# Multitriangulations and tropical Pfaffians

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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.

Category: seminars
Tags: Combi seminar combinatorics