On exchange spectra of valued quivers and cluster algebras

Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of the valued quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool.

First, we give relations between exchange spectrum of valued quivers and adjacency spectrum of their underlying valued graphs, and between exchange spectra of valued quivers and their full valued subquivers. The key point is to find some invariants from the spectrum theory under mutations of cluster algebras, which is the second part we discuss. We give two equivalent conditions for a quiver $Q$ without $3$-cycles and its mutation to be cospectral. In particular, we prove that $Q$ and $\mu_k (Q)$ are cospectral if and only if $k$ is a sink or a source. Following this discussion, the so-called cospectral subalgebras of cluster algebras are introduced.We study bounds of exchange spectrum radii of quivers and give a characterization of $2$-maximal quivers via the classification of oriented graphs of its mutation equivalence. Then as an application, we obtain that the preprojective algebra of a quiver of Dynkin type is representation-finite if and only if the quiver is $2$-maximal.

Keywords

cluster algebra, cluster quiver, skew-symmetrizable matrix, spectrum, mutation

2010 Mathematics Subject Classification

13F60, 58C40, 68R10

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This project is supported by the National Natural Science Foundation of China (No. 12071422) and the Zhejiang Provincial Natural Science Foundation of China (No. LY19A010023).

Received 7 March 2017

Accepted 9 April 2019

Published 17 February 2021