  The theory of random walks on finite graphs is well developed with numerous
applications. In quantum walks, the propagation is governed by quantum
mechanical rules; generalizing random walks to the quantum setting. They have
been successfully applied in the development of quantum algorithms. In
particular, to solve problems that can be mapped to searching or property
testing on some specific graph. In this paper we investigate the discrete time
coined quantum walk (DTCQW) model using tools from non-commutative Fourier
analysis. Specifically, we are interested in characterizing the DTCQW on Cayley
graphs generated by the symmetric group ($\sym$) with appropriate generating
sets. The lack of commutativity makes it challenging to find an analytical
description of the limiting behavior with respect to the spectrum of the
walk-operator. We determine certain characteristics of these walks using a path
integral approach over the characters of $\sym$.
