Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPositive configuration space
272
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang-Mills scattering amplitudes, which will be discussed in a sequel.
rate research
Read More
The well-known moment map maps the Grassmannian $Gr_{k+1,n}$ and the positive Grassmannian $Gr^+_{k+1,n}$ onto the hypersimplex $Delta_{k+1,n}$, which is a polytope of codimension $1$ inside $mathbb{R}^n$. Over the last decades there has been a great deal of work on matroid subdivisions (and positroid subdivisions) of the hypersimplex; these are closely connected to the tropical Grassmannian and positive tropical Grassmannian. Meanwhile any $n times (k+2)$ matrix $Z$ with maximal minors positive induces a map $tilde{Z}$ from $Gr^+_{k,n}$ to the Grassmannian $Gr_{k,k+2}$, whose image has full dimension $2k$ and is called the $m=2$ amplituhedron $A_{n,k,2}$. As the positive Grassmannian has a decomposition into positroid cells, one may ask when the images of a collection of cells of $Gr^+_{k+1,n}$ give a dissection of the hypersimplex $Delta_{k+1,n}$. By dissection, we mean that the images of these cells are disjoint and cover a dense subset of the hypersimplex, but we do not put any constraints on how their boundaries match up. Similarly, one may ask when the images of a collection of positroid cells of $Gr^+_{k,n}$ give a dissection of the amplituhedron $mathcal{A}_{n,k,2}$. In this paper we observe a remarkable connection between these two questions: in particular, one may obtain a dissection of the amplituhedron from a dissection of the hypersimplex (and vice-versa) by applying a simple operation to cells that we call the T-duality map. Moreover, if we think of points of the positive tropical Grassmannian $mbox{Trop}^+Gr_{k+1,n}$ as height functions on the hypersimplex, the corresponding positroidal subdivisions of the hypersimplex induce particularly nice dissections of the $m=2$ amplituhedron $mathcal{A}_{n,k,2}$. Along the way, we provide a new characterization of positroid polytopes and prove new results about positroidal subdivisions of the hypersimplex.
Any totally positive $(k+m)times n$ matrix induces a map $pi_+$ from the positive Grassmannian ${rm Gr}_+(k,n)$ to the Grassmannian ${rm Gr}(k,k+m)$, whose image is the amplituhedron $mathcal{A}_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${bfOmega}(mathcal{A}_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${bfOmega}(mathcal{A}_{n,k,4})$ encodes scattering amplitudes in $mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of ${bfOmega}(mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of $mathcal{A}_{n,k,m}$. However, while triangulations of polytopes are fully captured by their secondary polytopes, the study of triangulations of objects beyond polytopes is still underdeveloped. We initiate the geometric study of subdivisions of $mathcal{A}_{n,k,m}$ and provide a concrete birational parametrization of fibers of $pi: {rm Gr}(k,n)dashrightarrow {rm Gr}(k,k+m)$. We then use this to explicitly describe a rational top-degree form $omega_{n,k,m}$ (with simple poles) on the fibers and compute ${bfOmega}(mathcal{A}_{n,k,m})$ as a summation of certain residues of $omega_{n,k,m}$. As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when $n-k-1=m$ (even). We show that, in this case, each fiber of $pi$ is parametrized by a projective space and its volume form $omega_{n,k,m}$ has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes ${bfOmega}(mathcal{A}_{n,k,m})$ from $omega_{n,k,m}$. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Netos classification of degree-two foliations on projective space. Corresponding to the ``exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.
In 1960, Hoffman and Singleton cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (kappa - 1) I_n + J_n - A A^{rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $kappa$, respectively. If $A$ is an incidence matrix of some configuration $cal C$ of type $n_kappa$, then the left-hand side $Theta(A):= (kappa - 1)I_n + J_n - A A^{rm T}$ is an adjacency matrix of the non--collinearity graph $Gamma$ of $cal C$. In certain situations, $Theta(A)$ is also an incidence matrix of some $n_kappa$ configuration, namely the neighbourhood geometry of $Gamma$ introduced by Lef`evre-Percsy, Percsy, and Leemans cite{LPPL}. The matrix operator $Theta$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $Theta^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $kappa$, for $kappa = 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantors list cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Bettens list cite{BB99}.
Tight-spans of metrics were first introduced by Isbell in 1964 and rediscovered and studied by others, most notably by Dress, who gave them this name. Subsequently, it was found that tight-spans could be defined for more general maps, such as directed metrics and distances, and more recently for diversities. In this paper, we show that all of these tight-spans as well as some related constructions can be defined in terms of point configurations. This provides a useful way in which to study these objects in a unified and systematic way. We also show that by using point configurations we can recover results concerning one-dimensional tight-spans for all of the maps we consider, as well as extend these and other results to more general maps such as symmetric and unsymmetric maps.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
