EQUIVALENCES FOR BLOCKS OF THE WEYL GROUPS

Radha Kessar

Abstract. We describe the source algebras of the blocks of the Weyl groups of type B and type D in terms of the source algebras of the blocks of the symmetric groups. As a consequence, we show that Puig’s conjecture on the finiteness of the number of isomorphism classes of source algebras for blocks of finite groups with a fixed defect group holds for these classes of groups. We also show how certain isomorphisms between subalgebras of block algebras of the symmetric groups can be lifted to block algebras of the Weyl groups of type B.

1. Introduction

Let p be a prime number and let (O, K, k) be a p-modular system, i.e. O is a complete discrete valuation ring with maximal ideal J(O), k = O/J(O) is its residue field with characteristic p and K is its field of quotients with characteristic zero; we assume that k is algebraically closed. By a p-block of a finite group G, we mean a central idempotent of the group algebra OG. Recall that a source algebra of b is an algebra of the form iOGi, where P is a defect group of G and i is a primitive idempotent of (OGb)^{P}, such that b is in the image of the relative trace map Tr_{P}^{G}((OG)^{P} i(OG)^{P}). Equivalently, i is characterized as belonging to a local point of P on OGb. A source algebra iOGi of b is naturally an interior P -algebra via the homomorphism which maps an element x of P to the element ix of iOGi (for details see [T], §38).

The Weyl group B_{n} is the wreath product of the group of order 2 with S_{n}. We will find it convenient to use the following description for it. For a finite set U, we denote by E(U) the finite elementary abelian two group with a distinguished basis U, i.e.

E(U) =< u : u ∈ U, u^{2} = 1, uvuv = 1 >,

by E^{0}(U), the subgroup of E(U) of index two which is generated by elements uv, u, v ∈ U, and by S(U) the symmetric group of the set U.

Given E(U), the action of S(U) on the elements of U naturally extends to an action of S(U) on E(U) (via ^{σ}(uv) = ^{σ}u ^{σ}v) and this action leaves invariant the subgroup E^{0}(U). If we put

B(U) := E(U) ⋉ S(U)

and

D(U) := E^{0}(U) ⋉ S(U),

then B(U) and D(U) are respectively the Weyl groups W (B_{|U|}) and W (D_{|U|+1}).

Typeset by AMS-TEX

For a subset X of U, let us denote by τ_{X} the trivial character of E(X) and by τ_{X}^{∗} the irreducible character of E(X) which takes the value −1 on any element x of X. Let e_{X} and e_{X}^{∗} denote the central idempotents of KE(X) corresponding respectively to the character τ_{X} and the character τ_{X}^{∗}.

The block theory of B(U) divides naturally into the case where p = 2 and the case where p is odd. Since E(U) is a normal 2 subgroup of B(U) which is its own centralizer, B(U) has only one 2 block. On the other hand, when p is odd, Clifford theory easily yields a parametrization of the p-blocks of B(U) in terms of the pblocks of the symmetric groups (see [JK], [P1]).We state below the relevant results, which were first described in [O1] and [O2].

¿From now on, unless we explicitly state otherwise, we assume that p is an odd prime. Let U = U_{1} ∪ U_{2} be a partition of U into disjoint nonempty subsets. Since p is odd the central idempotents e_{U_{1}} and e_{U_{2}}^{∗} of KE(U_{1}) and KE(U_{2}) respectively are blocks of E(U_{1}) and E(U_{2}), hence e_{U_{1}}e_{U_{2}}^{∗} is a block of E(U) and further, the stabilizer of e_{U_{1}}e_{U_{2}}^{∗} under the conjugation action of B(U) on the algebra OE(U) is given by

Stab_{B(U)}(e_{U_{1}}e_{U_{2}}^{∗}) = B(U_{1}) × B(U_{2}).

For i = 1, 2 let b̃_{i} be a block of S(U_{i}). Clearly, since E(U) is abelian, the idempotent e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} is a block of B(U_{1}) × B(U_{2}) and it follows by the theory of covering blocks (see theorem 5.10, Chapter 5, of [NT]), that the idempotent

b := Tr_{B(U_{1})×B(U_{2})}^{B(U)}(e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2})

is a block of B(U) and that all blocks of B(U) can be described in this way.

It also follows that, if for i = 1, 2, P_{i} is a defect group of b̃_{i}, then P := P_{1} × P_{2} is a defect group of b.

