ON FOLDED CLUSTER PATTERNS OF AFFINE TYPE

Abstract

© 2022, Pacific Journal of Mathematics.All Rights Reserved.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