Yangians and degenerate affine Schur algebras

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