#### Vol. 6, No. 4, 2012

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) 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
##### 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 $Lie{\left(r\right)}_{B}$ of the Lie module $Lie\left(r\right)$ is projective, by a result of Erdmann and Tan, although $Lie\left(r\right)$ itself is projective only when $p\nmid r$. Write $r={p}^{m}k$, where $p\nmid k$, and let ${S}_{k}^{\ast }$ be the diagonal of a Young subgroup of ${S}_{r}$ isomorphic to ${S}_{k}×\cdots ×{S}_{k}$. We show that ${p}^{m}\phantom{\rule{0.3em}{0ex}}Lie{\left(r\right)}_{B}\cong {\left(Lie\left(k\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{↑}_{{S}_{k}^{\ast }}^{{S}_{r}}\right)}_{B}$. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of $Lie{\left(r\right)}_{B}$. Corresponding results are obtained, when $F$ is infinite, for the $r$-th Lie power ${L}^{r}\left(E\right)$ of the natural module $E$ for the general linear group ${GL}_{n}\left(F\right)$.

##### Keywords
Lie module, symmetric group, Lie power, Schur algebra, block
##### Mathematical Subject Classification 2010
Primary: 20C30
Secondary: 20G43, 20C20