On folded cluster patterns of affine type

Byung Hee An and Eunjeong Lee

Vol. 318 (2022), No. 2, 401–431

DOI: 10.2140/pjm.2022.318.401

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.

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