Yangians and degenerate affine Schur algebras

Abstract

Abstract Drinfeld’s degenerate affine analog of Schur–Weyl duality relates representations of the degenerate affine Hecke algebra $$\textrm{A}\hspace{-.2pt}\textrm{H}_r$$ A H r to representations of the Yangian $$\textrm{Y}(\mathfrak {gl}_n).$$ Y ( gl n ) . One way to understand the construction is to introduce an intermediate algebra $$\textrm{A}\hspace{-.2pt}\textrm{S}(n,r),$$ A S ( n , r ) , the degenerate affine Schur algebra , which appears both as the endomorphism algebra of an induced tensor space over $$\textrm{A}\hspace{-.2pt}\textrm{H}_r,$$ A H r , and as the image of a homomorphism $$\texttt{D}_{n,r}:\textrm{Y}(\mathfrak {gl}_n)\twoheadrightarrow \textrm{A}\hspace{-.2pt}\textrm{S}(n,r).$$ D n , r : Y ( gl n ) ↠ A S ( n , r ) . In this paper, we describe $$\texttt{D}_{n,r}$$ D n , r using a diagrammatic calculus. Then we use a theorem of Drinfeld to compute $$\ker \texttt{D}_{n,r}$$ ker D n , r when $$n > r,$$ n > r , thereby giving a presentation of $$\textrm{A}\hspace{-.2pt}\textrm{S}(n,r)$$ A S ( 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 $$\textrm{A}\hspace{-.2pt}\textrm{S}(n,r).$$ A S ( n , r ) .

The authors' abstract, as published at the source. Mathematische Zeitschrift, 2026 · DOI ↗