Vol. 6, No. 4, 2012

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Block components of the Lie module for the symmetric group

Roger M. Bryant and Karin Erdmann

Vol. 6 (2012), No. 4, 781–795

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) SkSr)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).

Lie module, symmetric group, Lie power, Schur algebra, block
Mathematical Subject Classification 2010
Primary: 20C30
Secondary: 20G43, 20C20
Received: 10 March 2011
Revised: 8 June 2011
Accepted: 6 July 2011
Published: 25 July 2012
Roger M. Bryant
School of Mathematics
University of Manchester
Oxford Road
M13 9PL
United Kingdom
Karin Erdmann
Mathematical Institute
University of Oxford
24-29 St. Giles
United Kingdom