Multitriangulations and tropical Pfaffians

TL;DR

This paper investigates the combinatorial and algebraic structures of $k$-associahedra and their relation to tropical geometry and Pfaffian varieties, revealing new connections between triangulations, matroids, and cluster algebras.

Contribution

It characterizes the Gr"obner cone associated with $k$-associahedra, links $k$-triangulations to algebraic matroids of Pfaffian varieties, and embeds these complexes into tropical and cluster algebra frameworks.

Findings

$k$-triangulations form bases in the algebraic matroid of $Pf_k(n)$

$Ass_k(n)$ embeds into the tropicalization of $Pf_k(n)$

For $k=1$, the associahedron is realized as a polytopal fan related to cluster algebras

Abstract

The $k$-associahedron $Ass_k(n)$ is the simplicial complex of $(k+1)$-crossing-free subgraphs of the complete graph with vertices on a circle. Its facets are called $k$-triangulations. We explore the connection of $Ass_k(n)$ with the Pfaffian variety $Pf_k(n)\subset {\mathbb K}^{\binom{[n]}2}$ of antisymmetric matrices of rank $\le 2k$. First, we characterize the Gr\"obner cone $Grob_k(n)\subset{\mathbb R}^{\binom{[n]}2}$ producing as initial ideal of $I(Pf_k(n))$ the Stanley-Reisner ideal of $Ass_k(n)$ (that is, the monomial ideal generated by $(k+1)$-crossings). This implies that $k$-triangulations are bases in the algebraic matroid of $Pf_k(n)$, a matroid closely related to low-rank completion of antisymmetric matrices. We then look at the tropicalization of $Pf_k(n)$ and show that $Ass_k(n)$ embeds naturally as the intersection of $\operatorname{trop}(Pf_k(n))$ and $Grob_k(n)$, and is contained in the totally positive part $\operatorname{trop}^+( Pf_k(n))$ of it. We show that for $k=1$ and for each triangulation $T$ of the $n$-gon, the projection of this embedding of $Ass_k(n)$ to the $n-3$ coordinates corresponding to diagonals in $T$ gives a complete polytopal fan, realizing the associahedron. This fan is linearly isomorphic to the $\mathbf g$-vector fan of the cluster algebra of type $A$, shown to be polytopal by Hohlweg, Pilaud and Stella in (2018).