A decomposition formula for the Bartholdi zeta function of a hypergraph covering

TL;DR

Extends decomposition results for Bartholdi/Zeta-type functions from graphs to hypergraphs via permutation voltage assignments. Builds a representation-theoretic and determinant-based framework around the derived bipartite graph to factor the Bartholdi zeta of a covering hypergraph into base zeta-L-functions associated with irreducible representations. The main result shows a product factorization under unitary representations, connecting the covering zeta to Bartholdi L-functions and providing an explicit example. This work broadens the toolkit for zeta-function decompositions in hypergraph theory and has implications for spectral and combinatorial zeta analyses.

Abstract

It is shown by Mizuno and Sato that the Bartholdi zeta function of a covering graph is decomposed as a product of Bartholdi zeta functions of a base graph that are associated with representations. In this paper, we extend their result to the case of a hypergraph covering.

Theorems & Definitions (70)

• Theorem 1.1: Li
• Theorem 1.2: Theorem \ref{['thm53']}
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8