Structures and Representations of Affine q-Schur Algebras

This paper provides a survey for the latest developments in the theory of affine q-Schur algebras and Schur–Weyl duality between affine quantum gln and affine type A Hecke algebras. More precisely, we will establish, on the one side, an isomorphism between the double Ringel–Hall algebra D△(n) of a cyclic quiver △(n) and the quantum loop algebra of gln, and establish, on the other side, explicit epimorphisms from the double Ringel–Hall algebra to affine q-Schur algebras S△(n, r). Then, by two commuting actions on an affine tensor space, we will establish, for n > r, a category equivalence between representations of affine q-Schur algebras and representations of affine Hecke algebras. In this way, we describe two classification theorems for simple representations of affine q-Schur algebras over C when the parameter q is not a root of unity. Further structures of affine q-Schur algebras will also be discussed as applications of the epimorphisms mentioned above.