A bideterminant basis for a reductive monoid

TL;DR

This paper develops a bideterminant basis for a normal reductive monoid and its noninvertible elements, extending to the coordinate ring of the general linear group and providing new proofs for key algebraic theorems.

Contribution

It introduces a bideterminant basis for reductive monoids and their coordinate rings, offering new proofs for the double centraliser theorem in algebra.

Findings

Bideterminant basis for reductive monoid and noninvertible elements

Bideterminant basis for the coordinate ring of GL(n) and its truncations

Alternative proof of the double centraliser theorem for rational Schur and walled Brauer algebras

Abstract

We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the general linear group and for all its truncations with respect to saturated sets. Finally, we deduce an alternative proof of the double centraliser theorem for the rational Schur algebra and the walled Brauer algebra over an arbitrary infinite base field which was first obtained by Dipper, Doty and Stoll.