EXAMPLES OF BROUÉ’S ABELIAN

DEFECT GROUP CONJECTURE IN

REPRESENTATION THEORY OF FINITE GROUPS

Shigeo KOSHITANI (越谷重夫)^{1}

Abstract. In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding p-block B of N_{G}(P) are derived equivalent. We survey in this article that Broué’s conjecture holds for nonprincipal 3-blocks A with elementary abelian defect group P of order 9 for several sporadic simple groups.

0. Introduction

I believe nobody would disagree that in modular representation theory of finite groups Richard Brauer (1901–77) was a real pioneer [13]. He actually gave a nice survey talk in early sixties [5]. I guess many people would agree that now in representation theory of finite groups three of the most important problems should be the following. Namely,

Conjecture 1. Alperin’s weight conjecture (1986) [3] Conjecture 2. Dade’s conjecture (1990) [15], [16], [17], [18], [19] Conjecture 3. Broué’s abelian defect group conjecture (1988) [6], [7], [8],

[9], [26]

These three conjectures have actually origins which had already appeared in the Brauer’s problems in [5]. We anyway need some notation and terminology.

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

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

This paper will not be submitted for publication anywher else. 1The 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, 2002–2003.

(0.1) Notation and assumption. Let k be an algebraically field of characteristic p > 0, G is always a finite group, and modules here mean always finitely generated right modules unless stated otherwise. Let A be a block algebra of the group algebra kG, namely, A is an indecomposable two-sided ideal of kG which is a direct summand of _{kG}kG_{kG}. Then, it is well-known that up to G-conjugacy a unique p-subgroup P of G is attached to the block algebra A, which is called a defect group of A, see for instance [43]. Now, let H = N_{G}(P). Then, by Brauer’s first main theorem, there uniquely exists a block algebra B of kH, which has the same defect group P and A and B correspond each other via so-called the Brauer homomorphism. Since many years ago, it has been predicted the following.

(0.2) Question. How do A and B resemble each other?

As a matter of fact, the above three conjectures are all on this question. In this survey note we concentrate on the third conjecture, that is, Broué’s abelian defect group conjecture. Then, what is Broué’s abelian defect group conjecture? To state it let’s move on to the next section.

1. Broué’s abelian defect group conjecture

(1.1) Broué’s abelian defect group conjecture ([6, 6.2.Question], [8, 4.9.Conjecture], [56, §9.2.4, Conjecture], [62, §5.2]). We use the notation and assumption in (0.1). Moreover, if the defect group P is abelian, then the block algebras A and B should be derived equivalent (Rickard equivalent), that is to say, the categories D^{b}(mod-A) and D^{b}(mod-B) are equivalent as triangulated categories, where D^{b}(mod-A) is a bounded derived category of a category mod-A of a finitely generated right kG-modules in A.

There are several (not many) cases where Broué’s abelian defect group conjecture has been checked. For example, the author proves the next with N. Kunugi by using many initiated results which are done by L. Puig, T. Okuyama, T. Okuyama and K. Waki, N. Kunugi, the author and H. Miyachi and the author and N. Kunugi, see [50], [44], [45], [47], [48], [49], [27], [32], [33], [34], [11]. Namely,

(1.2) Theorem (Koshitani-Kunugi [28]. If A is the principal block algebra with elementary abelian defect group P of order 9, then Broué’s abelian defect group conjecture is true.

It should be noted that to get the above result (1.2) the classification of finite simple groups is necessary.

For other results of Broué’s abelian defect group conjecture, see the following Web Page done by Jeremy Rickard:

      http://www.maths.bris.ac.uk/∼majcr/adgc/adgc.html

2. Broué’s abelian defect group conjecture for non-principal 3-blocks

Since Broué’s abelian defect group conjecture (1.1) has been checked for principal block algebras A with elementary abelian defect group P ≅ C_{3}×C_{3} of order 9, one of the next objects might be the case where A is a nonprincipal block algebra with the same defect group P ≅ C_{3} × C_{3}. On this direction the author with N. Kunugi and K. Waki proves that Broué’s abelian defect group conjecture holds in the following special cases. Namely,

(2.1) Theorem (Koshitani, Kunugi and Waki). Let A be a non-principal block algebra of kG with elementary abelian defect group P of order 9. Furthermore, if G is one of the following four sporadic simple groups, then Broué’s abelian defect group conjecture is true:

(i) G = O′N (O’Nan simple group)

(ii) G = HS (Higman-Sims simple group)

(iii) G = He (Held simple group)

(iv) G = Suz (sporadic Suzuki group).

Proof. For entire proofs see our papers [30] and [31].

(2.2) Final remark. In this article, perhaps the author has not been able to give a sufficient explanation. Therefore, the author would like the readers to consult papers listed below, especially, for instance [8], [13], [21], [26], [39], [42], [46], [55], [56], [57], [58], [59], [60], [61] and [62].

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