Algebra & Number Theory

Block components of the Lie module for the symmetric group

Roger Bryant and Karin Erdmann

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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 pr. Write r=pmk, where pk, 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).

Article information

Source
Algebra Number Theory, Volume 6, Number 4 (2012), 781-795.

Dates
Received: 10 March 2011
Revised: 8 June 2011
Accepted: 6 July 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729823

Digital Object Identifier
doi:10.2140/ant.2012.6.781

Mathematical Reviews number (MathSciNet)
MR2966719

Zentralblatt MATH identifier
1247.20011

Subjects
Primary: 20C30: Representations of finite symmetric groups
Secondary: 20G43: Schur and $q$-Schur algebras 20C20: Modular representations and characters

Keywords
Lie module symmetric group Lie power Schur algebra block

Citation

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


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