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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2203.04633 |
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| _version_ | 1866910419935821824 |
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| author | Ruiz, Luis Crespo Santos, Francisco |
| author_facet | Ruiz, Luis Crespo Santos, Francisco |
| contents | 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 $Pf_k(n)\subset {\mathbb K}^{\binom{[n]}2}$ 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). This implies that $k$-triangulations are bases in the algebraic matroid of $Pf_k(n)$, a matroid closely related to low-rank completion of antisymmetric matrices.
We then look at the tropicalization of $Pf_k(n)$ and show that $Ass_k(n)$ embeds naturally as the intersection of $\operatorname{trop}(Pf_k(n))$ and $Grob_k(n)$, and is contained in the totally positive part $\operatorname{trop}^+( Pf_k(n))$ of it.
We show that for $k=1$ and for each triangulation $T$ of the $n$-gon, the projection of this embedding of $Ass_k(n)$ 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04633 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multitriangulations and tropical Pfaffians Ruiz, Luis Crespo Santos, Francisco Combinatorics Algebraic Geometry 14T15, 05E45, 52B40 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 $Pf_k(n)\subset {\mathbb K}^{\binom{[n]}2}$ 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). This implies that $k$-triangulations are bases in the algebraic matroid of $Pf_k(n)$, a matroid closely related to low-rank completion of antisymmetric matrices. We then look at the tropicalization of $Pf_k(n)$ and show that $Ass_k(n)$ embeds naturally as the intersection of $\operatorname{trop}(Pf_k(n))$ and $Grob_k(n)$, and is contained in the totally positive part $\operatorname{trop}^+( Pf_k(n))$ of it. We show that for $k=1$ and for each triangulation $T$ of the $n$-gon, the projection of this embedding of $Ass_k(n)$ 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). |
| title | Multitriangulations and tropical Pfaffians |
| topic | Combinatorics Algebraic Geometry 14T15, 05E45, 52B40 |
| url | https://arxiv.org/abs/2203.04633 |