Multitriangulations and tropical Pfaffians. (English) Zbl 07849887
Summary: The \(k\)-associahedron \(\mathcal{A}\mathit{ss}_k(n)\) 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 \(\mathcal{A}\mathit{ss}_k(n)\) with the Pfaffian variety \(\mathcal{P}\mathit{f}_k(n)\) of antisymmetric matrices of rank \(\leq 2k\). First, we characterize the Gröbner cone \(Grob_k(n)\) for which the initial ideal of \(I(\mathcal{P}\mathit{f}_k(n))\) equals the Stanley-Reisner ideal of \(\mathcal{A}\mathit{ss}_k(n)\) (that is, the monomial ideal generated by \((k+1)\)-crossings). We then look at the tropicalization of \(\mathcal{P}\mathit{f}_k(n)\) and show that \(\mathcal{A}\mathit{ss}_k(n)\) embeds naturally as the intersection of \(trop(\mathcal{P}\mathit{f}_k(n))\) and \(Grob_k(n)\), and that it is contained in the totally positive part \(trop^+(\mathcal{P}\mathit{f}_k(n))\) of it. We show that for \(k=1\) and for each triangulation \(T\) of the \(n\)-gon, the projection of this embedding of \(\mathcal{A}\mathit{ss}_k(n)\) 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 \(g\)-vector fan of the cluster algebra of type \(A\), shown to be polytopal by Hohlweg, Pilaud, and Stella in 2018.
MSC:
| 14T15 | Combinatorial aspects of tropical varieties |
| 05E45 | Combinatorial aspects of simplicial complexes |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
Software:
GfanReferences:
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