (H_{x,00}^{+}H_{x,01}^{+}H_{x,1}^{+})_{p′} = (H_{x,1}^{+})_{p′},

Proposition (5D). Let λ be a self-dual partition of n with p-core λ_{0} and p-quotient Λ, and let x ∈ {x_{1}, x_{2}, …, x_{w}}. Then

d_{H^{+}}^{x}(R_{μ^{+}}((χ_{λ,+} − χ_{λ,−}))(y) = 0

unless |λ_{p^{∗}}| = |Ω_{x,1}| and y ∈ (H_{x,00}^{+}H_{x,01}^{+}H_{x,1}^{+})_{p′}. Moreover, if |λ_{p^{∗}}| = |Ω_{x,1}| and y = y_{00}y_{01}y_{1} with y_{00} ∈ (H_{x,00}^{+})_{p′}, y_{01} ∈ (H_{x,01}^{+})_{p′}, y_{1} ∈ (H_{x,1}^{+})_{p′}, then

\begin{aligned}d_{H^{+}}^{x}(R_{μ^{+}}(χ_{λ,+} − χ_{λ,−}))(y) & = σ_{λ}(−ϵ_{p})^{d_{λ}}ϵ(χ_{λ})(χ_{λ_{0},+} − χ_{λ_{0},−})(y_{00}) \\ & × (ψ_{Λ′,+} − ψ_{Λ′,−})(y_{01})(ψ_{Λ′′,+} − ψ_{Λ′′,−})(xy_{1}). \end{aligned}

Here ψ_{Λ′} and ψ_{Λ′′} are the irreducible characters of Y ≀ S(Ω_{x,0}) and Y ≀ S(Ω_{x,0}) parametrized by the p-regular and p-singular parts Λ′ and Λ′′ of Λ, and the factor (χ_{λ_{0},+} − χ_{λ_{0},−}) or (ψ_{Λ′,+} − ψ_{Λ′,−}) is interpreted as 1 when |λ_{0}| ≤ 1 or |Ω_{x,0}| = 0.

Proof. For y ∈ (H_{x}^{+})_{p′} we have

\begin{aligned}d_{H^{+}}^{x}(R_{μ^{+}}(χ_{λ,+} − χ_{λ,−}))(y) & = R_{μ^{+}}(χ_{λ,+} − χ_{λ,−})(xy) \\ & = (−ϵ_{p})^{d_{λ}}σ_{λ}ϵ(χ_{λ})(J^{+}(χ_{λ,+}) − J^{+}(χ_{λ,−}))(xy). \end{aligned}

Let y = y_{00}y_{01}y_{1}, where y_{00} ∈ (H_{x,00})_{p′}, y_{01} ∈ (H_{x,01})_{p′}, y_{1} ∈ (H_{x,1})_{p′}. Thus y_{00} ∈ (H_{0})_{p′}, y_{01}y_{1} ∈ (H_{1})_{p′}, and

(J^{+}(χ_{λ,+}) − J^{+}(χ_{λ,−}))(xy)

= \begin{cases}(χ_{λ_{0},+} − χ_{λ_{0},−})(y_{00})(ψ_{Λ,+} − ψ_{Λ,−})(xy_{01}y_{1}) & if xy ∈ H_{0}^{+}H_{1}^{+}, \\ 0 & if xy ∉ H_{0}^{+}H_{1}^{+} \end{cases}

by the labeling of J^{+}(χ_{λ,±}) in steps 8–10. Suppose d_{H^{+}}^{x}(R_{μ^{+}}(χ_{λ,+} − χ_{λ,−}))(y) ≠ 0. Then y_{00} ∈ (H_{x,00}^{+})_{p′}, y_{01}y_{1} ∈ (H_{1}^{+})_{p′}, and (ψ_{Λ,+} − ψ_{Λ,−})(xy_{01}y_{1}) ≠ 0. Let H′_{1} and H′′_{1} be the subgroups of H_{1} associated to Λ′ and Λ′′. By (4H) (i) there exist elements u′ in a p-regular splitting class of H′_{1}, u′′ in a p-singular splitting class of H′′_{1}, and h ∈ H_{1}^{+} such that (xy_{01}y_{1})^{h} = u′u′′. A comparison of the p-regular and p-singular cycle types of these elements as elements of S(Γ × Ω) shows that (y_{01})^{h} = u′, (xy_{1})^{h} = u′′. Thus |λ_{p^{∗}}| = |Ω_{x,1}|, y_{01} ∈ H_{x,01}^{+}, y_{1} ∈ H_{x,1}^{+}, and the first part of (5D) holds. Now suppose |λ_{p^{∗}}| = |Ω_{x,1}|, y_{00} ∈ (H_{x,00}^{+})_{p′}, y_{01} ∈ (H_{x,01}^{+})_{p′}, and y_{1} ∈ (H_{x,1}^{+})_{p′}. We can then choose Y ≀S(Ω_{x,0}) and Y ≀S(Ω_{x,1}) as the subgroups H′_{1} and H′′_{1} of H_{1} associated to Λ′ and Λ′′. The second part of (5D) now follows from (4H) (ii), the labelings in steps 8–9, and the definition of J^{+}(χ_{λ,±}).

Proposition (5E). The diagram

\begin{matrix}R_{K}(G^{+}, e^{+}) & ⟶^{R_{μ^{+}}} & R_{K}(H^{+}, f^{+}) \\ ↓^{d_{G^{+}}^{x}} & & ↓^{d_{H^{+}}^{x}} \\ CF_{p′}(G_{x}^{+}, e_{x}^{+}) & ⟶^{R_{μ^{+} x,p′}} & CF_{p′}(H_{x}^{+}, f_{x}^{+}) \end{matrix} (5.12)

is commutative for all x in {x_{1}, x_{2}, …, x_{w}}.

