Cluster algebras from surfaces and extended affine Weyl groups

TL;DR

This paper characterizes mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms, associating surfaces and triangulations with bases and extended affine Weyl groups.

Contribution

It introduces a geometric framework linking cluster algebra mutations to partial reflections and extended affine Weyl groups, extending to exceptional types.

Findings

Mutation-finite cluster algebras are characterized by positive semi-definite quadratic forms.

Triangulations correspond to bases in a quadratic space, with mutations as partial reflections.

Extended affine Weyl groups of type A are associated with triangulations and invariant under flips.

Abstract

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with every triangulation a basis in $V$, such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type $A$, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types $E$.