Let us fix a partition U = U_{1} ∪ U_{2} of U into disjoint subsets, and a block b of B(U) as defined above. We also fix a defect group P = P_{1} × P_{2} of b as above.

The associate of the block b is the block of B(U) given by

b^{a} := Tr_{B(U_{1})×B(U_{2})}^{B(U)}(e_{U_{1}}^{∗}e_{U_{2}}b̃_{1}b̃_{2}).

Note that b is equal to its own associate if and only if there is an involution σ of S(U) such that ^{σ}U_{1} = U_{2} and ^{σ}b̃_{1} = b̃_{2}. In the following theorem, whenever b is self associate, we will assume a fixed choice of such a σ. Also, we will assume in this case that P_{2} = ^{σ}P_{1}.

Theorem 1.1. (i) For i = 1, 2 let t_{i} belong to a local point of P_{i} on OS(U_{i})b̃_{i}. Then the interior P_{1} × P_{2}-algebra t_{1}OS(U_{1})t_{1} ⊗_{O} t_{2}OS(U_{1})t_{2} is isomorphic to a source algebra of the block b of B(U).

(ii) If b is not self associate, then b′ := b + b^{a} is a block of D(U) with defect group P, and there is an interior D(U)-algebra isomorphism

OD(U)b′ ≃ OB(U)b

which induces an interior P -algebra isomorphism between a source algebra for b and a source algebra of b′.

(iii) If b is self-associate, and P is not the identity group the idempotent b is a block of OD(U). Further, if t_{1} and t_{2} in (i) are such that t_{2} = ^{σ}t_{1} then a source algebra for b when b is considered as a block of D(U) is isomorphic as interior P - algebra to the crossed product of t_{1}OS(U_{1})t_{1} ⊗_{O} t_{2}OS(U_{1})t_{2} with the group algebra O < σ >. In other words, it is isomorphic to the algebra which as O-module is the module

t_{1}OS(U_{1})t_{1} ⊗_{O} t_{2}OS(U_{1})t_{2} ⊗_{O} O < σ >;

with multiplication defined by

(a_{1} ⊗ a_{2} ⊗ σ)(a1^{′} ⊗ a2^{′} ⊗ σ^{i}) = a_{1} ^{σ}a2^{′} ⊗ a_{2} ^{σ}a1^{′} ⊗ σσ^{i},

(a_{1} ⊗ a_{2} ⊗ 1_{<σ>})(a1^{′} ⊗ a2^{′}σ^{i}) = a_{1}a1^{′} ⊗ a_{2}a2^{′} ⊗ σ^{i},

for a_{1}, a1^{′} in A_{1}, a_{2}, a2^{′} in A_{2}, i = 1, 2 and where the interior P -algebra structure is given by

g_{1} × g_{2} → g_{1}1_{t_{1}OS(U_{1})t_{1}} ⊗ ^{σ}g_{2}1_{t_{2}OS(U_{1})t_{2}} ⊗ 1_{<σ>},

for g_{1} in P_{1} and g_{2} in P_{2}.

Corollary 1.2. Puig’s conjecture (see [B2]) is true for the blocks of the Weyl groups of type B and type D for all primes p. Hence, it holds for all Weyl groups. In other words, for a fixed p-group P there are only finitely many isomorphism classes of interior P -algebras that are source algebras for blocks of the Weyl groups which have defect groups isomorphic to P.

In [Sc] Scopes showed that the existence of a certain combinatorial relationship between two blocks of two symmetric groups which have isomorphic defect groups forces the corresponding block algebras to be Morita equivalent. In [P2], Puig extended these results to show that this combinatorial relationship in fact forces an explicit isomorphism between certain subalgebras of the block algebras of the concerned blocks. The next theorem is a Clifford theoretic “lift” of Puig’s result to the Weyl groups of type B. We will say that two blocks of two symmetric groups are Scopes related if they are connected to each other in the same way as the two blocks of Theorem 1.7 of [P2].

Let b be as before a block of B(U) and suppose that V_{1} is a subset of U_{1} such that c̃_{1} is a block of S(V_{1}) with P_{1} as defect group. Put V := V_{1} ∪ U_{2} and let c be the block of B(V) defined by

c := Tr_{B(V_{1})×B(U_{2})}^{B(V)}(e_{U_{1}} c̃_{1} ⊗ e_{U_{2}}^{∗}b̃_{2}).