Proof. Suppose χ^{+} = Res_{G^{+}}^{G}(χ) for χ ∈ (G, e)^{∨}. Then

d_{H^{+}}^{x}(R_{μ^{+}}(χ^{+})) = R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ^{+})),

that is, (5.12) is commutative for such χ^{+}. Indeed, d_{H}^{x} ∘ R_{μ} = R_{μ_{x},p′} ∘ d_{G}^{x} by (3B), and

R_{μ^{+}} ∘ Res_{G^{+}}^{G} = Res_{H^{+}}^{H} ∘R_{μ}, R_{μ_{x}^{+}} ∘ Res_{G_{x}^{+}}^{G_{x}} = Res_{H_{x}^{+}}^{H_{x}} ∘R_{μ_{x}},

since μ^{+} is μ-induced and μ_{x}^{+} is μ_{x}-induced. Thus

d_{H^{+}}^{x} ∘ R_{μ^{+}} ∘ Res_{G^{+}}^{G} = d_{H^{+}}^{x} ∘ Res_{H^{+}}^{H} ∘R_{μ}
= Res_{H_{x,p′}^{+}}^{H_{x,p′}} ∘ d_{H}^{x} ∘ R_{μ}
= Res_{H_{x,p′}^{+}}^{H_{x,p′}} ∘R_{μ_{x},p′} ∘ d_{G}^{x}
= R_{μ_{x}^{+},p′} ∘ Res_{G_{x,p′}^{+}}^{G_{x,p′}} ∘ d_{G}^{x}
= R_{μ_{x}^{+},p′} ∘ d_{G^{+}}^{x} ∘ Res_{G^{+}}^{G},

since μ_{x} and μ_{x}^{+} are perfect isometries.

Suppose χ^{+} = χ_{λ,+} − χ_{λ,−}, where χ_{λ} ∈ (G, e)^{∨}, λ is self-dual, and G_{x}^{+} > G_{x}^{++}. The commutativity of (5.12) for χ^{+} follows from the expressions for

R_{μ_{x}^{+},p′}(d_{G^{+}}^{x}(χ_{λ,+} − χ_{λ,−}))(y), d_{H^{+}}^{x}(R_{μ^{+}}(χ_{λ,+} − χ_{λ,−}))(y)

given in (5C) and (5D). Indeed, we may suppose that Λ′ is a p-regular splitting mapping of Y ≀ S(Ω_{x,0}) and Λ′′ is a p-singular splitting mapping of Y ≀ S(Ω_{x,1}). It remains to show that

σ_{λ}ϵ_{x,0}(χ_{λ′}) = (−ϵ_{p})^{d_{λ}}σ_{λ}ϵ(χ_{λ}). (5.13)

But ϵ(χ_{λ}) = v(λ)(−1)^{|λ_{p^{∗}} |} = (−1)^{d_{λ}} since λ is self-dual, ϵ_{x,0}(χ_{λ′}) = v(λ′) since λ′ is p-regular, and v(λ)v(λ′) = (ϵ_{p})^{d_{λ}}. Thus (5.13) holds.

Suppose χ^{+} = χ_{λ,+} − χ_{λ,−}, where χ_{λ} ∈ (G, e)^{∨}, λ is self-dual, and G_{x}^{+} = G_{x}^{++}. Then |Π_{0} ∪ (Γ × Ω_{x,0})| ≤ 1 or |Ω_{x,1}| = 1. If |Π_{0} ∪ (Γ × Ω_{x,0})| ≤ 1, then G_{x}^{+} = H_{x}^{+}, μ_{x}^{+} is the identity isometry, and (5.12) holds trivially. If |Ω_{x,1}| = 1, then

G_{x}^{+} = A(Π_{0} ∪ (Γ × Ω_{x,0}) × P, H_{x}^{+} = (S(Π_{0}) × (Y ≀ S(Ω_{x,0})))^{+} × P

The previous arguments apply if (ψ_{Λ′′,+} − ψ_{Λ′′,−})(y_{1}) and (ρ_{Λ′′,+} − ρ_{Λ′′,−})(y_{1}) are interpreted as 1 when |λ_{0}| ≤ 1 or |Ω_{x,0}| = 0. This completes the proof of (5E).

Theorem (5F). e^{+} and f^{+} are isotypic blocks of OG^{+} and OH^{+} with local system given by the translates of the μ_{x}^{+} for x in {x_{1}, x_{2}, …, x_{w}}.

Proof. The inclusion of D in G^{+} and H^{+} induces an equivalence of Br_{e^{+},D}(G^{+}) and Br_{f^{+},D}(H^{+}) by [1, (4.1)]. We claim the diagram (5.12) with x replaced by any generator of ⟨x⟩ is a commutative diagram. Indeed, the horizontal arrows in (5.12) depend only on ⟨x⟩ and not on x. The proofs remain valid when x is replaced by any generator of ⟨x⟩. Since x ∈ {x_{1}, x_{2}, …, x_{w}} and the w subgroups ⟨x_{j}⟩ form a set of representatives for the N_{G^{+}}(D, e_{D}^{+})-orbits of non-trivial cyclic subgroups of D, (5F) holds by transport of structure, as remarked in §1.

6. Compatibility

The Broué conjecture for the principal block holds for all finite groups if it holds for all finite simple non-abelian groups X and if the isotypies for X satisfy compatibility conditions with regard to extensions of X by p′-groups of automorphisms (see [10, (5E)]). We show that the isotypies constructed in §5 for alternating groups satisfy these compatibility conditions. We suppose p > 2 throughout this section.

Let G and G^{+} be the symmetric and the alternating groups of degree n, let e and e^{+} be the 1-blocks of OG and OG^{+}, and let e and e^{+} have the common non-trivial abelian defect group D. Moreover, let H = N_{G}(D) and H^{+} = N_{G^{+}}(D), and let f and f^{+} be the Brauer correspondents of e and e^{+}. In particular, e^{+} and f^{+} are stable under G and H. On the other hand, e = e^{∗} and f = f^{∗} hold if and only if |Π_{0}| ≤ 1. The maximal subpairs (D, e_{D}), (D, f_{D}), (D, e_{D}^{+}), (D, f_{D}^{+}) for e, f, e^{+}, f^{+} are unique since e, f, e^{+}, f^{+} are the 1-blocks. Let {μ_{P} } and {μ_{P}^{+}} be the local systems for the isotypies between e and f and between e^{+} and f^{+} constructed in §3 and §5.

