On the Affine Schur Algebra of Type A

Let n, r ∈ N. The affine Schur algebra S̃(n, r) (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type Ar−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the U(ĝl n ) action on the tensor space when K is the field of complex numbers. We show that S̃(n, r) can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra KG̃Ln,a (defined in Section 3) 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.