Studies on properties of Brauer-friendly modules and slash functors

概要

This thesis is based on [23] and [22].
Let p be a prime number and (K, O, k) a p-modular system such that k is algebraically closed. Throughout this thesis, RG-modules mean finitely generated RG-lattices, for R ∈{O, k}. In the modular representation theory of finite groups, the following philosophy exists such as the local-global principle : Representations of a finite group are controlled by represen- tations of p-local subgroups of the group. One of the specific formulations of this philosophy is known to be the following Brou´e’s conjecture.

Conjecture (Brou´e’s conjecture). Let G be a finite group, b a block of RG with a defect group P, and c the Brauer correspondent of b in RNG(P ). If P is abelian, then the block algebras RGb and RNG(P )c are derived equivalent.

This conjecture is one of the most important problems and has been studied by many re- searchers, in the modular representation theory of finite groups. It is known that the conjecture holds in many blocks(see [18, 5.2.2]). In many cases, Okuyama’s method, introduced in [17], played an important role. It is a method of constructing a derived equivalence from a stable equivalence of Morita type. The details of the method may be found in [17]. From the method, constructing a stable equivalence of Morita type between the block algebras RGb and RNG(P )c can be used to prove Brou´e’s conjecture. We review the gluing principle of constructing stable equivalences of Morita type for principal blocks and general blocks.

First, we consider the case where b is the principal block of RG. In this case, M. Brou´e introduced the following method which is useful for constructing a stable equivalence of Morita type.

Theorem 1.0.1 (Brou´e’s gluing principle [6, 6.3. Theorem]). Let G and H be finite groups having a common Sylow p-subgroup P such that FP (G) = FP (H). Let b and c be the principal blocks of RG and RH, respectively. For any subgroup Q of P, let bQ and cQ be the principalblocks of kCG(Q) and kCH(Q), respectively, and M = S(G×H, ∆P ) the Scott R(G×H)-module with vertex ∆P. Then the following are equivalent.
(i) The bimodule M and its dual M∗ induce a stable equivarence of Morita type between RGband RHc.
(ii) For each non-trivial subgroup Q of P, the bimodule BrQ(M ) and its dual BrQ(M )∗ induce a Morita equivalence between kCG(Q)bQ and kCH(Q)cQ.

In [11], R. Kessar, N. Kunugi, and N. Mitsuhashi introduced the notation of Brauer inde- composability, which plays a key role when we apply the principle to principal blocks.

Definition 1.0.2 ([11]). Let M be an indecomposable RG-module. We say that M is Brauer indecomposable if ResNG(Q)/QG(BrQ(M )) is indecomposable or 0, for any p-subgroup Q of G.

In [9], H. Ishioka and N. Kunugi gave an equivalent condition for Scott modules to be Brauer indecomposable.

Theorem 1.0.3 ([9, Theorem 1.3]). Let G be a finite group and P a p-subgroup of G. Let M = S(G, P ) and suppose that 7 = 7P (G) is saturated. Then the following conditions are equivalent.
(i) M is Brauer indecomposable.
(ii) ResNG(Q)Gof P.(S(NG(Q), NP (Q))) is indecomposable, for each fully 7-normalized subgroup Q

If one of the equivalent conditions is satisfied, then BrQ(M ) ~= S(NG(Q), NP (Q)) for each fully7-normalized subgroup Q of P.

Next, we consider the case where b is a general block of RG. M. Linckelmann generalized Brou´e’s gluing principle to general blocks.

Theorem 1.0.4 (Linckelmann’s gluing principle [14, Theorem 1.2]). Let G and H be finite groups and b and c blocks of RG and RH, respectively, with a common defect group P. Let i ∈ (RGb)∆P and j ∈ (RH)∆P be almost source idempotents. For any subgroup Q of P, denote by eQ and fQ the unique blocks of kCG(Q) and kCH(Q), respectively, satisfying Br∆Q(i)eQ /= 0 and Br∆Q(j)fQ /= 0. Denote by eˆQ and fˆQ the unique blocks of &CG(Q) and &CH(Q) lifting eQ and fQ, respectively. Suppose that 7(P,eˆP )(G, b) = 7(P,fˆP )(H, c), and write 7 = 7(P,eˆP )(G, b). Let V be an 7-stable indecomposable endo-permutation RP-module with vertex P, viewed as an R∆P-module through the canonical isomorphism ∆P ~= P. Let M be an indecomposable direct summand of the RGb-RHc-bimodule

RGi ⊗RP IndP ×P (V ) ⊗RP jRH.

