Presentation of rational Schur algebras

TL;DR

This paper provides an explicit presentation of rational Schur algebras over arbitrary fields, correcting previous work in characteristic zero and introducing the first such presentation for positive characteristic fields.

Contribution

It offers a new explicit description of the generators of the ideal defining rational Schur algebras over any ground field, including positive characteristic fields.

Findings

Explicit generators for the ideal I(n,r,s) over any field

Correction and completion of previous presentations in characteristic zero

First explicit presentation over fields of positive characteristic

Abstract

We present rational Schur algebra $S(n,r,s)$ over an arbitrary ground field $K$ as a quotient of the distribution algebra $Dist(G)$ of the general linear group $G=GL(n)$ by an ideal $I(n,r,s)$ and provide an explicit description of the generators of $I(n,r,s)$. Over fields $K$ of characteristic zero, this corrects and completes a presentation of $S(n,r,s)$ in terms of generators and relations originally considered by Dipper and Doty. The explicit presentation over ground fields of positive characteristics appears here for the first time.