Big Algebra in Type $A$ for the Coordinate Ring of the Matrix Space

TL;DR

This paper studies the big algebra associated with the $ ext{GL}_n$-action on matrix space coordinate rings, providing explicit formulas, establishing commutativity in type $A$, and connecting to Yangian Bethe subalgebras.

Contribution

It introduces explicit formulas for big algebra generators, proves their commutativity in type $A$, and links them to Yangian Bethe subalgebras, advancing understanding of algebraic structures in representation theory.

Findings

Big algebras in type $A$ are commutative.

Explicit formulas for big algebra generators are obtained.

Connections to Yangian Bethe subalgebras are established.

Abstract

In this paper, we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\mathrm{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type $A$ are commutative and relate them to the Bethe subalgebra in the Yangian $\operatorname{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$.