In this paper, we establish Schur-Weyl reciprocity between the quantum general super Lie algebra U σ q gl(m, n) and the Iwahori-Hecke algebra H Q(q),r (q). We introduce the sign q-permutation representation of H Q(q),r (q) on the tensor spaceThis action commutes with that of U σ q gl(m, n) derived from the vector representation on V. Those two subalgebras of End Q(q) (V ⊗r ) satisfy Schur-Weyl reciprocity. As special cases, we obtain the super case (q→1), and the quantum case (n = 0). Hence this result includes both the super case and the quantum case, and unifies those two important cases.

We define the new algebra. This algebra has a parameter q. The defining relations of this algebra at q s 1 coincide with the basic relations of the alternating group. We also give the new subalgebra of the Hecke algebra of type A which is isomorphic to this algebra. This algebra is free of rank half that of the Hecke algebra. Hence this algebra is regarded as a q-analogue of the alternating group.All the isomorphism classes of the irreducible representations of this algebra and the q-analogue of the branching rule between the symmetric group and the alternating group are obtained. ᮊ

Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We treat the right eigenvalue problem of quaternionic matrices to analysis the spectra of its transition matrix. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of the quaternionic quantum walk, we deduce some properties about it. Our main results, Theorem 5.3, determine all the right eigenvalues of a quaternionic quantum walk by use of those of the corresponding weighted matrix. In addition, we give some examples of quaternionic quantum walks and their right eigenvalues.Abbr. title: The discrete-time quaternionic quantum walk on a graph AMS 2010 subject classifications: 60F05, 05C50, 15A15, 11R52

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In this paper, we establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We define Hall-Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and investigate some properties of supersymmetric functions. Based on Schur-Weyl reciprocity between Iwahori-Hecke algebras and the general quantum super algebras, which was obtained in [8], we derive that one of several types of Hall-Littlewood supersymmetric functions, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case (n = 0) that was obtained in [10] and the classical super case (q→1).