Block components of the Lie module for the symmetric group - Bryant, R; Erdmann, K (2012)
Let F be a field of prime characteristic p and let B be a nonprincipal block of the group algebra F S r of the symmetric group S r. 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 {does not divide} r. Write r = p m k, where p {does not divide} k, and let S z.ast;k be the diagonal of a Young subgroup of S r isomorphic to S k × · · · × S k. We show that p m Lie(r) B = (Lie(k)↑ SrS z.ast;k)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 L r (E) of the natural module E for the general linear group GL n (F). © 2012 by Mathematical Sciences Publishers.