The extended Brauer quotient of N-interior G-algebras

Tiberiu Coconeţ^{∗}

Faculty of Economics and Business Administration, Babeş-Bolyai University, Str. Teodor Mihali, nr.58-60 , 400591 Cluj-Napoca, Romania

Constantin-Cosmin Todea

Department of Mathematics, Technical University of Cluj-Napoca, Str. G. Baritiu 25, Cluj-Napoca 400027, Romania

Abstract

We generalize the extended Brauer quotient defined by L. Puig and Y. Zhou in [3, Section 3] to the case of N-interior G-algebras, where N is a normal subgroup of a finite group G. We use this extended Brauer quotient on permutation algebras to establish some correspondences that generalize known results that hold in the group algebra case.

Keywords: G-algebra, Brauer quotient, pointed group 2000 MSC:20C20

1. Introduction

Throughout this paper N is a normal subgroup of the finite group G, p is a prime and k is the residue field of a discrete valuation ring O, with characteristic p. All modules, considered as left modules, and algebras are finite finitely generated. Recall that an Ninterior G-algebra A is a k-algebra endowed with two group homomorphisms

α : N → A^{×}

and

ϕ : G → Aut_{k}(A),

March 23, 2013

∗Corresponding author

Email addresses: tiberiu.coconet@math.ubbcluj.ro (Tiberiu Coconeţ), Constantin.Todea@math.utcluj.ro (Constantin-Cosmin Todea)

1The first author acknowledges the support of the Romanian PN-II-ID-PCE-2012-4-0100 project

where A^{×} is the subgroup of invertible elements of A and Aut_{k}(A) is the group of kalgebra automorphisms of A. For any g ∈ G,n ∈ N,a ∈ A we set a^{g} = α(g)(a),n^{g} = gng^{−1} . Moreover, we have a^{n}= α(n)aα(n)^{−1} . Note that since A is N-interior it carries a kN −kNbimodule structure. The algebra A serves for the construction of the interior G-algebra

Â:= A ⊗_{N}G,

where for a,b ∈ A and x,y ∈ G, the product is given by

(a ⊗ x)(b ⊗ y) = ab^{x}⊗ xy.

The extended Brauer quotient for interior G-algebras was defined and analyzed in [3] and in [4]. It turns out that this extended Brauer quotient is an important tool for generalizing some results regarding basic Morita equivalences and basic Rickard equivalences between blocks of finite groups. In [3], the definition of the extended Brauer quotient is stated using any subgroup of Aut(P), where P is a p-subgroup of G. For technical reasons, that are clarified later on this paper, we restrict ourselves to the set

K := {φ ∈ Aut(P) | φ(u)u^{−1} ∈ N,∀u ∈ P},

which is actually a subgroup of Aut(P). If φ ∈ K we denote by Δ_{φ} : P → P × P the φtwisted diagonal, that is the group homomorphism defined by Δ_{φ}(u) = (φ(u),u) for any u ∈ P. For any subgroup T of K the T -normalizer N_{G}^{T}(P) of P in G is the inverse image of T through the canonical homomorphism N_{G}(P) → Aut(P).

In Section 2 we make the necessary constructions for defining the extended Brauer quotients of an N-interior G-algebra. Note that Definition 2.1, although it is seems similar to that of [3], it yields an additional action of the whole N_{G}(P) on the extended Brauer quotient. The mentioned action is an important ingredient used in Section 3, since we obtain two correspondences that generalize the correspondence obtained in [3, Proposition 3.3]. Explicitly these are Theorem 3.1 and 3.2. Also for a subalgebra of Â, we obtained in Proposition 3.3 a relationship between the extended Brauer quotient defined in [3] and the one constructed in Section 2.

Our general assumptions and notations are standard. We refer the reader to [5] and [2] for Puig’s theory of G-algebras and pointed groups.

