Broué's Conjecture Holds for Principal 3-Blocks with Elementary Abelian Defect Group of Order 9

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 finite group G has an abelian Sylow p-subgroup P, then the derived categories of the principal p-blocks of G and of the normalizer N G P of P in G are equivalent. We prove in this paper that Broué's conjecture holds for the principal 3-block of an arbitrary finite group G with an elementary abelian Sylow 3-subgroup P of order 9, by using initiated works for the case where G is simple, which are due to Puig, Okuyama, Waki, Miyachi, and the authors. The result depends on the classification of finite simple groups.  2002 Elsevier Science (USA)

Eigenvector matrices of Cartan matrices for finite groups

“…The following result is due to Koshitani, Kunugi [7] and many authors' results (e.g. [8,9,12,13]) through proving Broué's abelian defect group conjecture to be true.…”

“…P Remark 1. There are small misprints in [7,Lemma 5.7]. In Cases 2 and 4 the small star marks should be the big star marks.…”

“…For Cartan matrices of these groups G, see "decomposition matrices" in "The Modular Atlas" homepage [18]. Note that [7,Lemmas 5.4,5.7,5.8] just suggest (a) → (e) to be true.…”

Eigenvalues of Cartan matrices of principal 2-blocks with abelian defect groups

“…We proved that Conjecture is true if B is a cyclic block or a tame block and we proved (b) and (c) are equivalent if G is a p-solvable group in [13]. Furthermore, in [27] we proved that Conjecture is true if p = 3 and G is a finite group with an elementary abelian Sylow 3-subgroup of order 9 and B is the principal 3-block of G by making use of some results of [15]. In this case, we can take the Brauer character table Φ b of b as a unimodular eigenvector matrix U B of C B over a complete discrete valuation ring R.…”

The Broué conjecture for the faithful 3-blocks of 4.M22

“…[15,21], where in many cases the appropriate stable equivalence is simply restriction. In particular, building on work of several authors it is shown in [9] that Broué's conjecture holds for principal 3-blocks in the case of elementary abelian 3-Sylow subgroups of order 9.…”