On the symmetric and exterior powers of Young permutation modules

TL;DR

This paper investigates the structure of symmetric and exterior powers of Young permutation modules over fields of positive characteristic, classifying projective and indecomposable summands, and explicitly describing Scott modules within these decompositions.

Contribution

It provides a comprehensive classification of projective and indecomposable summands of symmetric and exterior powers of Young permutation modules in positive characteristic.

Findings

Identified which symmetric and exterior powers are projective.

Classified all indecomposable exterior powers of $M^$.

Parameterizes Scott modules as summands and computes their multiplicities.

Abstract

Let $n$ be a positive integer and $\lambda$ be a partition of $n$. Let $M^\lambda$ be the Young permutation module labelled by $\lambda$. In this paper, we study symmetric and exterior powers of $M^\lambda$ in positive characteristic case. We determine the symmetric and exterior powers of $M^\lambda$ that are projective. All the indecomposable exterior powers of $M^\lambda$ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of $M^\lambda$. We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of $M^\lambda$ and determining their corresponding multiplicities explicitly.