David J. Hemmer

SPECHT FILTRATIONS FOR HECKE ALGEBRAS OF TYPE A

Let Hq (d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive lth root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l 4 the multiplicities in a Specht or dual Specht module filtration of an Hq (d)-module are well defined. A cohomological criterion is given for when an Hq (d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p 5. These results are false in general for p = 2 or 3.

Support varieties for modules over symmetric groups

1.1. In the late 1970s, Alperin [A] defined an invariant called the complexity of a module as a way to relate the modules with the complexes and resolutions that they admit. Several years later, Carlson [Ca1,Ca2] defined affine algebraic varieties corresponding to modules over group algebras. These varieties are subvarieties of the spectrum of the cohomology ring which was earlier described by Quillen [Q]. They are known in present day language as support varieties. It was discovered early on that the complexity of a module is equal to the dimension of the support variety of the module. Geometric methods involving support varieties have played a fundamental role in understanding the interplay between the modular representation theory and cohomology for finite groups. Despite substantial progress in this direction, there have been few explicit computations of support varieties for important classes of modules over certain groups. The goal of this paper is to introduce methods and techniques for computing support varieties for modules over the symmetric group Σd . In the process, we will provide explicit computations of support varieties for certain classes

Symmetric Group Modules with Specht and Dual Specht Filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology. Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.

Representation type of Schur superalgebras

Abstract Let S(m|n, d) be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup GL(m|n) of degree d. In this paper we determine the representation type of these algebras (that is, we classify the ones which are semisimple, have finite, tame and wild representation type). Moreover, we prove that these algebras are in general not quasi-hereditary and have infinite global dimension.

The complexity of certain Specht modules for the symmetric group

During the 2004–2005 academic year the VIGRE Algebra Research Group at the University of Georgia (UGA VIGRE) computed the complexities of certain Specht modules Sλ for the symmetric group Σd, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The UGA VIGRE Algebra Group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of λ is built out of p×p blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the p-weight of partitions and branching.

The Lie module and its complexity

The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group

Stable decompositions for some symmetric group characters arising in braid group cohomology

\Sigma_n

Irreducible Specht modules are signed Young modules

, the Lie module

THE GROUP OF ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS

\mathsf{Lie}(n)

On the cohomology of Young modules for the symmetric group

has attracted a great deal of interest in recent years. We prove here that the complexity of