Unimodality and Cluster Algebras from Surfaces

TL;DR

This work connects cluster algebras from surfaces with the combinatorics of fence posets to study unimodality phenomena in polynomial expansions. By interpreting $F$-polynomials via rank polynomials of fence posets and specializing coefficient variables $y_i$ to a single parameter $q$, the authors prove unimodality and almost interlacing for loop fence posets and extend these properties to tagged arcs. They further show unimodality for the single-lamination cluster expansions through the $c$-polynomial, and conjecture $\log$-concavity in this regime. The results illuminate structural regularities in the combinatorics of cluster variables on surfaces and suggest broader conjectural patterns for laminations beyond the single-lamination case, with potential implications for positivity and symmetry in cluster algebras from surfaces.

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.

Figures (17)

• Figure 1: A tagged arc $\gamma^{(p)}$ and its two hook replacements at the notched endpoint.
• Figure 2: Example of an ideal triangulation $T$ of octagon
• Figure 3: Defining the shear coordinates
• Figure 4: Example of lamination $L$ in triangulation $T$
• Figure 5: An elementary lamination $L_{\gamma}$ associated with a tagged arc $\gamma$, plain(left) and notched(right).

Theorems & Definitions (70)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: Theorem 3.1 of FZ2002
• Theorem 2: Corollary 0.4 of GHKK18
• Definition 5
• Definition 6
• Definition 7
• Remark 1