Schur-Weyl duality over finite fields

TL;DR

This paper extends Schur-Weyl duality to finite fields, establishing conditions under which the duality holds for tensor powers of vector spaces, with implications for representation theory over finite fields.

Contribution

It proves Schur-Weyl duality over finite fields when the field size exceeds the tensor power degree, and clarifies when the natural map is an isomorphism based on vector space dimension.

Findings

Schur-Weyl duality holds over finite fields if the field has at least r+1 elements.

The natural map from the group algebra to endomorphisms is an isomorphism when the vector space dimension exceeds r.

Failure of the isomorphism occurs if the vector space dimension is not strictly larger than r.

Abstract

We prove a version of Schur--Weyl duality over finite fields. We prove that for any field $k$, if $k$ has at least $r+1$ elements, then Schur--Weyl duality holds for the $r$th tensor power of a finite dimensional vector space $V$. Moreover, if the dimension of $V$ is at least $r+1$, the natural map $k\Sym_r \to End\_{GL(V)}(V^{\otimes r})$ is an isomorphism. This isomorphism may fail if $\dim_k V$ is not strictly larger than $r$.