Note that cOB(U)bc is naturally an interior B(V) × B(U − V)-algebra via the map which sends an element x of B(V) × B(U − V) to the element xbc of cOB(U)bc, and we shall assume this interior B(V)×B(U −V)-algebra structure for cOB(U)bc in the theorem below. Also, note that since S(U − V) is a quotient of B(U − V), OS(U − V) is naturally a B(U − V)) module, and this gives End_{O}(OS(U − V)) an interior B(U − V) algebra structure.

We are now ready to state our theorem.

Theorem 1.3. Suppose that b̃_{1} and c̃_{1} are Scopes related. Then we have an isomorphism of interior B(V) × B(U − V)-algebras

OB(V)c ⊗_{O} End_{O}(OS(U − V)) ≃ cOB(U)bc.  (1.1)

2. Some results on the induction of interior G-algebras

All the results in this section can either be found in section 16 of [T] or are easy consequences of results given there. Hence, except for Proposition 2.4, we omit giving proofs.

Let G be a finite group, H a subgroup of G and B an interior G-algebra. Recall that the algebra Ind_{H}^{G}(B) is an interior G-algebra which as an O-module is the O-module OG ⊗_{OH} B ⊗_{OH} OG with multiplication defined by

(x ⊗ t ⊗ y)(x′ ⊗ t′ ⊗ y′) = \{ \begin{matrix}x ⊗ tyx′t′ ⊗ y′ & if yx′ ∈ H \\ 0 & if yx′ ∉ H \end{matrix}

for x, x′, y, y′ in G and t in B; the interior G-algebra structure of Ind_{H}^{G}(B) is defined by the map

ϕ : G → Ind_{H}^{G}(B), g → ∑_{x∈G/H} gx ⊗ 1_{B} ⊗ x^{−1}.

The (OG, OG) bimodule structure on Ind_{H}^{G}(B) induced by left and right multiplication by elements ϕ(g) coincides with the natural (OG, OG) bimodule structure on OG ⊗_{OH} B ⊗_{OH} OG. In other words, we have

ϕ(g)(f ⊗ t ⊗ f′) = gf ⊗ t ⊗ f′ and (f ⊗ t ⊗ f′)ϕ(g) = gf ⊗ t ⊗ f′g.

For the next lemma, recall that if A is an interior G-algebra, X a subgroup of G and j an idempotent of A^{X}, then jAj is naturally an interior X-algebra via the map which sends the element x of X to the element xj of jAj.

Lemma 2.1. Let H be a subgroup of G and B an interior H-algebra. Put A = Ind_{H}^{G}(B). Further, suppose that X and Y are subgroups of G with X ∩H = Y, and that i is an idempotent of B^{Y}. Then a left transversal of Y in X can be extended to a left transversal of H in G. In A, let j = ∑_{x∈X/Y} x ⊗ i ⊗ x^{−1}. Then j is an idempotent of A^{X}, and further, there is a canonical isomprphism of interiorXalgebras,

Ind_{Y}^{X}(iBi) ≃ jAj, x ⊗ t ⊗ y → x ⊗ t ⊗ y, (x, y ∈ X, t ∈ iBi).

We also have

Lemma 2.2. Let D be a finite group, T an interior-D algebra and B an interior H-algebra. There is an isomorphism of interior G × D-algebras

Ind_{H×D}^{G×D}(B ⊗_{O} T) ≃ Ind_{H}^{G}(B) ⊗_{O} T, x ⊗ (t ⊗ s) ⊗ y → (x ⊗ t ⊗ y) ⊗ s

for x, y in G, t in B, and s in T.

A central point in the proof of Theorem 1.1 will be the following proposition, which is proposition 16.6 of [T].

Proposition 2.3. Let A be an interior G-algebra and let H be a subgroup of G. Assume that there exists an idempotent i of A^{H} such that 1_{A} = Tr_{H}^{G}(i) and i ^{g}i = 0 for all g ∈ G − H. Then there is an isomorphism of interior G-algebras

Ind_{H}^{G}(iAi) ≃ A, x ⊗ t ⊗ y → xty (x, y ∈ G, t ∈ iAi).

The above proposition immediately gives the following Clifford theoretic result.

Proposition 2.4. Let E be a normal subgroup of G, e a block of E and H the stabilizer of e in G. Let i be a block of H covering e and put b = Tr_{H}^{G}(i). Then there is an isomorphism of interior G-algebras

Ind_{H}^{G}(OHi) ≃ OGb, x ⊗ t ⊗ y → xty (x, y ∈ G, t ∈ OHi).