2. The Brauer quotient of N-interior G-algebras

We begin with some useful propositions. With the above notations we have:

Proposition 2.1. The k-algebra A is a kΔ_{φ}(P)-module via the action

(φ(u),u)a:= φ(u)u^{−1} a^{u},

for any u ∈ P,a ∈ A. Proof. Let u_{1},u_{2} ∈ P,a ∈ A. The following equalities are enough to prove the proposition

[(φ(u_{1}),u_{1})(φ(u_{2}),u_{2})]a = φ(u_{1}u_{2})u_{2}^{−1} u_{1}^{−1} a^{u_{1}u_{2}};

(φ(u_{1}),u_{1})[(φ(u_{2}),u_{2})a] = (φ(u_{1}),u_{1})[φ(u_{2})u_{2}^{−1} a^{u_{2}}] = φ(u_{1})u_{1}^{−1} (φ(u_{2})u_{2}^{−1} a^{u_{2}})^{u_{1}} = φ(u_{1})u_{1}^{−1} u_{1}φ(u_{2})u_{1}^{−1} (u_{2}^{−1} )^{u_{1}}a^{u_{1}u_{2}} = φ(u_{1}u_{2})u_{2}^{−1} u_{1}^{−1} a^{u_{1}u_{2}}.

We denote the φ-normalizer of P in A by

N_{A}^{φ}(P) := {a ∈ A | (φ(u),u)a = a,∀u ∈ P}.

Proposition 2.1 implies that this set is the k-subspace of Δ_{φ}(P)-fixed elements, also denoted A^{Δ_{φ} (P)}. As usual, we may consider the Brauer map, which is the canonical surjection

Br_{Δ_{φ} (P)}^{A}: N_{A}^{φ}(P) → N_{A}^{φ}(P)/( ∑_{R<P}Tr_{Δ_{φ} (R)}^{Δ_{φ} (P)}(N_{A}^{φ}(R))),

where Tr_{Δ_{φ} (R)}^{Δ_{φ} (P)} denotes the relative trace map, see [3, 2.5] for more details. We define the external direct sum

N_{A}^{K}(P) := ⨁_{φ∈K} N_{A}^{φ}(P).

Proposition 2.2. With the above notations the following hold.

i) For any φ,ψ ∈ K we have N_{A}^{ψ}(P)N_{A}^{φ}(P) ⊆ N_{A}^{ψ∘φ} (P); ii) The group N_{G}(P) acts on K in the following way. If (x,φ) ∈ N_{G}(P)×K then (x,φ) ↦

φ^{x}, where φ^{x}: P → P with

φ^{x}(u) := xφ(x^{−1} ux)x^{−1}

for any u ∈ P; iii) N_{N}^{K}(P) is a normal subgroup of N_{G}(P), hence the external direct sum N_{A}^{K}(P) is a

N_{N}^{K}(P)-interior N_{G}(P)-algebra.

Proof. i) Let a ∈ N_{A}^{ψ}(P), b ∈ N_{A}^{φ}(P) and u ∈ P. Then ψ(u)u^{−1} a^{u} = a and φ(u)u^{−1} b^{u} = b. Then

(ψ ∘ φ)(u)u^{−1} (ab)^{u} = ψ(φ(u))u^{−1} a^{u}b^{u} = ψ(φ(u))φ(u)^{−1} φ(u)u^{−1} a^{u}uφ(u^{−1} )φ(u)u^{−1} b^{u}
= ψ(φ(u))φ(u^{−1} )(a^{u})^{φ(u)u^{−1}} b = ψ(φ(u))φ(u)^{−1} a^{φ(u)}b = ab.

ii) If x ∈ N_{G}(P) then φ^{x} is clearly in Aut(P). Since for any u ∈ P we have

x^{−1} u^{−1} xφ(x^{−1} ux) ∈ N,

which is equivalent to u^{−1} φ^{x}(u) ∈ N, we obtain that φ^{x} is in K.

iii) Statement i) gives a k-algebra structure on N_{A}^{K}(P). So it is enough to prove that for any x ∈ N_{G}(P) and a ∈ N_{A}^{φ}(P) the element a^{x} belongs to N_{A}^{φ^{x}}(P), since this yields that N_{A}^{K}(P) is an N_{N}^{K}(P)-interior N_{G}(P)-algebra. For any u ∈ P there is u′∈ P such that x^{−1} ux = u′. We have

φ^{x}(u)u^{−1} (a^{x})^{u} = xφ(x^{−1} ux)x^{−1} u^{−1} xx^{−1} a^{ux}
= xφ(u′)u^{′−1}x^{−1} a^{xu′}= (φ(u′)u^{′−1}a^{u′})^{x} = a^{x}.

