On the decomposition map for symmetric groups

Abstract

Let R d be the ?-module generated by the irreducible characters of the symmetric group ${\mathcal{S}}_{d}$ . We determine bases for the kernel of the decomposition map. It is known that R d ? ? F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I br d is the kernel of the decomposition map for prime p then R d /I br d ? ? F is isomorphic to the radical quotient of the p-modular Solomon descent algebra.