Title: On the cohomology of Young modules for the symmetric group

 

Authors: Frederick R. Cohen, David J. Hemmer, Daniel K. Nakano

 

Status: Advances in Mathematics, to appear.

 

Abstract:

 

  The main result of this paper is an application of the topology of

the space Q(X) to obtain results for the cohomology of the symmetric

group on d letters, \Sigma_d, with `twisted' coefficients in various choices
of

Young modules and to show that these computations reduce to certain

natural questions in representation theory. The authors extend

classical methods for analyzing the homology of certain spaces

Q(X) with mod-p coefficients to describe the homology

\HH_{\bullet}(\Sigma_d, V^{\otimes d}) as a module for the general

linear group GL(V) over an algebraically closed field k of

characteristic p. As a direct application, these results provide a

method of reducing the computation of

\text{Ext}^{\bullet}_{\Sigma_{d}}(Y^{\lambda},Y^{\mu}) (where

Y^{\lambda}, Y^{\mu} are Young modules) to a representation

theoretic problem involving the determination of tensor products and

decomposition numbers. In particular, in characteristic two, for

many d, a complete determination of \Hs Y^\lambda) can be found.

This is the first nontrivial class of symmetric group modules where

a complete description of the cohomology in all degrees can be

given.

 

For arbitrary d the authors determine \HH^i(\Sigma_d,Y^\lambda)

for i=0,1,2. An interesting phenomenon is uncovered--namely a

stability result reminiscent of generic cohomology for algebraic

groups. For each i the cohomology \HH^i(\Sigma_{p^ad},

Y^{p^a\lambda}) stabilizes as a increases. The methods in this

paper are also powerful enough to determine, for any p and

\lambda, precisely when \HH^{\bullet}(\sd,Y^\lambda)=0. Such

modules with vanishing cohomology are of great interest in

representation theory because their support varieties constitute the

representation theoretic nucleus.