Suppose that M has ∆P as a vertex as an R[G × H]-module. Then for any non-trivial sub- group Q of P, there is a canonical kCG(Q)eQ-kCH(Q)fQ-module MQ satisfying Endk(MQ) ~= Br∆Q(EndR(eˆQM fˆQ)). Moreover, if for all non-trivial subgroups Q of P the bimodule MQ in- duces a Morita equivalence between kCG(Q)eQ and kCH(Q)fQ, then M and its dual M∗ induce a stable equivalence of Morita type between RGb and RHc.

In [7], E. C. Dade introduced slash constructions for endo-permutation modules. In [3], E. Biland defined Brauer-friendly modules and generalized slash constructions to slash functors for Brauer-friendly modules. Brauer-friendly modules are generalizations of (endo-)p-permutation modules. The module M which appears in the theorem above is a Brauer-friendly module, and the module MQ which appears in the theorem can be represented as Sl(∆Q,eˆQ⊗fˆQ)(M ) by usinga (∆Q, eˆQ ⊗ fˆQ)-slash functor Sl(∆Q,eˆ⊗fˆQ). For Brauer-friendly modules, the slash indecom-posability can be defined in a similar way as Brauer indecomposability (For Frobenius-friendlymodules (i.e. endo-p-permutation modules), the slash indecomposability has been defined by Feng-Li [8]). The slash indecomposability plays an important role in Linckelmann’s gluing principle.

Our first result is that we generalize Ishioka-Kunugi’s equivalent condition to an equivalent condition for Brauer-friendly modules to be slash indecomposable.

We have the following relation:
where AMod is the category of all A-modules. In general, the functor on the right side is not essentially surjective. There exist kG-modules which have corresponding &G-modules. These modules are said to be liftable, i.e. a kG-module M is said to be liftable if there exists an &G- module M such that k ⊗O M ~= M . In this situation, M is called a lift of M . Liftability is one of the properties that we want to be satisfied in order to study the structure of a kG-module. If a module is liftable, then we can construct the ordinary character corresponding to a lift of the module, from the above relation. Then we can examine the structure of the module using the ordinary character. Therefore, in the modular representation theory of finite groups, it is important to find a class of liftable modules. A few classes of liftable modules are known. For example, any p-permutation kG-module is liftable, in particular, any projective kG-module is liftable. Moreover, they lift to a p-permutation &G-module and a projective &G-module, respectively. In addition, any endo-permutation kG-module is liftable to an endo-permutation &G-module. More details on these examples may be found in [13, 1. Introduction]. In [21], J.-M. Urfer introduced endo-p-permutation modules, which are generalizations of endo-permutation modules. In [13], C. Lassueur and J. Th´evenaz proved that any endo-p-permutation kG-module is liftable to an endo-p-permutation &G-module.

Theorem 1.0.5 ([13, Theorem 4.2]). Let M be an indecomposable endo-p-permutation kG- module and P a vertex of M. Then there exists an indecomposable endo-p-permutation &G- module M^ with vertex P such that M^/pM^ ~= M.

By [21, Theorem 1.5], any endo-p-permutation RG-module has a G-stable endo-permutati- on source. Hence, in [13, Remark 4.3], C. Lassueur and J. Th´evenaz raised the question of whether or not kG-modules with an endo-permutation source which is not necessarily G-stable are liftable. In [3], E. Biland introduced Brauer-friendly modules, which are generalizations of endo-p-permutation modules. Any Brauer-friendly module has an endo-permutation source. From the question and since Brauer-friendly modules may induce a stable equivalence of Morita type between &Gb and &Hc by Linckelmann’s gluing principle, we want to know the liftability of Brauer-friendly modules.

Our second result is that we show that any indecomposable Brauer-friendly module sat- isfying certain condition is liftable to an indecomposable Brauer-friendly module, which is a generalization of the main theorem of [13].