论文标题
多媒体调查和热带PFAFFIAN
Multitriangulations and tropical Pfaffians
论文作者
论文摘要
$ k $ -associahedron $ ass_k（n）$是$（k+1）$ - 完整图的无交叉子图的简单复合物，带有圆上的顶点。它的方面称为$ k $三角形。 我们探索$ ass_k（n）$与pfaffian品种$ pf_k（n）\ subset {\ mathbb k}^{\ binom {\ binom {[n]} 2} $ cank $ \ le 2k $的反对称矩阵的连接。 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).这意味着$ k $ - 三角形是$ pf_k（n）$的代数矩阵中的基础，这是一种与低级别的反对称矩阵密切相关的矩阵。 然后，我们查看$ pf_k（n）$的热带化，并证明$ ass_k（n）$自然嵌入为$ \ permatatorName {trop}（trop}（pf_k（n））$和$ grob_k（n）$的相交，并包含在完全积极的部分$ \ operaTornArname $ \ operaTornArnAme $ n trop_____________k（pf）中。 我们表明，对于$ k = 1 $，对于$ n $ gon的每个三角剖分$ t $，将$ ass_k（n）$的嵌入到$ n-3 $坐标中的嵌入给对角线$ t $中的$ n-3 $坐标提供了完整的polytopal粉丝，并实现了协会。该风扇与$ a $ a $的群集代数的$ \ mathbf g $ - 向量的粉丝线性同构，该群体由Hohlweg，Pilaud和Stella（2018年）显示为多面有。
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).
