Multitriangulations and tropical Pfaffians

Abstract

The k-associahedron AsSk(n) is the simplicial complex of (fc + l)-crossing-free subgraphs of the complete graph with vertices on a circle. Its facets are called fc-triangulations. We explore the connection of Assk(n) with the Pfaffian variety Vf'k(n) of antisymmetric matrices of rank < 2k. First, we characterize the Grobner cone Grobk(n) for which the initial ideal of I(Vfk(n)) equals the Stanley-Reisner ideal of Assk(n) (that is, the monomial ideal generated by (k + l)-crossings). We then look at the tropicalization oiVfk{n) and show that Assk(n) embeds naturally as the intersection of trop(Vfk{n)) and Grobk(n), and that it is contained in the totally positive part trop+ (Pfk(n)) of it. We show that for fc = 1 and for each triangulation T of the n-gon, the projection of this embedding of Assk(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 g-vector fan of the cluster algebra of type A. shown to be polytopal by Hohlweg, Pilaud, and Stella in 2018.