Let G̃ = Aut(G^{+}), so that G̃ ≥ G and |G̃/G| = 1 or 2 according as n ≠ 6 or n = 6. Let H̃ = NG̃(D). The inclusion of H̃ in G̃ induces an isomorphism α: G̃/G^{+} → H̃/H^{+}, since G̃ = H̃G^{+}. The compatibility conditions from [10, §3] are of three types. The first type (F̃1), (F̃2′), (F̃3) relate to fusion:

(F̃1) G̃/G^{+} and H̃/H^{+} are p′-groups stabilizing e^{+} and f^{+}.

(F̃2′) Given x in NG̃(D), there exists y in NH̃(D) with α(xG^{+}) = yH^{+} such that x and y induce the same mapping on D. Similarly, given y in NH̃(D), there exists x in NG̃(D) with α(xG^{+}) = yH^{+} such that x and y induce the same mapping on D.

(F̃3) α(G̃_{P} G^{+}/G^{+}) = H̃_{P} H^{+}/H^{+} for all cyclic subgroups P of D.

(F̃1) holds since e^{+} and f^{+} are the 1-blocks. (F̃2′) and (F̃3) follow easily from G̃ = H̃G^{+} and the structure of G̃_{P} and H̃_{P}.

The second type (Ĩ1) imposes extendibility conditions on the isometries μ_{P}^{+} = (J_{P}^{+}, ϵ_{P}^{+}) for all cyclic P ≤ D. Namely, let α_{P} : G̃_{P} /G_{P}^{+} ⟶ H̃_{P} /H_{P}^{+} be the isomorphism defined by α_{P} (xG_{P}^{+}) = α(xG^{+}) ∩ H̃_{P} for x ∈ G̃_{P}, and set

\begin{aligned}Δ(α_{P}) & = \{(x, y) : x ∈ G̃_{P} , y ∈ H̃_{P} , α_{P} (xG_{P}^{+}) = yH_{P}^{+}\}, \\ Δ(μ_{P}^{+}) & = \{χ_{P}^{+} × ψ̄_{P}^{+} : χ_{P}^{+} ∈ (G_{P}^{+}, e_{P}^{+})^{∨}, ψ_{P}^{+} ∈ (H_{P}^{+}, f_{P}^{+})^{∨}, J_{P}^{+}(χ_{P}^{+}) = ψ_{P}^{+}\}. \end{aligned}

In particular, Δ(α_{P}) = Δ(α) ∩ (G̃_{P} × H̃_{P}). Then the following hold:

(Ĩ1.1) μ_{P}^{+} is stable under Δ(α_{P}). (Ĩ1.2) Each χ_{P}^{+} × ψ̄_{P}^{+} in Δ(μ_{P}^{+}) has a Δ(α_{P})-equivariant extension to a character

E_{P} (χ_{P}^{+} × ψ̄_{P}^{+}) of its stabilizer Δ(α_{P})_{χ_{P}^{+}×ψ̄_{P}^{+}} in Δ(α_{P}).

(Ĩ1.3) The extension of μ_{P}^{+} to Δ(α_{P}) given by

E_{P} (μ_{P}^{+}) = ∑_{χ_{P}^{+}×ψ̄_{P}^{+}∈Δ(μ_{P}^{+})} ϵ_{μ_{P}^{+}}(χ_{P}^{+})E_{P} (χ_{P}^{+} × ψ̄_{P}^{+}),

where E_{P} (χ_{P}^{+} × ψ̄_{P}^{+})(x, y) = 0 for (x, y) ∉ Δ(α_{P})_{χ_{P}^{+}×ψ̄_{P}^{+}}, is perfect in the sense that

(Ĩ1.3a) E_{P} (μ_{P}^{+}) separates p-singular and p-regular elements of Δ(α_{P}), and (Ĩ1.3b) E_{P} (μ_{P}^{+})(x, y) is divisible in O by |C_{G_{P}^{+}}(x)| and |C_{H_{P}^{+}}(y)| for all (x, y) in

Δ(α_{P}).

We verify (Ĩ1) when n ≠ 6. If e = e^{∗}, then μ_{P}^{+} is stable under Δ(α_{P}) by the construction in §5. If e ≠ e^{∗}, then each term χ_{P}^{+} × ψ̄_{P}^{+} in μ_{P}^{+} is even stable under Δ(α_{P}). Let χ_{P}^{+} and ψ_{P}^{+} be characters corresponding under J_{P}^{+}. Then Δ(α_{P})_{χ_{P}^{+}×ψ̄_{P}^{+}} is G_{P}^{+} × H_{P}^{+} or Δ(α_{P}). In the first case we set E_{P} (χ_{P}^{+} × ψ̄_{P}^{+}) = χ_{P}^{+} × ψ̄_{P}^{+}. In the second case χ_{P}^{+} and ψ_{P}^{+} extend to irreducible characters χ_{P} and ψ_{P} in (G_{P} , e_{P})^{∨} and (H_{P} , f_{P})^{∨} corresponding under μ_{P}, and we set

E_{P} (χ_{P}^{+} × ψ̄_{P}^{+}) = Res_{Δ(α_{P})}^{G_{P} ×H_{P}} (χ_{P} × ψ̄_{P}).

By (1D) and (2.3)

E_{P} (μ_{P}^{+})(x, y) = \begin{cases}μ_{P} (x, y) & if e ≠ e^{∗}, \\ \frac{1}{2}μ_{P} (x, y) + \frac{1}{2}Δ(μ_{P}^{+})(x, y) & if e = e^{∗}. \end{cases} (6.1)

where Δ(μ_{P}^{+}) is the generalized character of G_{P}^{+} × H_{P}^{+} of (2B) extended by zero to Δ(α_{P}). Since μ_{P} and Δ(μ_{P}^{+}) are perfect, it follows that all conditions of (Ĩ1) hold.

The third type (G̃1) and (G̃2) are glueing conditions on the extensions in (Ĩ1).

