Multitriangulations and tropical Pfaffians
Time: 15:00 -- Location: online
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 \(Pf_k(n)\subset K^\binom{n}{2}\) of antisymmetric matrices of rank \(\le 2k\). First, we characterize the Gröbner 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). We then look at the tropicalization of \(Pf_k(n)\) and show that \(Ass_k(n)\) embeds naturally as the intersection of \(trop(Pf_k(n))\) and \(Grob_k(n)\), and is contained in the totally positive part \(trop^+(Pfk(n))\) 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.