Presenting affine Schur algebras

Abstract

The universal enveloping algebra U ( g l ^ n ) \mathcal {U}(\widehat {\frak {gl}}_n) of g l ^ n \widehat {\frak {gl}}_n was realized in [A double Hall algebra approach to affine quantum Schur–Weyl theory, Cambridge University Press, Cambridge, 2012, Ch. 6] using affine Schur algebras. In particular some explicit multiplication formulas in affine Schur algebras were derived. We use these formulas to study the structure of affine Schur algebras. In particular, we give a presentation of the affine Schur algebra S △ ( n , r ) Q {\mathcal S}_{{\!\vartriangle }}(n,r)_{\mathbb Q} over Q \mathbb {Q} .