Yangians and degenerate affine Schur algebras

TL;DR

This paper explores the structure and representation theory of degenerate affine Schur algebras, connecting them with Yangians and Hecke algebras through diagrammatic calculus and algebraic presentations.

Contribution

It provides a diagrammatic description of the homomorphism from Yangians to degenerate affine Schur algebras and computes its kernel for certain parameters, offering new insights into their structure.

Findings

Computed the kernel of the homomorphism D_{n,r} for n > r

Presented a diagrammatic calculus for the algebra homomorphism

Developed aspects of the representation theory of AS(n,r)

Abstract

Drinfeld's degenerate affine analog of Schur-Weyl duality relates representations of the degenerate affine Hecke algebra $AH_r$ to representations of the Yangian $Y_n$. One way to understand the construction is to introduce an intermediate algebra $AS(n,r)$, the degenerate affine Schur algebra, which appears both as the endomorphism algebra of an induced tensor space over $AH_r$, and as the image of a homomorphism $D_{n,r}:Y_n \rightarrow AS(n,r)$. In this paper, we describe $D_{n,r}$ using a diagrammatic calculus. Then we use a theorem of Drinfeld to compute $\ker D_{n,r}$ when $n > r$, thereby giving a presentation of $AS(n,r)$ in these cases. We formulate a conjecture in the remaining cases. Finally, we apply results of Arakawa to develop some of the representation theory of $AS(n,r)$.