Proof. Let g ∈ G − H. Then

i ^{g}i = igig^{−1} = iegeig^{−1} = iegeg^{−1}gig^{−1} = 0

since H is the stablizer of e in G and i is a block of H covering e. Then applying Proposition 2.3 with A = OGb, we get immediately that there is an isomorphism of interior G-algebras

Ind_{H}^{G}(iOGbi) ≃ OGb, x ⊗ t ⊗ y → xty (x, y ∈ G, t ∈ iOGi).

But the algebra iOGbi is the algebra iOHi, since for any g ∈ G − H, we have shown above that igi = 0. This proves the proposition.

3. Proof of Theorem 1.1

We use the notation of Theorem 1.1. It is clear from Proposition 2.4 and from the description of b that there is an isomorphism of interior OB(U)-algebras

Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} ≃ OB(U)b,
x ⊗ t ⊗ y → xty, (x, y ∈ B(U), t ∈ O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2}.  (3.1)

Now, for i = 1, 2, let t_{i} be an element of a local point of P_{i} on OS(U_{i})b̃_{i}. Since

(OS(U_{1})b̃_{1} ⊗_{O} OS(U_{2})b̃_{2})^{P} ≃ (OS(U_{1})b̃_{1})^{P_{1}} ⊗_{O} (OS(U_{2})b̃_{2})^{P_{2}}

and

\begin{matrix}((OS(U_{1})b̃_{1})^{P_{1}} ⊗_{O} (OS(U_{2})b̃_{2})^{P_{2}})/J((OS(U_{1})b̃_{1})^{P_{1}} ⊗_{O} (OS(U_{2})b̃_{2})^{P_{2}}) ≃ \\ (OS(U_{1})b̃_{1})^{P_{1}})/J((OS(U_{1})b̃_{1})^{P_{1}}) ⊗_{k} (OS(U_{2})b̃_{2})^{P_{2}})/J((OS(U_{2})b̃_{2})^{P_{2}}), \end{matrix}

we see that t_{1}⊗t_{2} ia a primitive idempotent of (OS(U_{1})b̃_{1}⊗_{O}OS(U_{2})b̃_{2})^{P}. Similarly, we have an isomorphism of Brauer quotients

Br_{P} ((OS(U_{1})b̃_{1} ⊗_{O} OS(U_{2})b̃_{2})^{P}) ≃ Br_{P_{1}}((OS(U_{1})b̃_{1})^{P_{1}}) ⊗_{k} Br_{P_{2}}((OS(U_{2})b̃_{2})^{P_{2}}).

In other words, t_{1} ⊗ t_{2} belongs to a local point of P on OS(U_{1})b̃_{1} ⊗_{O} OS(U_{2})b̃_{2}; hence, using the canonical isomorphism

OS(U_{1})b̃_{1} ⊗_{O} OS(U_{2})b̃_{2} ≃ O(S(U_{1}) × S(U_{2}))b̃_{1}b̃_{2}, a_{1} ⊗ a_{2} → a_{1}a_{2},

we have that t_{1}t_{2} belongs to a local point of P on O(S(U_{1}) × S(U_{2}))b̃_{1}b̃_{2}. Further, multiplying the term on the left of the above isomorphism on both sides with t_{1} ⊗t_{2} and the term on the right on both sides with t_{1}t_{2} yields an isomorphism of interior P -algebras

t_{1}OS(U_{1})t_{1} ⊗_{O} t_{2}OS(U_{2})t_{2} ≃ t_{1}t_{2}O(S(U_{1}) × S(U_{2}))t_{1}t_{2}.  (3.2)

On the other hand, notice that if we consider O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} as an interior S(U_{1}) × S(U_{2})-algebra, then O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} is isomorphic to the algebra O(S(U_{1}) × S(U_{2}))b̃_{1}b̃_{2} since for any u in E(U), ue_{U_{1}}e_{U_{2}}^{∗} = ±e_{U_{1}}e_{U_{2}}^{∗}; in particular these are isomorphic as interior P -algebras. Hence, t_{1}t_{2} belongs to a local point of P on O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} and we have an isomorphism of interior P -algebras

t_{1}t_{2}O(S(U_{1}) × S(U_{2}))t_{1}t_{2} ≃ t_{1}t_{2}O(B(U_{1}) × B(U_{2}))t_{1}t_{2}.  (3.3)

