Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

Danz, Susanne; Külshammer, Burkhard

doi:10.1016/j.jalgebra.2009.03.035

Cited by 6 publications

(3 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First e 0 · Ind G N (1 1 ) = χ 14 + χ 15 + χ 19 , so that by the degree and character value criteria we obtain that Γ G (1 1 ) affords e 0 · Ind Now, the natural permutation kG-module on 23 points has a 22-dimensional composition factor S 22 , which must be a direct summand. Therefore S 22 is a trivial source module with vertex P and its kN-Green correspondent is simple (see [DK09,§3.7]). Thus it must have dimension at most 2.…”

Section: The Torsion Subgroup T T (G) Of T (G) In Odd Characteristicmentioning

confidence: 99%

Endotrivial modules for the sporadic simple groups and their covers

Lassueur

Mazza

2015

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

Section: The Torsion Subgroup T T (G) Of T (G) In Odd Characteristicmentioning

confidence: 99%

Endotrivial modules for the sporadic simple groups and their covers

Lassueur

Mazza

2015

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…Moreover, Ã has an irreducible ordinary character χ13 of degree 10, and Ã has a simple kM -module S of dimension 10 corresponding to χ13 . Now, it follows from[9, Proposition 3.19] that S has a trivial source. On the other hand, a computation in GAP[10] shows χ13 ↑ G = χ 32 .…”

mentioning

confidence: 98%

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

Koshitani¹,

Müller²,

Noeske³

2013

Journal of Algebra

View full text Add to dashboard Cite

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:

show abstract

“…Introduction 1.1. In [3] Danz and Külshammer, investigating the simple modules for the large Mathieu groups, have found two blocks with noncyclic defect groups of order 9 where all the simple modules have trivial sources and whose source algebras are isomorphic to the source algebras of the corresponding blocks of their inertial subgroups [3, Theorems 4.3 and 4.4] †.…”

mentioning

confidence: 99%