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

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.