Unimodality and Cluster Algebras from Surfaces

TL;DR

This paper proves unimodality of certain rank polynomials related to lattice structures in cluster algebras from surfaces, revealing symmetry and unimodality properties in these algebraic objects.

Contribution

It establishes the unimodality of rank polynomials in lattice of order ideals and cluster expansions, connecting combinatorial and algebraic structures in surface cluster algebras.

Findings

Rank polynomial of the lattice of order ideals of a loop fence poset is unimodal.

Rank polynomial of any tagged arc is unimodal and almost interlacing.

Cluster expansion evaluated at specific variables is also unimodal.

Abstract

We prove that the rank polynomial of the lattice of order ideals of a loop fence poset is unimodal. This poset arises as the poset of join-irreducibles in the lattice of good matchings of loop graphs associated with notched arcs. Equivalently, such polynomials can be obtained by evaluating all coefficient variables in an F-polynomial at a single variable q. We also conclude that the rank polynomial of any tagged arc, whether plain or notched, is not only unimodal but also satisfies a symmetry condition known as almost interlacing. Furthermore, when the lamination consists of a single curve, the cluster expansion-evaluated by setting all cluster variables to 1 and all coefficient variables to q-is also unimodal. We conjecture that polynomials in this case are log-concave.