Vol. 318, No. 2, 2022

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On folded cluster patterns of affine type

Byung Hee An and Eunjeong Lee

Vol. 318 (2022), No. 2, 401–431
Abstract

A cluster algebra is a commutative algebra whose structure is decided by a skew symmetrizable matrix or a valued quiver. When a skew symmetrizable matrix is invariant under an action of a finite group and this action is admissible, the folded cluster algebra is obtained from the original one. Any cluster algebra of nonsimply laced affine type can be obtained by folding a cluster algebra of simply laced affine type with a specific G-action. In this paper, we study the combinatorial properties of quivers in the cluster algebra of affine type. We prove that for any quiver of simply laced affine type, G-invariance and G-admissibility are equivalent. This leads us to prove that the set of G-invariant seeds forms the folded cluster pattern.

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Keywords
cluster patterns of affine type, folding, invariance, admissibility
Mathematical Subject Classification
Primary: 13F60, 05E18
Secondary: 17B67
Milestones
Received: 22 July 2021
Revised: 24 March 2022
Accepted: 21 May 2022
Published: 20 August 2022
Authors
Byung Hee An
Department of Mathematics Education
Kyungpook National University
Daegu
South Korea
Eunjeong Lee
Department of Mathematics
Chungbuk National University
Cheongju
South Korea
Center for Geometry and Physics
Institute for Basic Science
Pohang
South Korea