Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$

TL;DR

This paper derives new formulas for decomposing symmetric powers of specific modules in modular representation theory, showing they have Specht filtrations and identifying vertices of summands, with a novel lifting construction for splitting maps.

Contribution

It introduces explicit decomposition formulas for symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$ in characteristic $p$, and develops a general lifting method for splitting maps.

Findings

Symmetric powers have Specht filtrations.

Explicit decomposition formulas are established.

Vertices of indecomposable summands are identified.

Abstract

Let $p$ be a prime and $n\geq 2$ be a positive integer. We establish new formulae for the decompositions of the first $p-1$ symmetric powers of the Specht module $S^{(n-1,1)}$ and the irreducible module $D^{(n-1,1)}$ in characteristic $p$ as direct sums of Young permutation modules. As an application of the formulae, we show that these symmetric powers have Specht filtration and find the vertices of their indecomposable summands. Our main tool, constructed in this paper, is a lift of a splitting map of a short exact sequence of certain symmetric powers to a splitting map of a short exact sequence of higher symmetric powers. This is a general construction, which can be applied to a broader family of modules.