(G̃1) E_{⟨1⟩}(μ_{⟨1⟩}^{+})(xx′, yy′) = 0 for all (xx′, yy′) ∈ Δ(α) such that x ∈ G_{p}^{+}, y ∈

H_{p}^{+}, x′ ∈ G̃_{x,p′}, y′ ∈ H̃_{y,p′}, and x ≁ y in G^{+}. (G̃2) E_{⟨1⟩}(μ_{⟨1⟩}^{+})(zx′, zy′) = E_{⟨z⟩}(μ_{⟨z⟩}^{+})(x′, y′) for all (zx′, zy′) ∈ Δ(α) such that

z ∈ D, x′ ∈ CX̃(z)_{p′}, y′ ∈ CỸ (z)_{p′}.

We verify (G̃1) and (G̃2) when n ≠ 6. Suppose e ≠ e^{∗}. Then E_{⟨1⟩}(μ_{⟨1⟩}^{+}) = Res_{Δ(α)}^{G×H}(μ) and E_{P} (μ_{P}^{+}) = Res_{Δ(α_{P})}^{G_{P} ×H_{P}} (μ_{P}), so (G̃2) holds. Let (xx′, yy′) ∈ Δ(α) be as in (G̃1). Since G_{x} ≰ G^{+} when |Π_{0}| ≥ 3, it follows that x ≁ y in G, and (G̃1) holds. Suppose e = e^{∗}. Let (xx′, yy′) ∈ Δ(α) be as in (G̃1). If x ≁ y in G, then μ_{⟨1⟩}(xx′, yy′) = 0. Moreover, Δ(μ_{⟨1⟩}^{+})(xx′, yy′) = 0, since each summand

(χ_{+} − χ_{−})(xx′) (\overline{ψ}_{+} − \overline{ψ}_{−})(yy′)

in Δ(μ_{⟨1⟩}^{+})(xx′, yy′) is zero by (4A), (4F), (4G), or (4H). Thus E_{⟨1⟩}(μ_{⟨1⟩}^{+})(xx′, yy′) = 0 by (6.1), and (G̃1) holds. If x ∼ y in G, then n = p + 1 or p; otherwise G_{x} > G_{x}^{+} and x ∼ y in G^{+}. In this case (G, e)^{∨} = {χ_{1}, χ_{2}, …, χ_{p}}, where χ_{i} is parametrized by the partition λ_{i} of n with a p-hook of leg length i − 1. So Res_{G^{+}}^{G}(χ_{i}) = χ_{i}^{+} is irreducible for 1 ≤ i ≤ p^{∗} − 1, Res_{G^{+}}^{G}(χ_{p^{∗}}) = χ_{p^{∗},+} + χ_{p^{∗},−} is reducible, and

(G^{+}, e^{+})^{∨} = {χ_{1}^{+}, χ_{2}^{+}, …, χ_{p^{∗}−1}^{+}, χ_{p^{∗},+}, χ_{p^{∗},−}}.

(H, f)^{∨} may be identified with Y^{∨} = {ξ_{1}, ξ_{2}, …, ξ_{p}}. So Res_{H^{+}}^{H}(ξ_{i}) = ξ_{i}^{+} is irreducible for 1 ≤ i ≤ p^{∗} − 1, Res_{H^{+}}^{H}(ξ_{p^{∗}}) = ξ_{p^{∗},+} + ξ_{p^{∗},−} is reducible, and

(H^{+}, f^{+})^{∨} = {ξ_{1}^{+}, ξ_{2}^{+}, …, ξ_{p^{∗}−1}^{+}, ξ_{p^{∗},+}, ξ_{p^{∗},−}}.

The isometries μ and μ^{+} are

\begin{matrix}χ_{i} & ↦ (−1)^{i+1}ξ_{i} for i ≠ p^{∗}, \\ χ_{i}^{+} & ↦ (−1)^{i+1}ξ_{i}^{+} for i ≠ p^{∗} \end{matrix} \begin{matrix}χ_{p^{∗}} & ↦ (−1)^{p^{∗}}ξ_{p^{∗}} \\ χ_{p^{∗},±} & ↦ −ϵ_{p}ξ_{p^{∗},±}, \end{matrix}

A direct calculation gives (G̃1).

Suppose (zx′, zy′) ∈ Δ(α), where z ∈ D, x′ ∈ G_{z,p′}, y′ ∈ H_{z,p′}. If x′ ∉ G^{+} and y′ ∉ H^{+}, then

E_{⟨1⟩}(μ_{⟨1⟩}^{+})(zx′, zy′) = E_{⟨z⟩}(μ_{⟨z⟩}^{+})(x′, y′) (6.2) by (6.1), since μ_{⟨1⟩}(zx′, zy′) = μ_{⟨z⟩}(x′, y′). If x′ ∈ G^{+} and y′ ∈ H^{+}, then (6.2) is just μ_{⟨1⟩}^{+}(zx′, zy′) = μ_{⟨z⟩}^{+}(x′, y′). Thus (G̃2) holds.

We now suppose n = 6. Then G^{+} ≃ A(6) ≃ PSL(2, 9), G̃ = Aut G^{+} ≃ PΓL(2, 9), and p = 3 or 5. Let G_{1}, G_{2}, G_{3} be the maximal subgroups of G̃ containing G^{+}, labeled so that

G_{1} ≃ S(6), G_{2} ≃ PGL(2, 9), G_{3} ≃ M_{10}.

