# ALGEBRAIC COMBINATORICS

Graph coverings and twisted operators
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 75-94.

Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator, uniquely defined up to conjugacy. The main result of this article is the fact that this operator behaves in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if $\stackrel{˜}{\Gamma }$ is a finite covering graph of a connected graph $\Gamma$ endowed with edge-weights $\mathsf{x}={\left\{{\mathsf{x}}_{\mathsf{e}}\right\}}_{\mathsf{e}}$, then the spanning tree partition function of $\Gamma$ divides the one of $\stackrel{˜}{\Gamma }$ in the ring $ℤ\left[\mathsf{x}\right]$. Several other consequences are obtained, some known, others new.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.258
Classification: 05C30, 05C50, 82B20
Keywords: graph coverings, linear representation, determinantal partition functions, polynomial identities
Cimasoni, David 1; Kassel, Adrien 2

1 Section de mathématiques Université de Genève rue du Conseil-Général 7-9 1205 Genève (Switzerland)
2 CNRS & Unité de Mathématiques Pures et Appliquées ENS de Lyon site Monod 46 allée d’Italie 69364 Lyon Cedex 07 (France)
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Cimasoni, David; Kassel, Adrien. Graph coverings and twisted operators. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 75-94. doi : 10.5802/alco.258. https://alco.centre-mersenne.org/articles/10.5802/alco.258/

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