Polynomial recognition of cluster algebras of finite type

TL;DR

This paper introduces a polynomial-time algorithm to determine if a cluster algebra is of finite type by verifying specific graph and matrix conditions, addressing a previously infeasible problem.

Contribution

It develops an efficient algorithm for recognizing finite type cluster algebras and proves the NP membership of a related matrix problem.

Findings

Algorithm operates in polynomial time for finite type recognition.

Deciding if a matrix has a positive quasi-Cartan companion is in NP.

Provides a practical method for classifying cluster algebras.

Abstract

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky (2006) presented an easier way to verify if a given algebra is of finite type, by testing that all chordless cycles of the graph related to the algebra are cyclically oriented and that there exists a positive quasi-Cartan companion of the skew-symmetrizable matrix related to the algebra. We develop an algorithm that verifies these conditions and decides whether or not a cluster algebra is of finite type in polynomial time. The second part of the algorithm is used to prove that the more general problem to decide if a matrix has a positive quasi-Cartan companion is in NP.