We follow Atlas notation [7, page 5] for the character table of G̃:

Table (6.3). G̃^{\begin{aligned}360 & 8 & 9 & 9 & 4 & 5 & 5 & & 24 & 24 & 4 & 3 & 3 & & 10 & 4 & 4 & 5 & 5 & & 2 & 4 & 4 \\ 1A & 2A & 3A & 3B & 4A & 5A & B^{∗} & & 2B & 2C & 4B & 6A & 6B & & 2D & 8A & B^{∗} & 10A & B^{∗} & & 4C & 8C & D^{∗∗} \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & : & 1 & 1 & 1 & 1 & 1 & : & 1 & 1 & 1 & 1 & 1 & : & 1 & 1 & 1 \\ 5 & 1 & 2 & −1 & −1 & 0 & 0 & : & 3 & −1 & 1 & 0 & −1 & • & 0 & 0 & 0 & 0 & 0 & • & 0 & 0 & 0 \\ 5 & 1 & −1 & 2 & −1 & 0 & 0 & : & −1 & 3 & 1 & −1 & 0 & • & 0 & 0 & 0 & 0 & 0 & • & 0 & 0 & 0 \\ 8 & 0 & −1 & −1 & 0 & −b5 & ∗ & • & 0 & 0 & 0 & 0 & 0 & : & 2 & 0 & 0 & b5 & ∗ & • & 0 & 0 & 0 \\ 8 & 0 & −1 & −1 & 0 & ∗ & −b5 & • & 0 & 0 & 0 & 0 & 0 & : & 2 & 0 & 0 & ∗ & b5 & • & 0 & 0 & 0 \\ 9 & 1 & 0 & 0 & 1 & −1 & −1 & : & 3 & 3 & −1 & 0 & 0 & : & −1 & 1 & 1 & −1 & −1 & : & 1 & −1 & −1 \\ 10 & −2 & 1 & 1 & 0 & 0 & 0 & : & 2 & −2 & 0 & −1 & 1 & : & 0 & r2 & −r2 & 0 & 0 & : & 0 & i2 & −i2 \end{aligned}}

The columns correspond to conjugacy classes C of G̃, the first two entries in each column being |C_{G^{+}}(g)| for g in C and the Atlas label of the class. Also b5 = \frac{1}{2}(−1 + \sqrt{5}), r2 = \sqrt{2}, i2 = i\sqrt{2}.

The natural inclusion of H̃ = NG̃(D) in G̃ induces an isomorphism α: G̃/G^{+} → H̃/H^{+} since G̃ = H̃G^{+}. Also H^{+} ≃ 3^{2} : 4, H̃ ≃ 3^{2} : [2^{4}] if p = 3, and H^{+} ≃ D_{10}, H̃ ≃ 10: 4 if p = 5. Let H_{i} = N_{G_{i}}(D), so that

\begin{aligned}H_{1} & ≃ 3^{2} : D_{8}, \\ H_{1} & ≃ 5: 4, \end{aligned} \begin{aligned}H_{2} & ≃ 3^{2} : 8, \\ H_{2} & ≃ D_{20}, \end{aligned} \begin{aligned}H_{3} & ≃ 3^{2} : Q_{8} \\ H_{3} & ≃ 5: 4 \end{aligned} \begin{matrix}if p = 3, \\ if p = 5. \end{matrix}

Let α_{i} be the restriction of α to G_{i}/G^{+} for i = 1, 2, 3, and set

Δ(α_{i}) = {(x, y) : x ∈ G_{i}, y ∈ H_{i}, α_{i}(xG^{+}) = yH^{+}}.

We verify the compatibility conditions (Ĩ1), (G̃1), and (G̃2) for the two cases p = 3 and p = 5.

Suppose p = 3. Then D ≃ 3^{2}, H^{+} ≃ 3^{2} : 4, and H̃ ≃ 3^{2} : [2^{4}]. The characters 1, 5_{1}, 5_{2}, 8_{1}, 8_{2}, 10 in (G^{+}, e^{+})^{∨} have stabilizers G̃, G_{1}, G_{1}, G_{2}, G_{2}, G̃ in G̃. Only the characters 1, 5_{1}, 5_{2}, 8_{1}, 8_{2} extend to their stabilizers. Let E(1), E(5_{1}), E(5_{2}), E(8_{1}), E(8_{2}) be the extensions given in (6.3). The following is the character table of H̃:

Table (6.4). H̃, p = 3^{\begin{aligned}36 & 4 & 4 & 4 & 9 & 9 & & 6 & 6 & 3 & 3 & & 4 & 4 & 4 & 4 & & 2 & 2 \\ 1A & 2A & 4A & 4A & 3A & 3B & & 2B & 2C & 6A & 6B & & 8A & 8A & 8B & 8B & & 4C & 4C \\ 1 & 1 & 1 & 1 & 1 & 1 & : & 1 & 1 & 1 & 1 & : & 1 & 1 & 1 & 1 & : & 1 & 1 \\ 1 & 1 & −1 & −1 & 1 & 1 & : & 1 & −1 & 1 & −1 & : & i & −i & i & −i & : & 1 & -1 \\ 1 & −1 & i & −i & 1 & 1 & • & 0 & 0 & 0 & 0 & : & z_{8} & z_{8}^{7} & z_{8}^{5} & z_{8}^{3} & • & 0 & 0 \\ 1 & −1 & −i & i & 1 & 1 & • & 0 & 0 & 0 & 0 & : & z_{8}^{7} & z_{8} & z_{8}^{3} & z_{8}^{5} & • & 0 & 0 \\ 4 & 0 & 0 & 0 & 1 & −2 & : & 0 & 2 & −1 & 0 & • & 0 & 0 & 0 & 0 & • & 0 & 0 \\ 4 & 0 & 0 & 0 & −2 & 1 & : & 2 & 0 & 0 & −1 & • & 0 & 0 & 0 & 0 & • & 0 & 0 \end{aligned}}

The columns correspond to conjugacy classes C of H̃, the first two entries being |C_{H^{+}}(h)| for h in C and the Atlas name of the G̃-class containing C. The characters 1_{1}, 1_{2}, 1_{3}, 1_{4}, 4_{1}, 4_{2} in (H^{+}, f^{+})^{∨} have stabilizers H̃, H̃, H_{2}, H_{2}, H_{1}, H_{1} in H̃. Only the characters 1_{1}, 1_{3}, 1_{4}, 4_{1}, 4_{2} extend to their stabilizers. Let E(1_{1}), E(1_{3}), E(1_{4}), E(4_{1}), E(4_{2}) be the extensions given in (6.4). The isometry μ^{+} = (J^{+}, ϵ^{+})

1 ↦ 1_{1}, 10 ↦ 1_{2}, 5_{1} ↦ −4_{2}, 5_{2} ↦ −4_{1}, 8_{1} ↦ −1_{3}, 8_{2} ↦ −1_{4}

is clearly stable under Δ(α).