Lemma 2.3. Let R,R′≤ P and φ,φ′∈ K. If a = Tr_{Δ_{φ} (R)}^{Δ_{φ} (P)}(c) and a′= Tr_{Δ_{φ′}(R′)}^{Δ_{φ′}(P)}(c′) then

aa′= Tr_{Δ_{φ∘φ′}(R′)}^{Δ_{φ∘φ′}(P)}(ac′) = Tr_{Δ_{φ∘φ′}(R′)}^{Δ_{φ∘φ′}(P)}(ca′).

Proof. We know that

a = ∑_{(φ(u),u)∈Δ_{φ} (P)/Δ_{φ} (R)}φ(u)u^{−1} c^{u};

a′= ∑_{(φ′(u),u)∈Δ_{φ′}(P)/Δ_{φ′}(R′)}φ′(u)u^{−1} (c′)^{u}.

The following equalities hold

aa′= ∑_{(φ′(u),u)∈Δ_{φ′}(P)/Δ_{φ′}(R′)}aφ′(u)u^{−1} (c′)^{u}

= ∑_{(φ′(u),u)∈Δ_{φ′}(P)/Δ_{φ′}(R′)}φ(φ′(u))φ′(u^{−1} )a^{φ′(u)}φ′(u)u^{−1} (c′)^{u}

= ∑_{(φ′(u),u)∈Δ_{φ′}(P)/Δ_{φ′}(R′)}φ(φ′(u))u^{−1} uφ′(u^{−1} )a^{φ′(u)}φ′(u)u^{−1} (c′)^{u}

= ∑_{(φ′(u),u)∈Δ_{φ′}(P)/Δ_{φ′}(R′)}φ(φ′(u))u^{−1} (a^{φ′(u)})^{uφ′(u^{−1} )}(c′)^{u}

= ∑_{(φ(φ′(u)),u)∈Δ_{φ∘φ′}(P)/Δ_{φ∘φ′}(R′)} φ(φ′(u))u^{−1} (ac′)^{u}

= Tr_{Δ_{φ∘φ′}(R)}^{Δ_{φ∘φ′}(P)}(ac′).

Similar computations give the second equality from the conclusion of the lemma.

Note that if R ≠ P or R′ ≠ P Lemma 2.3 implies that aa′∈ Ker(Br_{Δ_{φ∘φ′}(P)}^{A}), hence

⨁ _{φ∈K} Ker(Br_{Δ_{φ} (P)}^{A}) is a two-sided ideal of N_{A}^{K}(P). This allows us to consider the following definition.

Definition 2.1. The extended Brauer quotient is the following N_{N}^{K}(P)-interior N_{G}(P)- algebra

\overline{N}_{A}^{K}(P) = N_{A}^{K}(P)/(⨁_{φ∈K} Ker(Br_{Δ_{φ} (P)}^{A}).

The following two results contain some useful properties of the extended Brauer quotient and its computation for a particular N-interior G-algebras. For more details regarding N-interior G-algebras see [2]. The proof of the next proposition is similar to [3, Proposition 3.4] and is left as an exercise for the reader.

Proposition 2.4. Let A,B be two N-interior G-algebras and f : A → B a homomorphism of N-interior G-algebras. Then f induces two homomorphisms of N_{N}^{K}(P)-interior N_{G}(P)- algebras N_{f}^{K}: N_{A}^{K}(P) → N_{B}^{K}(P) and \overline{N}_{f}^{K}: \overline{N}_{A}^{K}(P) → \overline{N}_{B}^{K}(P). Moreover if f is an embedding

then N_{f}^{K} and \overline{N}_{f}^{K} are embeddings too.

