Block components of the Lie module for the symmetric group

Bryant, Roger; Erdmann, Karin

doi:10.2140/ant.2012.6.781

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Block components of the Lie module for the symmetric group

Roger M. Bryant and Karin Erdmann

Vol. 6 (2012), No. 4, 781–795

DOI: 10.2140/ant.2012.6.781

Abstract

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 $L i e {(r)}_{B}$ of the Lie module $L i e (r)$ is projective, by a result of Erdmann and Tan, although $L i e (r)$ itself is projective only when $p ∤ r$ . Write $r = p^{m} k$ , where $p ∤ k$ , and let $S_{k}^{*}$ be the diagonal of a Young subgroup of $S_{r}$ isomorphic to $S_{k} \times \dots \times S_{k}$ . We show that $p^{m} L i e {(r)}_{B} ≅ {(L i e (k) ↑_{S_{k}^{*}}^{S_{r}})}_{B}$ . Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of $L i e {(r)}_{B}$ . 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 ${G L}_{n} (F)$ .

Keywords

Lie module, symmetric group, Lie power, Schur algebra, block

Mathematical Subject Classification 2010

Primary: 20C30

Secondary: 20G43, 20C20

Milestones

Received: 10 March 2011

Revised: 8 June 2011

Accepted: 6 July 2011

Published: 25 July 2012

Authors

Roger M. Bryant
School of Mathematics University of Manchester Oxford Road Manchester M13 9PL United Kingdom

Karin Erdmann
Mathematical Institute University of Oxford 24-29 St. Giles Oxford OX1 3LB United Kingdom
http://www.maths.ox.ac.uk/~erdmann

Vol. 6, No. 4, 2012