Let χ^{+} and ψ^{+} be characters in (G^{+}, e^{+})^{∨} and (H^{+}, f^{+})^{∨} corresponding under J^{+}, and set

E(χ^{+} × \overline{ψ^{+}}) = \begin{cases}E(χ^{+}) × \overline{E(ψ^{+})}, \\ sgn_{Δ(α_{1})}(E(χ^{+}) × \overline{E(ψ^{+})}), \end{cases}

according as (χ^{+}, ψ^{+}) = (1, 1_{1}), (8_{1}, 1_{3}), (8_{2}, 1_{4}) or (χ^{+}, ψ^{+}) = (5_{1}, 4_{2}), (5_{2}, 4_{1}). Here the characters on the right-hand side are to be restricted to the stabilizer of χ^{+} × \overline{ψ^{+}} in Δ(α). The character 10 × 1_{2} extends to a character E(10 × 1_{2}) of Δ(α). Indeed, 10 and 1_{2} extend to characters E_{1}(10) and E_{1}(1_{2}) of G_{1} and H_{1}, where E_{1}(10) and E_{1}(1_{2}) are the extensions given in Tables (6.3) and (6.4). Then

E_{1}(10 × 1_{2}) = Res_{Δ(α_{1})}^{G_{1}×H_{1}}(E_{1}(10) × E_{1}(1_{2}))

is Δ(α)-stable, so that E_{1}(10 × 1_{2}) extends to a character E(10 × 1_{2}) of Δ(α). The assignment χ^{+} × \overline{ψ^{+}} ↦ E(χ^{+} × \overline{ψ^{+}}) is Δ(α)-equivariant, and we may define

E(μ^{+}) = ∑_{χ^{+}∈(G^{+},e^{+})^{∨}} ϵ^{+}(χ^{+})E(χ^{+} × \overline{ψ^{+}}).

E(μ^{+})(x, y) satisfies conditions (Ĩ1.3a) and (Ĩ1.3b) for (x, y) ∈ Δ(α)−(G^{+} ×H^{+}). Indeed, it suffices to take (x, y) ∈ Δ(α_{1})−(G^{+} ×H^{+}), since x in G̃−G^{+} commutes with a non-trivial 3-element only if x ∈ G_{1}.

Let Q = ⟨z⟩ be a non-trivial cyclic subgroup of D. Then

G_{Q}^{+} = H_{Q}^{+} = Q × Q′ ≃ C_{3} × C_{3}, G̃_{Q} = H̃_{Q} = Q × Q′⟨t⟩ ≃ C_{3} × S_{3},

where t inverts Q′. In particular, G̃_{Q} ≤ G_{1} and H̃_{Q} ≤ H_{1}. Moreover, e_{Q}^{+} = f_{Q}^{+} = 1, μ_{Q}^{+} = ∑_{ζ∈(Q×Q′)^{∨}}(ζ × \overline{ζ}) is Δ(α_{Q})-stable, and μ_{Q}^{+}(1, 1) = 9. If ζ^{t} ≠ ζ, then Q′ ≰ ker ζ and Δ(α_{Q})_{ζ×\overline{ζ}} = G_{Q}^{+} × H_{Q}^{+}. If ζ^{t} = ζ, then Q′ ≤ ker ζ and Δ(α_{Q})_{ζ×\overline{ζ}} = Δ(α_{Q}). In this case ζ × \overline{ζ} extends to a character E_{Q}(ζ × \overline{ζ}) containing (Q′ × Q′)⟨(t, t)⟩ in its kernel. It follows easily that conditions (Ĩ1.3a) and (Ĩ1.3b) hold. Moreover, E_{Q}(μ_{Q}^{+})(t, t) = 3, so that (G̃1) and (G̃2) also hold.

Suppose p = 5. Then D ≃ 5, H^{+} ≃ D_{10}, and H̃ ≃ 10: 4. The characters 1, 9, 8_{1}, 8_{2} in (G^{+}, e^{+})^{∨} have stabilizers G̃, G̃, G_{2}, G_{2} in G̃ and extend to their stabilizers. Let E(1), E(9), E(8_{1}), E(8_{2}) be the extensions given in (6.3). The following is the character table of H̃:

Table (6.5). H̃, p = 5^{\begin{aligned}10 & 2 & 5 & 5 & & 2 & 2 & & 10 & 2 & 5 & 5 & & 2 & 2 \\ 1A & 2A & 5A & B^{∗} & & 4B & 4B & & 2D & 2A & 10A & B^{∗} & & 4C & 4C \\ 1 & 1 & 1 & 1 & : & 1 & 1 & : & 1 & 1 & 1 & 1 & : & 1 & 1 \\ 1 & −1 & 1 & 1 & : & i & −i & : & 1 & −1 & 1 & 1 & : & i & -i \\ 2 & 0 & b5 & ∗ & • & 0 & 0 & : & −2 & 0 & −b5 & ∗ & • & 0 & 0 \\ 2 & 0 & ∗ & b5 & • & 0 & 0 & : & −2 & 0 & ∗ & −b5 & • & 0 & 0 \end{aligned}}

The characters 1_{1}, 1_{2}, 2_{1}, 2_{2} in (H^{+}, f^{+})^{∨} have stabilizers H̃, H̃, H_{2}, H_{2} in H̃ and extend to their stabilizers. Let E(1_{1}), E(1_{2}), E(2_{1}), E(2_{2}) be the extensions given in (6.5). The isometry μ^{+} = (J_{μ^{+}}, ϵ_{μ^{+}}) maps

1 ↦ 1_{1}, 9 ↦ −1_{2}, 8_{1} ↦ −2_{1}, 8_{2} ↦ −2_{2}.

