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

JOURNAL OF ALGEBRA

卷:509

Article

Benkart, Georgia¹  Moon, Dongho² 

[1] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[2] Sejong Univ, Dept Math, Seoul 05006, South Korea

关键词: McKay quiver;    Tensor invariants;    Centralizer algebra;    Generalized hyperbolic function;   

DOI  :  10.1016/j.jalgebra.2018.05.009

来源: Elsevier

【 摘 要 】

The number of walks of k steps from the node 0 to the node lambda on the McKay quiver determined by a finite group G and a G-module V is the multiplicity of the irreducible G-module G(lambda) in the tensor power V-circle times k, and it is also the dimension of the irreducible module labeled by lambda for the centralizer algebra Z(k)(G) = End(G)(V-circle times k). This paper explores ways to effectively calculate that number using the character theory of G. We determine the corresponding Poincare series. The special case lambda = 0 gives the Poincare series for the tensor invariants T(V)(G) = circle plus(infinity)(k=0)(V-circle times 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free