Let K′denote the group of interior automorphisms of P induced by conjugation with the elements of N. Clearly it is a subgroup of K.

Proposition 2.5. Let A = kN as an N-interior G-algebra. There exists an isomorphism of N_{N}(P)-interior N_{G}(P)-algebras

ρ^{K′}: kN_{N}(P) → \overline{N}_{A}^{K′}(P).

Proof. It is known that the groups K′and N_{N}(P)/C_{N}(P) are isomorphic. Definition 2.1 assures the existence of an algebra homomorphism

kN_{N}(P) → N̄_{A}^{K′}(P),

which is actually a homomorphism of K′-graded algebras since there is a bijection between K′and any set of representatives of N_{N}(P)/C_{N}(P). Indeed, if x_{φ} ↦ φ denotes the mentioned bijection we have

kN_{N}(P) = ⨁_{φ∈K′}kC_{N}(P)x_{φ},

which is an N_{N}(P)-interior algebra such that

kC_{N}(P)x_{φ} ⊆ A^{Δ_{φ} (P)},

for any x_{φ}. Now, denoting by ρ_{φ}^{K′}the restriction on kC_{N}(P)x_{φ} of the homomorphism of modules

Br_{Δ_{φ} (P)}^{A}: A^{Δ_{φ} (P)} → A(Δ_{φ}(P)),

we see that ρ^{K′}:= ⊕_{φ∈K′}ρ_{φ}^{K′}is, by [1, Proposition 2.12], an N_{G}(P)-algebra isomorphism.

3. The correspondences in the extended Brauer quotient

We begin this section with the following theorem which may be regarded as an extension of [3, Proposition 3.3].

Theorem 3.1. With the above notation we assume that P ≤ N_{G}^{K}(P) and that P ∩N ≠ {1}. If A is an N-interior p-permutation G-algebra which is projective viewed as k(N ×1)-module and as k(1 × N)-module, we have

\overline{N}_{A}^{K}(P)^{P} ⊂ A(P) + J(k(P ∩ N))\overline{N}_{A}^{K}(P).

Moreover, for any subgroup H of G that contains P there is a bijective correspondence between the set of pointed groups H_{β} on A with defect group P and the set of pointed groups N_{H}(P)β̂ on \overline{N}_{A}^{K}(P) with defect group P such that Br_{P}(β) ⊂ β̂. Further, if P is normal in H, setting

Q = {Q ≤ H | Q is a p-subgroup such that P ≤ Q},

then the above bijection restricts to a bijection that preserves the defect groups in Q.

Proof. Similar arguments as the ones proving [2, (3.2.1)] give

J(k(P ∩ N))\overline{N}_{A}^{K}(P) = \overline{N}_{A}^{K}(P)J(k(P ∩ N)) ⊂ J(\overline{N}_{A}^{K}(P)).

Following the same steps as in the proof of [3, Proposition 3.3] we obtain

\overline{N}_{A}^{K}(P)^{H} = A(P)^{H}⨁ ( ⨁_{φ∈K\\{id_{P}\}} J(k(P ∩ N))\overline{N}_{A}^{φ}(P))^{H},

for any subgroup H contained in N_{G}(P). So that any point of H on \overline{N}_{A}^{K}(P) is actually the (\overline{N}_{A}^{K}(P)^{H})^{×} -conjugacy class of a primitive idempotent belonging to A(P)^{H}. Further, if P ≤ Q ≤ H ≤ N_{G}(P), the inclusion

A(P)_{Q}^{H} ⊆ \overline{N}_{A}^{K}(P)_{Q}^{H} = A(P)_{Q}^{H}⨁ ( ⨁_{φ∈K\\{id_{P}\}} J(k(P ∩ N))\overline{N}_{A}^{φ}(P))_{Q}^{H}

shows that the above bijection preserves the defect group Q.

