Walks on graphs and their connections with tensor invariants and centralizer algebras

The number of walks of k steps from the node 0 to the node λ on the McKay quiver determined by a finite group G and a G-module V is the multiplicity of the irreducible G-module Gλ in the tensor power V⊗k, and it is also the dimension of the irreducible module labeled by λ for the centralizer algebra Zk(G)=EndG(V⊗k). This paper explores ways to effectively calculate that number using the character theory of G. We determine the corresponding Poincaré series. The special case λ=0 gives the Poincaré series for the tensor invariants T(V)G=⨁k=0∞(V⊗k)G and a tensor analog of Molien's formula for polynomial invariants. When 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 McKay quivers, and the methods presented here provide efficient ways to compute the number of walks on them.