Algebra & Number Theory
- Algebra Number Theory
- Volume 6, Number 4 (2012), 781-795.
Block components of the Lie module for the symmetric group
Let be a field of prime characteristic and let be a nonprincipal block of the group algebra of the symmetric group . The block component of the Lie module is projective, by a result of Erdmann and Tan, although itself is projective only when . Write , where , and let be the diagonal of a Young subgroup of isomorphic to . We show that . Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of . Corresponding results are obtained, when is infinite, for the -th Lie power of the natural module for the general linear group .
Algebra Number Theory, Volume 6, Number 4 (2012), 781-795.
Received: 10 March 2011
Revised: 8 June 2011
Accepted: 6 July 2011
First available in Project Euclid: 20 December 2017
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Bryant, Roger; Erdmann, Karin. Block components of the Lie module for the symmetric group. Algebra Number Theory 6 (2012), no. 4, 781--795. doi:10.2140/ant.2012.6.781. https://projecteuclid.org/euclid.ant/1513729823