Now let H denote any subgroup of G containing P. Since

Br_{P}(A_{P}^{H}) = A(P)_{P}^{N_{H}(P)},

by lifting idempotents and the above bijection, the first assertion follows. Next, if P is normal in H, for any p-subgroup Q which verifies P ≤ Q ≤ H we have

Br_{P}(A_{Q}^{H}) = A(P)_{Q}^{H},

thus the second assertion holds.

We denote by P̄the group PN/N.

Theorem 3.2. Let A be an N-interior p-permutation G-algebra having an N × N-stable basis as k-module such that it is projective viewed as k(N × 1)-module and as k(1 × N)- module. Let H be any subgroup of G such that P ≤ H and set Ã:= A⊗_{N}N_{G}(P̄). Then there is a bijective correspondence preserving P as defect group between the points of H on Â and the points of N_{H}(P) on N̄K ; Ã(P). Further, if P is normal in H then the above bijection preserves the defect groups in

Q = {Q ≤ H | Q is a p-subgroup such that P ≤ Q}.

Proof. Let B denote a basis of A which is G-stable and N × N-stable. It is easy to verify that the set

{b ⊗ x | b ∈ B and x ∈ [N_{G}(P̄)/N]}

is a N_{G}(P̄) × N_{G}(P̄)-invariant basis of Ã. If ã∈ Nφ ; Ã(P) is in a P × P-stable basis, we have ã= ∑ b⊗x, for some b in B and x ∈ [N_{G}(P̄)/N]. If R is the subgroup of P fixing ã, for any u ∈ R we have

ã= ã^{u} = uφ(u)^{−1} · ã,

which is equivalent to

b ⊗ x = uφ(u)^{−1} · b ⊗ x,

for any pair b ⊗ x appearing in the decomposition of ã, thus φ(u) = u. Also note that Â(P) = Ã(P) as N_{N}^{K}(P)-interior N_{G}^{K}(P)-algebra, see [2, (9.5.4)]. It follows that Tr_{R}^{P}(ã) belongs to J(NK ; Ã(P)) if φ is not the unity element of K. At this point, the surjective homomorphism of p-permutation N_{N}^{K}(P)-interior N_{G}(P)-algebras

NK ; Ã(P) → N̄K ; Ã(P)

forces

( ⨁_{φ∈K\{id_{P}}} N̄φ ; Ã(P))^{P} = ( ⨁_{φ∈K\{id_{P}}} J(k(N ∩ P))N̄φ ; Ã(P))^{P} ⊆ J(N̄K ; Ã(P)).

To complete the proof is suffices to follow the steps of [3, Proposition 3.3] and of Theorem 3.1.

Remark 3.1. Let A = kN as N-interior G-algebra and let H = G. In this case Theorem 3.2 gives a bijective correspondence between the blocks of kG with defect P and the primitive idempotents of kN_{G}^{K}(P)^{N_{G}(P)}with defect P. Moreover since kN_{G}^{K}(P)^{N_{G}(P)}is a subalgebra in the center of kN_{G}(P), and this inclusion preserves the defect groups we obtain the First Main Theorem of Brauer.

Let Aut_{G}(P) be the group of interior automorphisms of P induced by conjugation with elements from G. For the rest of the section (by abuse of notation) we still denote by K the following subgroup

K := {φ ∈ Aut_{G}(P) | φ(u)u^{−1} ∈ N,∀u ∈ P}.

We define the following N_{G}^{K}(P)-interior algebra (and also N_{G}(P)-algebra in this case)

Ā:= A ⊗_{kN}N_{G}^{K}(P)N,

with the multiplication given as before Theorem 3.2. The extended Brauer quotient of this algebra can be given more explicitly. Proposition 3.3. With the above notations we have

\overline{N}K ; Ā(P) = \overline{N}_{A}^{K}(P) ⊗_{N_{N}^{K}(P)}N_{G}^{K}(P),

as N_{G}^{K}(P)-interior N_{G}(P)-algebras.

