arXiv subjects

Luis Crespo Ruiz

Publications and source records attributed to Luis Crespo Ruiz.

4 recordsLinked to original sources

arXiv2024

Realizations of multiassociahedra via rigidity

Let $Δ_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{[n]}{2}$. Here $k,n\in \mathbb N$ and $n\ge 2k+1$. Jonsson (2003) proved that (neglecting the short edges that cannot be part of any $(k+1)$-crossing), $Δ_k(n)$ is a shellable sphere of dimension $k(n-2k-1)-1$, and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (2004) on subword complexes. Despite considerable effort, the only values of $(k,n)$ for which the conjecture is known to hold are $n\le 2k+3$ (Pilaud and Santos, 2012) and $(2,8)$ (Bokowski and Pilaud, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $Δ_k(n)$ as a polytope for $(k,n)\in \{(2,9), (2,10) , (3,10)\}$. We also realize it as a simplicial fan for all $n\le 13$ and arbitrary $k$, except the pairs $(3,12)$ and $(3,13)$. Finally, we also show that for $k\ge 3$ and $n\ge 2k+6$ no choice of points can realize $Δ_k(n)$ via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

math.CO↗

arXiv2023

Realizations of multiassociahedra via bipartite rigidity

Let $Ass_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{n}{2}$. Here $k,n\in \mathbb{N}$ and $n\ge 2k+1$. It is conjectured that this simplicial complex is polytopal (Jonsson 2005). However, despite several recent advances, this is still an open problem. In this paper we attack this problem using as a vector configuration the rows of a rigidity matrix, namely, hyperconnectivity restricted to bipartite graphs. We see that in this way $Ass_k(n)$ can be realized as a polytope for $k=2$ and $n\le 10$, and as a fan for $k=2$ and $n\le 13$, and for $k=3$ and $n\le 11$. However, we also prove that the cases with $k\ge 3$ and $n\ge \max\{12,2k+4\}$ are not realizable in this way. We also give an algebraic interpretation of the rigidity matroid, relating it to a projection of determinantal varieties with implications in matrix completion, and prove the presence of a fan isomorphic to $Ass_{k-1}(n-2)$ in the tropicalization of that variety.

math.CO↗

arXiv2023

Bar-and-joint rigidity on the moment curve coincides with cofactor rigidity on a conic

We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the $C^{d-2}_{d-1}$-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.

math.CO↗

arXiv2022

Multitriangulations and tropical Pfaffians

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).

math.CO↗