The Brauer characters of the sporadic simple Harada-Norton group and its automorphism group in characteristics 2 and 3

We determine the 2-modular and 3-modular character tables of the sporadic simple Harada-Norton group and its automorphism group.Comment: 29 page

Brou\'e's abelian defect group conjecture holds for the double cover of the Higman-Sims sporadic simple group

In the representation theory of finite groups, there is a well-known and important conjecture, due to Brou\'e 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 N_G(P) of P in G are derived equivalent. We prove in this paper, that Brou\'e'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.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1011.442

Brou\'e's abelian defect group conjecture holds for the Harada-Norton sporadic simple group HNHN

In representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. 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 block $B$ of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent (Rickard equivalent). This conjecture is called Brou\'e's abelian defect group conjecture. We prove in this paper that Brou\'e's abelian defect group conjecture is true for a non-principal 3-block $A$ with an elementary abelian defect group $P$ of order 9 of the Harada-Norton simple group $HN$. It then turns out that Brou\'e's abelian defect group conjecture holds for all primes $p$ and for all $p$-blocks of the Harada-Norton simple group $HN$.Comment: 36 page

Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects

The mini-workshop focussed on the modular representation theory of the symmetric group and other closely related objects, including Hecke algebras and Schur algebras. The topics and problems discussed include computations of support varieties, vertices and sources for natural choices of symmetric group modules such as simple modules, Specht modules, and Lie modules, results on Carter–Payne homomorphisms and irreducible Specht modules, connections of symmetric group cohomology with algebraic group cohomology and algebraic topology, and positions of natural symmetric group modules in the Auslander–Reiten quiver

Vertices of Lie Modules

Let Lie(n) be the Lie module of the symmetric group S_n over a field F of characteristic p>0, that is, Lie(n) is the left ideal of FS_n generated by the Dynkin-Specht-Wever element. We study the problem of parametrizing non-projective indecomposable summands of Lie(n), via describing their vertices and sources. Our main result shows that this can be reduced to the case when n is a power of p. When n=9 and p=3, and when n=8 and p=2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers.Comment: 26 page