Presenting affine Schur Algebras

TL;DR

This paper studies the structure of affine Schur algebras, providing explicit multiplication formulas and a presentation, building on their connection to the universal enveloping algebra of affine gl_n.

Contribution

It offers a new presentation of affine Schur algebras and utilizes explicit multiplication formulas to analyze their structure.

Findings

Derived explicit multiplication formulas for affine Schur algebras

Provided a presentation of affine Schur algebra ${ m extbf{S}}_{ riangle}(n,r)_{f Q}$

Enhanced understanding of the algebraic structure of affine Schur algebras

Abstract

The universal enveloping algebra ${\mathcal U}({\widehat{\frak{gl}}_n})$ of ${\widehat{\frak{gl}}_n}$ was realized in \cite[Ch. 6]{DDF} using affine Schur algebras. In particular some explicit multiplication formulas in affine Schur algebras were derived. We use these formulas to study the structure of affine Schur algebras. In particular, we give a presentation of the affine Schur algebra ${\mathcal S}_{{\!\vartriangle}}(n,r)_{\mathbb Q}$.