Hideo Mitsuhashi

The discrete-time quaternionic quantum walk on a graph

Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We deal with the right eigenvalue problem of quaternionic matrices in order to study spectra of the transition matrix of a quaternionic quantum walk. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of a quaternionic quantum walk, we deduce some remarkable properties of it. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using those of the corresponding weighted matrix. In addition, we give some examples of quaternionic quantum walks and their right eigenvalues.

A generalized Bartholdi zeta function for a general graph

We define an -variable Bartholdi zeta function and an -variable Bartholdi -function of a graph , and give determinant expressions of them. We present a decomposition formula for the -variable Bartholdi zeta function of a regular covering of . Furthermore, we express the -variable Bartholdi zeta function of a regular covering of G as a product of its -variable Bartholdi L-functions.

A matrix-weighted zeta function of a graph

We define a matrix-weighted -function of a graph , and give a determinant expression of it. As a corollary, we present a decomposition formula for the matrix-weighted zeta function of a regular covering of by a product of matrix-weighted -functions of .

New theory of diffusive and coherent nature of optical wave via a quantum walk

We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno, Quantum Inf. Proc. (2010) 9, 405-418), we numerically show that the randomized localized matrices suppress the coherence and give diffusive nature.

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

For a graph with at most one loop at each vertex, we define a discrete-time quaternionic quantum walk on the graph, which can be viewed as a quaternionic extension of the Grover walk on the graph. We derive the unitary condition for the transition matrix of the quaternionic Grover walk, and discuss the relationship between the right spectra of the transition matrices and zeta functions of graphs.

The quaternionic weighted zeta function of a graph

We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product, which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.

A super Frobenius formula for the characters of Iwahori–Hecke algebras

In this article, 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 give some properties of supersymmetric functions. Based on Schur–Weyl reciprocity between Iwahori–Hecke algebras and the general quantum super algebras, which was obtained in Mitsuhashi [H. Mitsuhashi, Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra, Algeb. Represent. Theor. 9 (2006), pp. 309–322.], we derive that the 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 that was obtained in Ram [A. Ram, A Frobenius formula for the characters of the Hecke algebra, Invent. Math. 106 (1991), pp. 461–488.] and the classical super case.