Multitriangulations and Tropical Pfaffians

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