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

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 ∈ {𝔸 _n ; 𝔻 _n ; 𝔹 _n ; ℂ _n ; 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.