In an interacting continuous time quantum walk, while the walker (the cursor) is moving on a graph, computational primitives (unitary operators associated to the edges) are applied to ancillary qubits (the register). The model with one walker was originally proposed by R. Feynman, who thus anticipated many features of the Continuous Time Quantum Walk (CTWQ) computing paradigm. In this note we e...

In [12], Stanley associated with a graph G a symmetric function XG which reduces to G’s chromatic polynomial XG(n) under a certain specialization of variables. He then proved various theorems generalizing results about XG(n), as well as new ones that cannot be interpreted on the level of the chromatic polynomial. Unfortunately, XG does not satisfy a Deletion-Contraction Law which makes it diffi...

In this paper we continue to explore the connection between tensor algebras and displacement structure. We focus on recursive orthonormalization and we develop an analogue of the Szegö type theory of orthogonal polynomials in the unit circle for several noncommuting variables. Thus, we obtain the recurrence equations and Christoffel-Darboux formulas for Szegö polynomials in several noncommuting...

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient H of H and show that it has a basis parametrized by a certain subset Wc of the Coxeter group w. Specifically, Wc consists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommu...