Authors

Roger M. Bryant and Karin Erdmann

Title

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 non-principal block of the group algebra $FS_r$ of the
symmetric group $S_r$. The block component ${\rm Lie}(r)_B$ of the
Lie module ${\rm Lie}(r)$ is projective, by a result of Erdmann
and Tan, although ${\rm Lie}(r)$ itself is projective only
when $p \nmid r$. Write $r = p^mk$, where $p \nmid k$, and let
$S_k^*$ be the diagonal of a Young subgroup of $S_r$ isomorphic
to $S_k \times \cdots \times S_k$. We show
that  $p^m\,{\rm Lie}(r)_B \cong ({\rm Lie}(k)\upa_{S_k^*}^{S_r})_B$.
Hence we obtain a formula for the multiplicities of the projective
indecomposable modules in a direct sum decomposition of ${\rm Lie}(r)_B$.
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 ${\rm GL}_n(F)$.


Status:  submitted.