μ^{+} is clearly Δ(α)-stable, and

E(μ^{+}) = E(1) × E(1_{1}) − E(9) × E(1_{2}) − E(8_{1}) × E(2_{1}) − E(8_{2}) × E(2_{2}),

where the right-hand side is suitably interpreted on Δ(α). Since Δ(α_{1}) − (G × H) and Δ(α_{3}) − (G × H) contain no elements centralizing a 5-element, the conditions (Ĩ1.3a) and (Ĩ1.3b) on E(μ^{+}) need only be checked on (g, h) in Δ(α_{2}). This is straightforward.

The only non-trivial cyclic subgroup of D is D itself. Then

G_{D}^{+} = H_{D}^{+} = D, G̃_{D} = H̃_{D} = D × ⟨t⟩ ≃ C_{10}.

In particular, G̃_{D} ≤ G_{2} and H̃_{D} ≤ H_{2}. Moreover, e_{D}^{+} = f_{D}^{+} = 1, μ_{D}^{+} is the identity isometry, μ_{D}^{+} is Δ(α_{D})-stable, and μ_{D}^{+}(1, 1) = 5. We may choose E_{Q}(μ_{Q}^{+}) as ∑_{ζ}(ζ × \overline{ζ}), where ζ runs over the characters of D × ⟨t⟩ with ⟨t⟩ in their kernels. Then E_{D}(μ_{D}^{+})(t, t) = 5. The conditions (Ĩ1), (G̃1), (G̃2) now follow.

7. The isotypy conjecture for alternating groups for p = 2

Let G = S(Π) = S_{n} and G^{+} = A(Π) = A_{n} be the symmetric group and alternating group on the set Π of size n, and let p = 2. The 2-blocks of OG are self-dual since 2-cores are self-dual. Thus each block of OG^{+} is covered by a unique block of OG. Suppose e is a block of OG parametrized by the 2-core λ_{0}. If e has defect 0, then Res_{G^{+}}^{G}(χ_{λ_{0}}) = χ_{λ_{0},+} + χ_{λ_{0},−} and e covers two blocks of OG^{+} of defect 0, namely the blocks e_{±} containing χ_{λ_{0},±}. If e has positive defect, then (G, e)^{∨} contains characters χ_{λ} which do not split on G^{+}, and e covers a unique block e^{+} of OG^{+}. Thus a block of OG of positive defect covers a unique G-stable block of OG^{+}. Conversely a G-stable block of OG^{+} is covered by a unique block of OG of positive defect.

We consider a block of OG^{+} with abelian defect group. If the block is not Gstable, then it has defect 0 and is e_{±} for some block e of OG of defect 0. If the block is G-stable, then it is e^{+} for a unique block e of OG of positive defect. Let w be the weight of e and let ∑_{i≥1} 2^{i}w_{i} be the 2-adic expansion of 2w. Then e has defect group

D = ∏_{i≥1}(T_{i})^{w_{i}},

where T_{i} = Z_{2} ≀ Z_{2} ≀ ⋯ ≀ Z_{2} (i times) is a Sylow 2-subgroup of S_{2^{i}}, and e^{+} has defect group D^{+} = D ∩ G^{+} < D.

Lemma (7A). Suppose D^{+} is abelian. Then either D^{+} = 1 and D = T_{1}, or D^{+} ≃ Z_{2} × Z_{2} and D = T_{2}.

Proof. T_{i} is non-abelian for i ≥ 2 and [T_{i}, T_{i}] is non-abelian for i ≥ 3. Thus D is T_{1}, T_{2}, or T_{1} × T_{2}. The case D = T_{1} × T_{2} is impossible since D^{+} would have a non-abelian section isomorphic to T_{2}. Thus D = T_{1} or T_{2}.

Let e^{+} be a block of OG^{+} with abelian defect group D^{+} > 1, let e be the unique block of OG covering e^{+}, and let D be a defect group of e such that D^{+} = D ∩ G^{+}. So (7A) implies that D = T_{2} and D^{+} ≃ Z_{2} × Z_{2}. Let H = N_{G}(D^{+}) and H^{+} = N_{G^{+}}(D^{+}). Let λ_{0} be the 2-core parametrizing e. If |λ_{0}| = 0, then n = 4, G = H = S_{4}, G^{+} = H^{+} = A_{4}, and the isotypy conjecture holds trivially. If |λ_{0}| = 1, then n = 5, G = S_{5}, G^{+} = A_{5}, H = S_{4}, H^{+} = A_{4}, and the isotypy conjecture holds by [4, Appendice A1.1]. So we suppose |λ_{0}| ≥ 3 and n ≥ 7. Let Π = Π_{0} ∪ Π_{1}, where Π_{0} and Π_{1} are the fixed-point set and the support of D. Let H_{0} = S(Π_{0}) and H_{1} = S(Π_{1}) = S_{4}. Then H = H_{0} × H_{1}, H^{+} = (H_{0} × H_{1})^{+}, and

\begin{aligned}C_{G}(D^{+}) & = C_{H}(D^{+}) = H_{0} × D^{+}, \\ C_{G^{+}}(D^{+}) & = C_{H^{+}}(D^{+}) = H_{0}^{+} × D^{+}. \end{aligned}

Let χ_{λ_{0}} be the character of H_{0} parametrized by λ_{0}, let 1_{H_{1}} be the 1-character of H_{1}, and let f be the block of OH containing χ_{λ_{0}} × 1_{H_{1}}. Then f covers a unique block f^{+} of OH^{+} since Res_{H^{+}}^{H}(χ_{λ_{0}} × 1_{H_{1}}) is irreducible, and f and f^{+} are the Brauer correspondents of e and e^{+}. We take as maximal e^{+}-subpair (D^{+}, e_{D^{+}}^{+}) and maximal f^{+}-subpair (D^{+}, f_{D^{+}}^{+}) the subpair (D^{+}, e_{λ_{0},+}), where e_{λ_{0},+} is the block of O(H_{0}^{+} × D^{+}) containing χ_{λ_{0},+} × 1_{D^{+}}.

The partitions λ with 2-core λ_{0} have the following 2-quotients:

(2, ∅), (1^{2}, ∅), (1, 1), (∅, 2), (∅, 1^{2}).

