Yangians and degenerate affine Schur algebras

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 $AS(n,r)$.

💡 Research Summary

This paper investigates the relationship between the degenerate affine Hecke algebra (AH_r) and the Yangian (Y_n) through an intermediate object called the degenerate affine Schur algebra (AS(n,r)). The authors first construct a strict monoidal category (\mathcal{AS}chur) whose objects are compositions of the integer (r) and whose morphisms are string diagrams. Strings may have arbitrary thickness and can be decorated by symmetric polynomials in the variables attached to the string. The basic generating morphisms are merges, splits, and thick crossings, subject to a compact set of local relations. A key new relation allows elementary symmetric polynomials to be moved past crossings for any degree, extending earlier work that only treated the case where the degree equals the string thickness.

Using this diagrammatic framework, the paper gives an explicit description of Drinfeld’s homomorphism \