A Schur-Weyl Duality Approach to Walking on Cubes

Abstract

Walks on the representation graph $\mathcal R_{\mathsf{V}}(\mathsf{G})$ determined by a group $\mathsf{G}$ and a $\mathsf{G}$-module $\mathsf{V}$ are related to the centralizer algebras of the action of $\mathsf{G}$ on the tensor powers $\mathsf{V}^{\otimes k}$ via Schur-Weyl duality. This paper explores that connection when the group is $\mathbb{Z}_2^n$ and the module $\mathsf{V}$ is chosen so the representation graph is the $n$-cube. We describe a basis for the centralizer algebras in terms of labeled partition diagrams. We obtain an expression for the number of walks by counting certain partitions and determine the exponential generating functions for the number of walks