Yangians and degenerate affine Schur algebras

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 $AS(n,r)$.