We label the χ_{λ} by the 2-quotients of λ as χ_{(2,∅)}, χ_{(1^{2},∅)}, χ_{(1,1)}, χ_{(∅,2)}, χ_{(∅,1^{2})}. In particular, χ_{(2,∅)}^{∗} = χ_{(∅,1^{2})}, χ_{(1^{2},∅)}^{∗} = χ_{(∅,2)}, χ_{(1,1)}^{∗} = χ_{(1,1)}. So (G^{+}, e^{+})^{∨} consists of the four characters

χ_{(2,∅)}^{+} = Res_{G^{+}}^{G}(χ_{(2,∅)}), χ_{(1^{2},∅)}^{+} = Res_{G^{+}}^{G}(χ_{(1^{2},∅)}), χ_{(1,1),±},

where χ_{(1,1),±} are the constituents of Res_{G^{+}}^{G}(χ_{(1,1)}).

The irreducible characters in (H, f)^{∨} are

χ_{λ_{0}} × χ_{(4)}, χ_{λ_{0}} × χ_{(3,1)}, χ_{λ_{0}} × χ_{(2^{2})}, χ_{λ_{0}} × χ_{(2,1^{2})}, χ_{λ_{0}} × χ_{(1^{4})},

where the second factors are the irreducible characters of S(Π_{1}). Since the partitions (4), (3, 1), (2^{2}), (2, 1^{2}), (1^{4}) have 2-quotients (2, ∅), (1^{2}, ∅), (1, 1), (∅, 2), (∅, 1^{2}), we also label these characters in (H, f)^{∨} as ψ_{(2,∅)}, ψ_{(1^{2},∅)}, ψ_{(1,1)}, ψ_{(∅,2)}, ψ_{(∅,1^{2})}. In particular, ψ_{(2,∅)}^{∗} = ψ_{(∅,1^{2})}, ψ_{(1^{2},∅)}^{∗} = ψ_{(∅,2)}, ψ_{(1,1)}^{∗} = ψ_{(1,1)}. So (H^{+}, f^{+})^{∨} consists of the four characters

ψ_{(2,∅)}^{+} = Res_{H^{+}}^{H}(ψ_{(2,∅)}), ψ_{(1^{2},∅)}^{+} = Res_{H^{+}}^{H}(ψ_{(1^{2},∅)}), ψ_{(1,1),±},

where ψ_{(1,1),±} are the constituents of Res_{H^{+}}^{H}(ψ_{(1,1)}).

Let μ^{+} = (J^{+}, ϵ^{+}) be the isometry in R_{K}((G^{+}, e^{+}), (H^{+}, f^{+})) given by

χ_{(2,∅)}^{+} ↦ ψ_{(2,∅)}^{+}, χ_{(1^{2},∅)}^{+} ↦ −ψ_{(1^{2},∅)}^{+}, χ_{(1,1),±} ↦ −ψ_{(1,1),±}.

Each non-trivial cyclic subgroup P = ⟨x⟩ of D^{+} is Z(D)^{t} for some t in A(Π_{1}). Thus

G_{x} = H_{x} = S(Π_{0}) × D^{t}, G_{x}^{+} = H_{x}^{+} = (S(Π_{0}) × D^{t})^{+}

and the subpairs (⟨x⟩, e_{x}^{+}) and (⟨x⟩, f_{x}^{+}) coincide. We take μ_{x}^{+} = (J_{x}^{+}, ϵ_{x}^{+}) as the identity isometry. The diagram

CF(G^{+}, e^{+}, K) ⟶^{R_{μ{+ \atop ⟨1⟩}}} CF(H^{+}, f^{+}, K)
↓_{d_{G^{+}}^{(x,e^{+}_{x})}}  ↓_{d_{H^{+}}^{(x,f^{+}_{x})}}
CF_{p′}(G_{P}^{+}, e_{x}^{+}, K) ⟶^{R_{μ^{+} x,p′}} CF_{p′}(H_{P}^{+}, f_{x}^{+}, K)

(7.1)

is then commutative. Indeed, it suffices to show that

χ^{+}(xy) = R_{μ_{⟨1⟩}^{+}}(χ^{+})(xy)

for all χ^{+} in (G^{+}, e^{+})^{∨} and all y in H_{x,2′}^{+}. Such y’s lie in A(Π_{0}). Now

\begin{aligned}χ_{(2,∅)}^{+}(xy) & = χ_{(2,∅)}(xy) = χ_{λ_{0}}(y), \\ χ_{(1^{2},∅)}^{+}(xy) & = χ_{(1^{2},∅)}(xy) = χ_{λ_{0}}(y), \end{aligned}

by the Murnaghan-Nakayama formula. On the other hand,

\begin{aligned}ψ_{(2,∅)}^{+}(xy) & = ψ_{(2,∅)}(xy) = χ_{λ_{0}}(y), \\ ψ_{(1^{2},∅)}^{+}(xy) & = ψ_{(1^{2},∅)}(xy) = −χ_{λ_{0}}(y), \end{aligned}

since χ_{(4)}(x) = 1 and χ_{(3,1)}(x) = −1. Lastly,

χ_{(1,1),±}(xy) = \frac{1}{2}χ_{(1,1)}(xy) = −χ_{λ_{0}}(y).

The first equality holds since xy has cycle type unequal to η(λ), where λ is the partition with 2-core λ_{0} and 2-quotient (1, 1). The second equality holds by the Murnaghan-Nakayama formula. We may suppose with suitable labeling that

\begin{aligned}ψ_{(1,1),+}(xy) & = χ_{λ_{0},+}(y)χ_{(2^{2}),+}(x) + χ_{λ_{0},−}(y)χ_{(2^{2}),−}(x), \\ ψ_{(1,1),−}(xy) & = χ_{λ_{0},+}(y)χ_{(2^{2}),−}(x) + χ_{λ_{0},−}(y)χ_{(2^{2}),+}(x). \end{aligned}

Both equal χ_{λ_{0}}(y), since χ_{(2^{2}),±}(x) = 1. Thus (7.1) is commutative. It now follows as in §5 that e^{+} and f^{+} are isotypic blocks.

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Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60680

Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455