Further, by parts (d) and (f) of Proposition 15.1 of [T], 1 ⊗ t_{1}t_{2} ⊗ 1 belongs to a local point of P on Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2}. Since there is clearly an interior P -algebra isomorphism

1⊗t_{1}t_{2}⊗1(Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1})×B(U_{2})))1⊗t_{1}t_{2}⊗1 ≃ t_{1}t_{2}O(B(U_{1})×B(U_{2}))t_{1}t_{2},

by multiplying on both sides the term on the left of (3.1) with 1 ⊗ t_{1}t_{2} ⊗ 1 and the term on the right of (3.1) with t_{1}t_{2}, and using the isomorphisms (3.2) and (3.3) we get part (i) of Theorem 1.1

We put e := e_{U_{1}}e_{U_{2}}^{∗} + e_{U_{1}}^{∗}e_{U_{2}}. Then it is clear that e is a block of E^{0}(U). For a finite group G, and a block b of G, recall that Irr(G, b) denotes the set of ordinary irreducible characters of G that belong to the block b.

By the definition of b it is clear that every element χ of Irr(B(U), b) is the induced character

Ind_{B(U_{1})×B(U_{2})}^{B(U)}(τ_{U_{1}}τ_{U_{2}}^{∗}χ_{1}χ_{2})

for some elements χ_{1} of Irr(S(U_{1}), b̃_{1}) and χ_{2} of Irr(S(U_{2}), b̃_{2}). For the character χ as above let

χ^{a} := Ind_{B(U_{1})×B(U_{2})}^{B(U)}(τ_{U_{1}}^{∗}τ_{U_{2}}χ_{1}χ_{2})

By Mackey decomposition the restriction of χ to D(U) is given by

Res|_{D(U)}(χ) = Ind_{B(U_{1})×B(U_{2})∩D(U)}^{D(U)}(τ_{U_{1}}τ_{U_{2}}^{∗}χ_{1}χ_{2})
= Ind_{E^{0}(U)⋉(S(U_{1})×S(U_{2}))}^{D(U)}τ_{U_{1}}τ_{U_{2}}^{∗}(χ_{1}χ_{2})
= Ind_{E^{0}(U)⋉(S(U_{1})×S(U_{2}))}^{D(U)}τ_{U_{1}}^{∗}τ_{U_{2}}(χ_{1}χ_{2})
= Ind_{B(U_{1})×B(U_{2})∩D(U)}^{D(U)}(τ_{U_{1}}^{∗}τ_{U_{2}}χ_{1}χ_{2})
= Res|_{D(U)}(χ^{a})  (3.4)

Also, a character χ of B(U) belongs to the block b of B(U) if and only if χ^{a} belongs to the block b^{a}. So, if b ≠ b^{a}, then in particular χ ≠ χ^{a} for any χ in Irr(B(U), b) and using Clifford theory it follows by (3.4) that Res_{D(U)}(χ) = Res_{D(U)}(χ^{a}) is irreducible for all χ in Irr(B(U), b). In particular, b and b^{a} each cover exactly one block of D(U).

On the other hand, it is clear that

b′ := b + b^{a} = Tr_{S(U_{1})×S(U_{2})}^{S(U)}eb̃_{1}b̃_{2}

is a central idempotent of D(U). This proves that b′ is a block of D(U); it is clear that P is a defect group of b′.

It also follows from the above analysis that

Irr(D(U), b′) = {Res|_{D(U)}χ : χ ∈ Irr(B(U), b).

In other words, restriction provides a one to one correspondence between Irr(B(U), b) and Irr(D(U), b′). So, by Theorem 0.1 of [B1], it follows that the block algebras OD(U)b′ and OB(U)b are isomorphic via the map which sends an element a of OD(U)b′ to the element ab of OB(U)b. This is clearly an isomorphism of interior D(U)-algebras and in particular, it induces an interior P -algebra isomorphism between a source algebra of b′ and a source algebra of b. This proves part (ii) of Theorem 1.1.

The proof of the third part of Theorem 1.1 depends heavily on the following result which is a special case of Theorem 5.5 of [P2].

