The non-projective part of the Lie module for the symmetric group

TL;DR

This paper investigates the structure of the Lie module for symmetric groups and free Lie algebra components, showing non-projective parts are contained within the principal block under certain conditions, revealing new block membership properties.

Contribution

It establishes that non-projective summands of the Lie module and free Lie algebra components belong to the principal block in specific algebraic settings, extending understanding of their block structure.

Findings

Non-projective summands belong to the principal block of $FS_n$ when $p$ divides $n$.

Summands of $L^n(V)$ not being tilting modules are in the principal block of $S(m,n)$ for $m geq n$.

Results connect the structure of Lie modules with block theory in symmetric groups and Schur algebras.

Abstract

The Lie module of the group algebra $FS_n$ of the symmetric group is known to be not projective if and only if the characteristic $p$ of $F$ divides $n$. We show that in this case its non-projective summands belong to the principal block of $FS_n$. Let $V$ be a vector space of dimension $m$ over $F$, and let $L^n(V)$ be the $n$-th homogeneous part of the free Lie algebra on $V$; this is a polynomial representation of $GL_m(F)$ of degree $n$, or equivalently, a module of the Schur algebra $S(m,n)$. Our result implies that, when $m \geq n$, every summand of $L^n(V)$ which is not a tilting module belongs to the principal block of $S(m,n)$, by which we mean the block containing the $n$-th symmetric power of $V$.