Block components of the Lie module for the symmetric group

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 $S_k\times \cdots \times S_k$. We show that $p^{m}Lie{\left(r\right)}_B\cong {\left(Lie\left(k\right){\uparrow }_{S_k^{\ast }}^{S_r}\right)}_B$. 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 ${GL}_n\left(F\right)$.