Polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion

Article type: Research Article

Authors: Pérez, Claudia | Rivera, Daniel^{; *}

Affiliations: Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Cuernavaca, Mor. Mexico. darivera@uaem.mx

Correspondence: [*] Address for correspondence: Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Cuernavaca, Mor. Mexico.

Abstract: Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Δ ∈ {𝔸n; 𝔻n; 𝔹n; ℂn; 𝔼6; 𝔼7; 𝔼8; 𝔽4; 𝔾2} of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B, if it has one.

Keywords: Cluster algebra, positive quasi-Cartan companion, polynomial algorithm, biconnected component, triconnected component

DOI: 10.3233/FI-2021-2061

Journal: Fundamenta Informaticae, vol. 181, no. 4, pp. 313-337, 2021