Brouéʼs abelian defect group conjecture holds for the double cover of the Higman–Sims sporadic simple group

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture, due to Broué', saying 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 block B of the normaliser NG(P ) of P in G are derived equivalent. We prove in this paper, that Broué's abelian defect group conjecture, and even Rickard's splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman-Sims sporadic simple group. It then turns out that both conjectures hold for all primes p and for all p-blocks of 2.HS.Keywords: Broué's conjecture; abelian defect group; splendid derived equivalence, double cover of the Higman-Sims sporadic simple group.There are several cases where the conjectures 1.1 and 1.2 have been verified, albeit the general conjecture is widely open; for an overview, containing suitable references, see [7]. As for general results concerning blocks with a fixed defect group, by [28,45,49,50] conjectures 1.1 and 1.2 are proved for blocks with cyclic defect groups in arbitrary characteristic.Moreover, in [17, (0.2)Theorem] it is shown that 1.1 and 1.2 are true for the principal block algebra of an arbitrary finite group when the defect group is elementary abelian of order 9. In view of the strategy used in [17], and of a possible future theory reducing 1.1 and 1.2 to the quasi-simple groups, it seems worth-while to proceed with this class of groups, as far as nonprincipal 3-blocks with elementary abelian defect group of order 9 are concerned. Indeed, for these cases there are partial results already known, see [13,20,21,22,24,26,38] for instance. The present paper is another step in that programme, our main result being the following: Theorem 1.3. Let G be the double cover 2.HS of the Higman-Sims sporadic simple group, and let (K, O, k) be a splitting 3-modular system for all subgroups of G. Suppose that A is the faithful block algebra of OG with elementary abelian defect group P = C 3 × C 3 of order 9, and that B is a block algebra of ON G (P ) such that B is the Brauer correspondent of A. Then, A and B are splendidly derived equivalent, hence the conjectures 1.1 and 1.2 of Broué and Rickard hold.As an immediate corollary we get:Corollary 1.4. Broué's abelian defect group conjecture 1.1, and even Rickard's splendid equivalence conjecture 1.2 are true for all primes p and for all block algebras of OG.Our strategy to prove 1.3 is similar to the ones pursued, for example, for the Janko sporadic simple group J 4 in [22, 1.6.Theorem] or the Harada-Norton sporadic simple group HN in [24, 1.3.Theorem]. Our starting point was actually to realise that the 3-decomposition matrix of A coincides (up to a suitable order of rows and columns) with the 3-decomposition matrix of the principal 3-block A ′ of the alternating group A 8 on 8 letters:

Broué's abelian defect group conjecture and 3-decomposition numbers of the sporadic simple Conway group Co_1

“…Hence, recalling the notion from [27,Remark 1.6(c)], this shows that A is pseudo-principal. Moreover, as an immediate corollary we get:…”

“…In view of the strategy used in [18], and of a possible future theory reducing conjectures 1.1 and 1.2 to the quasi-simple groups, it seems worthwhile to proceed with this class of groups, as far as non-principal 3-blocks with elementary abelian defect group of order 9 are concerned. Indeed, for these cases there are partial results already known, see [14,21,22,23,25,30,27,28,41] for instance; all the non-principal 3-blocks with elementary abelian defect group of order 9 for the sporadic simple groups and their covering groups are given in [43].…”

Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

“…In case p = 2 or |D| = 9, the conjecture is known to hold for principal blocks (see [12,9,20]). Moreover, Broué's Conjecture holds for p-solvable groups G and some blocks of sporadic groups G (see [14,21,26,25,24,31,23,22,15]).…”