Walks on Graphs and Their Connections with Tensor Invariants and Centralizer Algebras

TL;DR

This paper connects walks on representation graphs of finite groups with tensor invariants and centralizer algebras, providing methods to compute walk counts using character theory and special functions.

Contribution

It introduces effective techniques for calculating walk multiplicities on representation graphs via character theory and explores their relation to tensor invariants and hyperbolic functions.

Findings

Derived formulas for walk counts using character theory.

Established Poincaré series for tensor invariants.

Showed exponential generating functions as hyperbolic function products for abelian groups.

Abstract

The number of walks of $k$ steps from the node $\mathsf{0}$ to the node $\lambda$ on the representation graph (McKay quiver) determined by a finite group $\mathsf{G}$ and a $\mathsf{G}$-module $\mathsf{V}$ is the multiplicity of the irreducible $\mathsf{G}$-module $\mathsf{G}_\lambda$ in the tensor power $\mathsf{V}^{\otimes k}$, and it is also the dimension of the irreducible module labeled by $\lambda$ for the centralizer algebra $\mathsf{Z}_k(\mathsf{G}) = {\mathsf{End}}_\mathsf{G}(\mathsf{V}^{\otimes k})$. This paper explores ways to effectively calculate that number using the character theory of $\mathsf{G}$. We determine the corresponding Poincar\'e series. The special case $\lambda = \mathsf{0}$ gives the Poincar\'e series for the tensor invariants $\mathsf{T}(\mathsf{V})^\mathsf{G} = \bigoplus_{k =0}^\infty (\mathsf{V}^{\otimes k})^\mathsf{G}$. When $\mathsf{G}$ is abelian, we show that the exponential generating function for the number of walks is a product of generalized hyperbolic functions. Many graphs (such as circulant graphs) can be viewed as representation graphs, and the methods presented here provide efficient ways to compute the number of walks on them.