A Schur-Weyl Duality Approach to Walking on Cubes

TL;DR

This paper investigates walks on the n-cube graph through the lens of Schur-Weyl duality, describing centralizer algebras with labeled partitions and deriving formulas for counting walks.

Contribution

It introduces a basis for centralizer algebras related to the n-cube using labeled partition diagrams and provides generating functions for walk enumeration.

Findings

Basis for centralizer algebras in terms of labeled partitions

Explicit formulas for counting walks on the n-cube

Exponential generating functions for walk enumeration

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