Schur-Weyl reciprocity between the quantum superalgebra and the Iwahori-Hecke algebra

In the representation theory, the classification and the construction of the irreducible representations are essential themes. In the first half of the twentieth century, I. Schur[11] introduced a prominent method to obtain the finite dimensional irreducible representations of the general linear group GL(n,C), or equivalently of its Lie algebra gl(n,C), which we call Schur-Weyl reciprocity at present. Schur applied this method to the permutation action of the symmetric group Sr and the diagonal action of GL(n,C) on the tensor powers V ⊗r of the n dimensional complex vector space V . After this work, Schur-Weyl reciprocity has been extended to various groups and algebras. Brauer[1] obtained the centralizer algebra of the orthogonal Lie group O(n). Sergeev[12] and Berele-Regev[4] extended the Schur’s result to the general super Lie algebra gl(m,n). Jimbo[6] extended it to the qanalogue case. He established Schur-Weyl reciprocity between the quantum enveloping algebra Uq(gln+1) and the Iwahori-Hecke algebra of type A. As in the book of Curtis-Reiner[2], the representation theory of Iwahori-Hecke algebras is an important part in representation theories of finite groups of Lie type. Hence we will focus on the representation theory of Iwahori-Hecke algebras.