Yangians and degenerate affine Schur algebras
Jonathan Brundan, Viacheslav Ivanov
TL;DR
The paper develops a diagrammatic framework for the degenerate affine Schur algebra AS(n,r) by defining the degenerate affine Schur category AS_chur and realizing AS(n,r) as an endomorphism algebra of an induced AH_r-tensor space. It constructs and analyzes Drinfeld's homomorphism D_{n,r}:Y(gl_n)→AS(n,r), computes its kernel for n>r, and uses this to present AS(n,r) in those cases, while formulating conjectures for the remaining cases. The authors establish a monoidal, diagrammatic presentation of the related categories, determine the centers, and connect the representation theory to polynomial representations of the Yangian via Drinfeld's functor, aided by Arakawa's results. They also provide explicit diagrammatic expressions for the Drinfeld homomorphism and extend the framework to graded/current versions, culminating in a classification of irreducible polynomial Y(gl_n)-modules of degree r in characteristic 0 and a Kazhdan-Lusztig–style description of standard modules for 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 $AS(n,r)$.