On the affine Schur algebra of type A

TL;DR

This paper explores the affine Schur algebra of type A, providing new equivalent definitions and relations to other algebraic structures, thereby enhancing understanding of its properties and dualities.

Contribution

It introduces two new equivalent definitions of the affine Schur algebra of type A and establishes relations with other algebraic structures, re-proving aspects of affine Schur-Weyl duality.

Findings

Affine Schur algebra can be defined via semigroup algebra representations.

It is the dual of a formal coalgebra related to the semigroup.

Many relations between Schur algebras and affine Schur algebras are established.

Abstract

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref{S:semigroups}) action on the tensor space and it equals to the 'dual' of a certain formal coalgebra related to this semigroup. By these approaches we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur-Weyl duality.