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On folded cluster patterns of affine type

A cluster algebra is a commutative algebra whose structure is decided by a skew-symmetrizable matrix or a 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 non-simply-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.