Proposition 1.3. Let L be a normal subgroup of N and let f be a block of L normalized by N. Assume that C_{N}(P) ⊂ L where P is a defect group of f. Let γ be a local point of P on OLf and H_{β} a pointed group on OLf such that P_{γ} ⊂ H_{β}. Then β is contained in a point of H on ON, i.e. elements of β remain primitive in the algebra (ON)^{H}.

Proof. See [P2], Theorem 5.5 (i) and Remark 5.6.

Note that if we take H = L, the above theorem yields that if C_{N}(P) ⊂ L, then f is a block of N.

Let us assume from now on that b is self associate and let σ be as in part (iii) of Theorem 1.1. Clearly, ^{σ}(e_{U_{1}}e_{U_{2}}^{∗}) = e_{U_{1}}^{∗}e_{U_{2}}; hence if we denote by T the group (S(U_{1}) × S(U_{2}))⋉ < σ >, then it is clear from the description of b that

b = Tr_{E^{0}(U)⋉T}^{D(U)} (eb̃_{1}b̃_{2}).  (3.5)

Now the idempotent eb̃_{1}b̃_{2} is a block of the normal subgroup E^{0}(U)⋉ < S(U_{1})× S(U_{2}) > of E^{0}(U) ⋉ T, and it has P as a defect group. Since we are under the assumption that P = P_{1} × P_{2} is not the identity group, we certainly have

C_{E^{0}(U)⋉T} (P) = C_{E^{0}(U)⋉<S(U_{1})×S(U_{2})>}(P)  (3.6)

and hence by Proposition 3.1 applied with L = E^{0}(U)⋉ < S(U_{1}) × S(U_{2}) >, N = E^{0}(U) ⋉ T, f = eb̃_{1}b̃_{2} and H = L, we have that eb̃_{1}b̃_{2} is a block of E^{0}(U) ⋉ T. Since E^{0}(U) ⋉ T is the stabilizer in D(U) of the block e of E^{0}(U), it follows by Theorem 5.5, Chapter 5 of [NT] and by (3.5), that b is a block of D(U).

So, applying Proposition 2.4 with E = E^{0}(U), H = E^{0} ⋉ T, and i = eb̃_{1}b̃_{2}, we get that there is an isomorphism of interior D(U)-algebras

Ind_{E^{0}(U)⋉T}^{D(U)} eb̃_{1}b̃_{2}O(E^{0}(U) ⋉ T) ≃ OD(U)b.

In particular, it follows just as for (i) that a source algebra for b as a block of D(U) is isomorphic as interior P -algebra to a source-algebra for the block eb̃_{1}b̃_{2} of E^{0}(U) ⋉ T, and this in turn is clearly ismorphic to a source algebra for the block b̃_{1}b̃_{2} of the group T.

On the other hand, let t_{1} and t_{2} = ^{σ}t_{1} be as in Theorem 1.1 (iii). Then, just as for the proof of (i), t_{1}t_{2} belongs to a local point of P on S(U_{1}) × S(U_{2}). Now, in view of (3.6), it follows by Proposition 3.1 applied with L = S(U_{1}) × S(U_{2}), f = b̃_{1}b̃_{2}, N = T and H = P that t_{1}t_{2} belongs to a local point of P on OT b̃_{1}b̃_{2}. In other words, t_{1}t_{2}OT t_{1}t_{2} is a source algebra for the block b̃_{1}b̃_{2} of the group T.

Define the O-linear map

ϕ : t_{1}OS(U_{1})t_{1} ⊗_{O} t_{2}OS(U_{2})t_{2} ⊗_{O} O < σ >→ t_{1}t_{2}OT t_{1}t_{2}, a ⊗ a′ ⊗ σ^{i} → a a′σ^{i}

for a ∈ t_{1}OS(U_{1})t_{1}, a′ ∈ t_{2}OS(U_{2})t_{2} and i = 1, 2. Then, since t_{2} = ^{σ}t_{1}, it is easy to check that ϕ is an isomorphism of interior P -algebras. This completes the proof of part (iii) of Theorem 1.1.

