ON INDECOMPOSABILITY OF A MODULE

GIVEN BY BRAUER CONSTRUCTION

Shigeo KOSHITANI (越谷重夫)

Abstract. In representation theory of finite groups one of the most important and well-known conjectures is Broué’s abelian defect group conjecture. In this note we introduce a sort of technical result which is useful to prove Broué’s abelian defect group conjecture for a few examples which are new.

1. Introduction

In representation theory of finite groups, there has been a very important problem, namely,

How are representations of a finite group G over a field k of prime characteristic p similar to those of its subgroup H containing a Sylow p-subgroup P of G?

representations of G over k ↕ representations of H over k

An origin of this problem is, of course, due to Richard Brauer (1901–77). From this point of view, one of the most important and interesting (and also well-known) conjectures in representation theory of finite groups is Broué’s abelian defect group conjecture. Actually, Michel Broué conjectures the following.

(1.1) Broué’s abelian defect group conjecture, (see [2], [3] and [5]) . For any prime p, if a block algebra A of OG has an abelian defect group P then A and its Brauer corresponding block algebra B in ON_{G}(P) should be derived equivalent, that is,

D^{b}(mod-A) ≃ D^{b}(mod-B)
equivalent

1991 Mathematics Subject Classification. primary 20C20, 20C05; secondary 20C34, 20C15.

Key words and phrases. Brauer construction, block, Broué’s abelian defect group conjecture, modular representation theory, finite group, derived equivalence, splendid Rickard equivalence.

The author was in part supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for JSPS Fellows 01016, 2002–2003; and the JSPS (Japan Society for Promotion of Science), Grant-in-Aid for Scientific Research C(2) 14540009, 2002–2004.

The detailed final vertion of this paper will be published, see [7].

as triangulated categories, where O is a complete discrete valuation ring whose residue field is an algebraically closed field k of characteristic p, and mod-A is the category of finitely generated right A-modules, and D^{b}(𝔄) is the bouded derived category of an abelian category 𝔄.

As well-known, there is a beautiful result of Jeremy Rickard ([9], [10]), which characterizes such a derived equivalence completely and which is a generalization of a Morita equivalence from modules to complexes. As a matter of fact, in (1.1) above, a stronger conclusion is expected. Namely, derived equivalent should be replaced by splendidly Rickard equivalent, which is due to Jeremy Rickard [11].

In this note we present a theorem which is a sort of technical one, but also useful to prove Broué’s abelian defect group conjecture, which is used in a work of the author, Kunugi and Waki [6].

Actually, we have a result which is a joint work with Markus Linckelmann. We present it in the next section.

2. The main result

First we introduce notation and terminology we need to state our main result.

(2.1) Notation and assumption. Let G be a finite group, and let O and k be as in (1.1). We call an OG-lattice of finite rank a finitely generated OG-module. For any subgroup L of G let ΔL = {(ℓ, ℓ) ∈ L × L | ℓ ∈ L}, and we consider OG as a right O[ΔL]-module via a(ℓ, ℓ) = ℓ^{−1}aℓ for a ∈ OG and ℓ ∈ L. Then, we next define (OG)^{ΔL} as (OG)^{ΔL} = {a ∈ OG | a(ℓ, ℓ) = afor∀ℓ ∈ L} = {a ∈ OG | ℓ^{−1}aℓ = afor∀ℓ ∈}. For a p-subgroup R of G let Br_{ΔR} be the Brauer map (homomorphism) with respect to R. Namely, Br_{ΔR} is defines as

Br_{ΔR} : (OG)^{ΔR} ⟶ kC_{G}(R), ∑_{g∈G} α_{g}g ↦ ∑_{g∈C_{G}(R)} ᾱ_{g}g,

where α_{g} ∈ O and ᾱ_{g} is its image in k

Note the Brauer map Br_{ΔR} is a surjective algebra-homomorphism. For a p-subgroup R which is contained in both of G and L and an (OG, OL)-bimodule Y we define the Brauer construction (quotient) Y (ΔR) by

Y (ΔR) = (Y^{ΔR}/ ∑_{R′≨R} Tr↑_{ΔR′}^{ΔR}(Y^{ΔR′})) ⊗_{O} k

where Tr is the transfer (trace) map. Clearly, the Brauer construction ?(ΔR) induces an additive functor from the category OG-mod-OL of the finitely generated (OG, OL)-bimodules to the category kC_{G}(R)-mod-kC_{L}(G). In particular if Y = OG we get Y (ΔR) = (OG)(ΔR) ≅ kC_{G}(R), and we identify these since we have a commutative diagram

(OG)^{ΔR} ⟶^{Br_{ΔR}} kC_{G}(R)
\begin{matrix}canonical \\ epimorphism \end{matrix} ↓  ∥
(OG)(ΔR) ⟶_{≃} kC_{G}(R)

First, let (P, e) be a Brauer pair in G, that is, P is a p-subgroup of G and e is a block of kC_{G}(P). Then, by results of Alperin-Broué [1, Theorem 3.4] and Broué-Puig [4, Theorem 1.8], it follows that, for any subgroup Q of P there uniquely exists a block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⩽ (P, e). For other notation and terminology, see books of Nagao-Tsushima [8] and Thévenaz [12].

Now, we can state the main result of this note; that is,

(1.2) Theorem(Koshitani-Linckelmann, see [7]). Let b be a block (block idempotent) of a block algebra A of OG (so that A = bOG), and let (P, e) be a maximal b-Brauer pair in G, and hence P is a defect group of b and e is a block of kC_{G}(P) such that Br_{ΔP} (b)e = e. Next, let H = N_{G}(P, e) = {g ∈ N_{G}(P) | g^{−1}eg = e} be the inertial group of e in N_{G}(P). For each subgroup Q let e_{Q} be the same as above, and let f_{Q} be a block of kC_{H}(Q) such that (Q, f_{Q}) ⩽ (P, e) since C_{G}(P) = C_{H}(P). Moreover, let f be a primitive idempotent of (bOG)^{ΔH} such that Br_{ΔP} (f)·e = e, and let X = OGf. Then we have the following.

(i) X is an indecomposable O[G × H]-module with vertex ΔP. (ii) If Q is a subgroup of Z(P), the center of P, then e_{Q}·X(ΔQ)·f_{Q} is a unique indecomposable direct summand of e_{Q}kC_{G}(Q)f_{Q} with vertex ΔP as a (kC_{G}(Q), kC_{H}(Q))- bimodule.

References

[1] J. L. Alperin, M. Broué, Local methods in block theory, Ann. of Math. (2) 110 (1979), 143–157.

[2] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181/182 (1990), 61–92.

[3] M. Broué, Equivalences of blocks of group algebras, in: ”Finite Dimensional Algebras and Related Topics”, V. Dlab and L. L. Scott (eds), Kluwer Acad. Pub. (1994), 1–26.

[4] M. Broué, L. Puig, Characters and local structure in G-algebras, J. Algebra 73 (1980), 306–317.

[5] S. König, A. Zimmermann, Derived equivalences for group rings, Springer Lecture Notes in Math. 1685 (1998).

[6] S. Koshitani, N. Kunugi, K. Waki, Broué’s abelian defect group conjecture for the Held group and the sporadic Suzuki group, J.?Algebra 279 (2004), 638–666.

[7] S. Koshitani, M. Linckelmann, The indecomposability of a certain bimodule given by the Brauer construction, J.?Algebra, in press (2005).

[8] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1988.

[9] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.

[10] J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331–358.

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

Department of Mathematics and Informatics, Faculty of Science, Chiba University, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan koshitan@math.s.chiba-u.ac.jp