Affine quantum Schur algebras and the Schur–Weyl reciprocity
Like the quantum Schur algebra, the affine quantum Schur algebra has several equivalent definitions. We first present the geometric definition, given by Ginzburg–Vasserot and Lusztig, which uses cyclic flags and the convolution product. We then discuss the two Hecke algebra definitions given by R. Green and by Varagnolo–Vasserot. The former uses q-permutation modules, while the latter uses tensor spaces. Both versions are related by the Bernstein presentation for Hecke algebras of affine type A.