The Schur functor on tensor powers

TL;DR

This paper investigates how the Schur functor acts on tensor powers of modules over Schur algebras, revealing its effect on various algebraic constructions like Lie, symmetric, and exterior powers.

Contribution

It provides a detailed description of the Schur functor's action on tensor powers and related algebraic structures, extending understanding of module transformations.

Findings

Schur functor maps tensor powers to bimodules involving symmetric groups

Explicit description of Schur functor on Lie, symmetric, and exterior powers

Enhances understanding of module transformations under Schur functor

Abstract

Let $M$ be a left module for the Schur algebra $S(n,r)$, and let $s \in \mathbb{Z}^+$. Then $M^{\otimes s}$ is a $(S(n,rs), F\mathfrak{S}_s)$-bimodule, where the symmetric group $\mathfrak{S}_s$ on $s$ letters acts on the right by place permutations. We show that the Schur functor $f_{rs}$ sends $M^{\otimes s}$ to the $(F\mathfrak{S}_{rs},F\mathfrak{S}_s)$-bimodule $F\mathfrak{S}_{rs} \otimes_{F(\mathfrak{S}_r \wr \mathfrak{S}_s)} ((f_rM)^{\otimes s} \otimes F\mathfrak{S}_s)$. As a corollary, we obtain the effect of the Schur functor on the Lie power $L^s(M)$, symmetric power $S^s(M)$ and exterior power $\bigwedge^s(M)$ of $M$.