It remains to prove Corollary 1.2. Since the groups B(U) and D(U) each have only one 2 block we only need to prove the result when p is odd. Let us fix a non identity p-group P and let b be a block of B(U). Then, by Theorem 1.1 there are p-groups P_{1} and P_{2} such that P = P_{1} × P_{2} and a source algebra for b is isomorphic to the tensor product A_{1} ⊗ A_{2} of the source algebras A_{1} and A_{2} of blocks of two symmetric groups which have P_{1} and P_{2} respectively as defect groups. By paragraph 1.8 of [P2], there are only finitely many possibilities (upto interior P_{i}-algebra isomorphism) for each A_{i},and since, for a fixed P there are only finitely many possibilities for P_{1} and P_{2} the result holds for the p blocks of the groups B(U). The result for the groups D(U) follows in the same way since by parts (ii) and (iii) the source algebras of a block of D(U) are isomorphic either to a tensor product of sou! ! ! ! rce algebras of blocks of symmetric groups or to a crossed product of the tensor product of these source algebras with the group algebra of the group of order 2. Since the only infinite families of Weyl groups are the symmetric groups and the groups of type B and D, it follows that Puig’s conjecture is true for the Weyl groups.

4. Proof of Theorem 1.3

By the description of b and c, we get that in OB(U),

bc = bTr_{B(V_{1})×B(U_{2})}^{B(V)}e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}

= Tr_{B(V_{1})×B(U_{2})}^{B(V)}be_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}

= Tr_{B(V_{1})×B(U_{2})}^{B(V)}(Tr_{B(U_{1})×U_{2}}^{B(U)}(e_{U_{1}}e_{U_{2}}^{∗})b̃_{1}b̃_{2})e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}

= Tr_{B(V_{1})×B(U_{2})}^{B(V)}( ∑_{x∈S(U)/S(U_{1})×S(U_{2})}(e _{^{x}U_{1}}e_{^{x}U_{2}}^{∗}) ^{x}b̃_{1} ^{x}b̃_{2})e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}

= Tr_{B(V_{1})×B(U_{2})}^{B(V)}(  ∑_{x∈S(U)/S(U_{1})×S(U_{2})} ^{x}b̃_{1} ^{x}b̃_{2}e _{^{x}U_{1}}e_{^{x}U_{2}}^{∗}e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}) (4.1)

Now for any x in S(U), e _{^{x}U_{1}}e_{^{x}U_{2}}^{∗}e_{V_{1}}e_{U_{2}}^{∗} = e _{^{x}U_{1}}e_{^{x}U_{2}}^{∗} if V_{1} ⊂ ^{x}U_{1} and U_{2} ⊂ ^{x}U_{2} and is zero otherwise. Moroever, U_{2} ⊂ ^{x}U_{2} if and only if x ∈ S(U_{1})×S(U_{2}). Hence, from (4.1) we get that

bc = Tr_{B(V_{1})×B(U_{2})}^{B(V)}(e_{U_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{1}b̃_{2}).  (4.2)

It is clear from (4.2) that in OB(U)b, the idempotent bc is the image under the isomorphism in (3.1) of the element

j := ∑_{x∈B(V)/B(V_{1})×B(U_{2})} x ⊗ e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2} ⊗ x^{−1}

of Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2} and that j is centralized by the group B(V) × B(U − V). Hence, multiplying the algebra on the left of (3.1) on both sides by the idempotent j and the algebra on the right of (3.1) on both sides by the idempotent bc, we get an isomorphism of interior B(V)×B(U −V)-algebras

j(Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2})j ≃ cOB(U)bc.  (4.3)

On the other hand, by Lemma 2.1 applied with G = B(U), H = B(U_{1}) × B(U_{2}), B = O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2}), X = B(V) × B(U − V), Y = B(V_{1}) × B(U − V) × B(U_{2}) and i = e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2}, we get that there is an isomorphism of interior B(V) × B(U − V)-algebras

j(Ind_{B(U_{1})×B(U_{2})}^{B(U)}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}b̃_{2})j ≃
Ind_{B(V_{1})×B(U−V)×B(U_{2})}^{B(V)×B(U−V)}c̃_{1}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2} (4.4)

and hence, composing the inverse of this isomorphism with the isomorphism of (4.3), we get an isomorphism of interior B(V) × B(U − V) -algebras

Ind_{B(V_{1})×B(U−V)×B(U_{2})}^{B(V)×B(U−V)}c̃_{1}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2} ≃ cOB(U)bc. (4.5)

Also, note that there is a canonical interior B(V_{1}) × B(U − V) × B(U_{2})-algebra isomorphism

c̃_{1}O(B(U_{1}) × B(U_{2}))e_{U_{1}}e_{U_{2}}^{∗}b̃_{1}c̃_{1}b̃_{2} ≃ c̃_{1}OB(U_{1})e_{U_{1}}b̃_{1}c̃_{1} ⊗_{O} OB(U_{2})e_{U_{2}}^{∗}b̃_{2}. (4.6)