Proof. Let φ ∈ K. Then there is z^{−1} ∈ N_{G}^{K}(P) such that φ(u) = u^{z^{−1}} for any u ∈ P. First we characterize the elements from Nφ ; Ā(P). Let a ⊗ x be an homogenous element from Ā which is in Nφ ; Ā(P) where x ∈ N_{G}^{K}(P). The definition of Āas N_{G}^{K}(P)-interior algebra assure us that the next statement is true

a^{φ(u)} ⊗ φ(u)xu^{−1} = a ⊗ x.  (1)

We denote by u_{1} the element xux^{−1} ∈ P, hence xu^{−1} = u_{1}^{−1} x. Denote by ψ ∈ K the automorphism of P induced by the conjugation with xz. It is trivially to check that φ(u) = ψ^{−1} (u_{1}). With the above notations, from (1) we obtain

a^{φ(u)} ⊗ φ(u)xu^{−1} = a^{φ(u)} ⊗ φ(u)u_{1}^{−1} x = a^{ψ^{−1} (u_{1})} ⊗ ψ^{−1} (u_{1})u_{1}^{−1} x = a ⊗ x.  (2)

The last equality is equivalent to a^{ψ^{−1} (u_{1})}ψ^{−1} (u_{1})u_{1}^{−1} = a, that is a ∈ N_{A}^{ψ^{−1}} (P). Let c_{x^{−1}} ∈ Aut_{G}(P) be the automorphism induced by the conjugation with x^{−1} ∈ N_{G}^{K}(P). With this notation we have ψ^{−1} = φ ∘ c_{x^{−1}}. The above computation respects direct sums, hence the following equalities are true

⨁_{φ∈K} Nφ ; Ā(P)= ⨁_{φ∈K} ( ⨁_{x∈[N_{G}^{K}(P)/N_{N}^{K}(P)]}N_{A}^{φ∘c_{x^{−1}}} (P) ⊗ x)

= ⨁_{x∈[N_{G}^{K}(P)/N_{N}^{K}(P)]}( ⨁_{φ∈K} N_{A}^{φ∘c_{x^{−1}}} (P) ⊗ x)

= ⨁_{x∈[N_{G}^{K}(P)/N_{N}^{K}(P)]}N_{A}^{K}(P) ⊗ x

To complete the proof it is enough to prove that Tr_{Δ_{φ} (Q)}^{Δ_{φ} (P)}(a ⊗ x) = Tr_{Δ_{φ} (x^{−1} Qx)}^{Δ_{φ} (P)}(a) ⊗ x for any Q < P, since this forces the requested equality. We have

Tr_{Δ_{φ} (Q)}^{Δ_{φ} (P)}(a ⊗ x) = ∑_{(φ(u),u)∈[Δ_{φ} (P)/Δ_{φ} (Q)]}a^{φ(u)} ⊗ φ(u)u^{−1} x^{u}

= ∑_{(ψ^{−1} (u_{1}),u_{1})∈[Δ_{φ} (P)/Δ_{φ} (x^{−1} Qx)]}a^{ψ^{−1} (u_{1})} ⊗ ψ^{−1} (u_{1})u_{1}^{−1} x

= ∑_{(ψ^{−1} (u_{1}),u_{1})∈[Δ_{φ} (P)/Δ_{φ} (x^{−1} Qx)]}a^{ψ^{−1} (u_{1})}ψ^{−1} (u_{1})u_{1}^{−1} ⊗ x

= Tr_{Δ_{φ} (x^{−1} Qx)}^{Δ_{φ} (P)}(a) ⊗ x

For the second equality we used the first part of (2).

References

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[2] L. Puig, Blocks of finite groups: The Hyperfocal subalgebra of a Block, Springer Verlag Berlin Heidelberg, 2002.

[3] L. Puig, Y. Zhou, A local property of basic Morita equivalences, Math. Zeitschrift. 256 (2007), 551-562.

[4] L. Puig, Y. Zhou, A local property of basic Rickard equivalences, J. Algebra. 322 (2009), 1946-1973.

[5] J. Thévenaz, G-algebras and modular representation theory, Oxford Math. Mon. Clarendon Press, Oxford(1995).