Schur-Weyl reciprocity for the q-analogue of the alternating group

In this paper, we establish Schur-Weyl reciprocity for the q-analogue of the alternating group. We analyze the sign q-permutation representation of the Hecke algebra HQ(q),r(q) on the rth tensor product of Z2-graded Q-vector space V = V0⊗V1 in detail, and examine its restriction to the qanalogue of the alternating group H1Q(q),r(q). In consequence, we find out that if dimV0 = dimV1, then the centralizer of H1Q(q),r(q) is a Z2-crossed product of the centralizer of HQ(q),r(q) and obtain Schur-Weyl reciprocity between H1Q(q),r(q) and its centralizer. Though the structure of the centralizer is more complicated for the case dimV0 6=dimV1, we obtain some results about the case. When q = 1, Regev has proved Schur-Weyl reciprocity for alternating groups in [12]. Therefore, our result can be regarded as an extension of Regev’s work.