Now, since b̃_{1} and c̃_{1} are Scopes related, by Theorem 1.7 of [P2], we have that there is an interior S(V_{1}) × S(U − V) -algebra isomorphism

OS(V_{1})c̃_{1} ⊗_{O} End_{O}(OS(U − V)) ≃ c̃_{1}OS(U_{1})b̃_{1}c̃_{1}.  (4.7)

For any x in E(V_{1}) × E(U − V), xe_{U_{1}} = e_{U_{1}} and xe_{V_{1}} = e_{V_{1}} ; hence the isomorphism (4.7) yields an interior B(V_{1}) × B(U − V)-algebra isomorphism

OB(V_{1})e_{V_{1}} c̃_{1} ⊗_{O} End_{O}(OS(U − V)) ≃ c̃_{1}OB(U_{1})e_{U_{1}}b̃_{1}c̃_{1}.  (4.8)

Then (4.5), (4.6) and (4.8) yield an interior B(V) × B(U − V)-algebra isomorphism

Ind_{B(V_{1})×B(U−V)×B(U_{2})}^{B(V)×B(U−V)}(OB(V_{1})e_{U_{1}} c̃_{1}⊗_{O}End_{O}(OS(U−V))⊗_{O}OB(U_{2})e_{U_{2}}^{∗}b̃_{2}) ≃ cOB(U)bc. (4.9)

Now, applying Lemma 2.2 with T = End_{O}(OS(U − V)), H = B(V_{1}) × B(U − V) × B(U_{2}), G = B(V) × B(U − V), and B = OB(V_{1})e_{U_{1}} c̃_{1} ⊗_{O} OB(U_{2})e_{U_{2}}^{∗}b̃_{2}, (4.10) yields interior B(V) × B(U − V)-algebra isomorphisms

cOB(U)bc ≃ (Ind_{B(V_{1})×B(U−V)×B(U_{2})}^{B(V)×B(U−V)}OB(V_{1})e_{U_{1}} c̃_{1} ⊗_{O} OB(U_{2})e_{U_{2}}^{∗}b̃_{2}) ⊗_{O} End_{O}(OS(U − V))

≃ (Ind_{B(V_{1})×B(U_{2})}^{B(V)}OB(V_{1})e_{V_{1}} c̃_{1} ⊗_{O} OB(U_{2})e_{U_{2}}^{∗}b̃_{2}) ⊗_{O} End_{O}(OS(U − V))

≃ (Ind_{B(V_{1})×B(U_{2})}^{B(V)}O(B(V_{1}) × B(U_{2}))e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2}) ⊗_{O} End_{O}(OS(U − V))(4.10)

On the other hand, as for (3.1), we have that

Ind_{B(V_{1})×B(U_{2})}^{B(V)}O(B(V_{1}) × B(U_{2}))e_{V_{1}}e_{U_{2}}^{∗} c̃_{1}b̃_{2} ≃ OB(V)c  (4.11)

as interior B(V)-algebras.

Then the isomorphisms (4.10) and (4.11) immediately yield the isomorphism desired in (1.1)

References

[B1] M. Broué, Isométries de caracteres et équivalences de Morita ou dérivées, Publ. Math. I.H.E.S. 71 (1990), 45-63.

[B2] , Equivalences of blocks of group algebras, Publ. LMENS-93-4, Paris (1993).

[JK] G.D. James and A. Kerber, The representation theory of the symmetric group. Enc. of Math. 16, Addison & Wesley, London, 1981.

[NT] H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press, London, 1989.

[O1] M. Osima, On the representations of the generalized symmetric group, Math. J. Okayama Univ. 4 (1955-56), 39-56.

[O2] , On the representations of the generalized symmetric group II, Math. J. Okayama Univ. 6 (1956-57), 81-97.

[P1] L. Puig, The Nakayama conjecture and the Brauer pairs, Séminaire sur les Groupes Finis, Pub. Math. Univ. Paris VII, 25, 1987.

[P2] , On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups, Algebra Colloq. 1:1 (1994), 25-55.

[S] J.Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441–455.

[T] J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford University Press, Oxford, 1995.

Department of Mathematics, University of Minnesota, 206, Church Street, S.E, Minneapolis, MN 55455

   E-mail address: kessar@math.umn.edu