Expansion Posets for Polygon Cluster Algebras

Abstract

Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion formula. We introduce an involution on several of the interrelated combinatorial objects and constructions associated to type A surface cluster algebras, including certain classes of arcs, triangulations, and distributive lattices. We use these involutions to formulate a dual version of skein relations for arcs, and dual versions of three existing expansion posets. In particular, this leads to two new cluster expansion formulas, and recovers the lattice path expansion of Propp et al. We provide an explicit, structure-preserving poset isomorphism between an expansion poset and its dual version from the dual arc. We also show that an expansion poset and its dual version constructed from the same arc are dual in the sense of distributive lattices.We show that any expansion poset is isomorphic to a closed interval in one of the lattices L(m, n) of Young diagrams contained in an m × n grid, and that any L(m, n) has a covering by such intervals. In particular, this implies that any expansion poset is isomorphic to an interval in Young's lattice.We give two formulas for the rank function of any lattice path expansion poset, and prove that this rank function is unimodal whenever the underlying snake graph is built from at most four maximal straight segments. This gives a partial solution to a recent conjecture by Ovsienko and Morier-Genoud. We also characterize which expansion posets have symmetric rank generating functions, based on the shape of the underlying snake graph.

$q$-Rationals and Finite Schubert Varieties

“…Remark 2.1.1. The connection between these posets and F -polynomials in cluster algebras has been noted in [MSW11], [Rab18], [BG21], and [Cla20]. These posets were called "fence posets" in [MSS21] and "piece-wise linear posets" in [BG21].…”

Double Dimer Covers on Snake Graphs from Super Cluster Expansions

“…Duality of Snake Graphs. In this section, we describe an involution on the set of all snake graphs which was described in [Pro05], and also discussed extensively in [Cla20]. Under this involution, dimer covers are taken to lattice paths.…”

“…An example of two dual snake graphs is pictured in Figure 19. There is a bijection, described in [Pro05] and [Cla20], between perfect matchings (dimer covers) of G and lattice paths in G going from the bottom left to the top right corner. With the labelling convention described above for G, this bijection is weight-preserving, where the weight of a lattice path is the product of the weights of the edges in the path.…”

“…For a poset P , let J(P ) denote the lattice of order ideals in P (ordered by inclusion). For the example of the lattice of dimer covers of a snake graph G, a description of this poset was given in [MSW13] (and studied also in [Cla20] and [MSS21]).…”

“…There have been other combinatorial models (e.g. [Yur19], [LLM15], [Cla20]), but these two will be our main focus in this article.…”