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Graph coverings and twisted operators
Cimasoni, David1; Kassel, Adrien2
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)
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 Γ ˜ is a finite covering graph of a connected graph Γ endowed with edge-weights x={x e } e , then the spanning tree partition function of Γ divides the one of Γ ˜ in the ring ℤ[x]. Several other consequences are obtained, some known, others new.

Received:
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)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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