Block components of the Lie module for the symmetric group

Abstract

Let F be a field of prime characteristic p and let B be a nonprincipal block of the group algebra FSr of the symmetric group Sr. The block component Lie(r)B of the Lie module Lie(r) is projective, by a result of Erdmann and Tan, although Lie(r) itself is projective only when p∤r. Write r=pmk, where p∤k, and let Sk∗ be the diagonal of a Young subgroup of Sr isomorphic to Sk×⋯×Sk. We show that pmLie(r)B≅(Lie(k)↑Sk∗Sr)B. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of Lie(r)B. Corresponding results are obtained, when F is infinite, for the r-th Lie power Lr(E) of the natural module E for the general linear group GLn(F).