Multitriangulations and Tropical Pfaffians

. The \ (k\) -associahedron \ (A-0ptss₊ (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 \ (A-0ptss₊ (n) \) with the Pfaffian variety \ (P-1ptf-1pt₊ (n) \) of antisymmetric matrices of rank \ (2k\). First, we characterize the Gröbner cone \ (Grobₖ (n) \) for which the initial ideal of \ (I (P-1ptf-1pt₊ (n) ) \) equals the Stanley–Reisner ideal of \ (A-0ptss₊ (n) \) (that is, the monomial ideal generated by \ ( (k+1) \) -crossings). We then look at the tropicalization of \ (P-1ptf-1pt₊ (n) \) and show that \ (A-0ptss₊ (n) \) embeds naturally as the intersection of \ (trop (P-1ptf-1pt₊ (n) ) \) and \ (Grobₖ (n) \), and that it is contained in the totally positive part \ (trop^+ (P-1ptf-1pt₊ (n) ) \) of it. We show that for \ (k=1\) and for each triangulation \ (T\) of the \ (n\) -gon, the projection of this embedding of \ (A-0ptss₊ (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. Keywordsmultitriangulationstropical geometryPfaffiansassociahedronMSC codes14T1505E4552B40