Related papers: The positive orthogonal Grassmannian
We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I…
Symmetry and antisymmetry are fundamental concepts in many strict sciences. Pairwise comparisons (PC) matrices are fundamental tools for representing pairwise relations in decision making. In this theoretical study, we present a novel…
We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by $\lambda_3\phi^{3}\ +\lambda_4…
The vector space of symmetric matrices of size $n$ has a natural map to a projective space of dimension $2^n-1$ given by the principal minors. This map extends to the Lagrangian Grassmannian ${\rm LG}(n,2n)$ and over the complex numbers the…
Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive…
The stratification of the Grassmannian by positroid varieties has been the subject of extensive research. Positroid varieties are in bijection with a number of combinatorial objects, including $k$-Bruhat intervals and bounded affine…
Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form - the canonical…
It is well known that the Pl\"ucker ideal defining the Grassmannian is generated by quadratic Pl\"ucker relations. These relations form a reverse lexicographic Gr\"obner basis and endow the Pl\"ucker algebra with the structure of an algebra…
We introduce the moduli space of generic piecewise circular $n$-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when $n=2k$, this moduli space contains a connected component homeomorphic to…
We investigate PBW deformations H of k[x,y]#G where G is the cyclic group of order p and k also has characteristic p; in these deformations, [x,y] takes a value in kG. These algebras are versions of symplectic reflection algebras that only…
The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the…
The biadjoint scalar partial amplitude, $m_n(\mathbb{I},\mathbb{I})$, can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an…
Starting with the seminal work of Arkani-Hamed et al arXiv:1711.09102, in arXiv:1811.05904, the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless $\phi^{4}$ theory. In this paper we show that the program can…
In this paper, we introduce `Pl\"{u}cker weight vector' and establish the definition of a weighted Grassmann orbifold ${\rm Gr}_{\bf b}(k,n)$, corresponding to a Pl\"{u}cker weight vector `${\bf b}$'. We achieve an explicit classification…
The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.
We introduce the totally nonnegative Lagrangian Grassmannian $\rm{LG}_{\geq 0}^R (n,2n)$, a new subset of the totally nonnegative Grassmannian consisting of subspaces isotropic with respect to a certain bilinear form $R$. We describe its…
We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence…
In this paper we consider the integral orthogonal group with respect to the quadratic form of signature $(2,3)$ given by $\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right) \perp \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0…
In this work, we define an orthogonal graph on the set of equivalence classes of $(2\nu + \delta)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $\nu$ are positive integers and $\delta = 0, 1$ or $2$. We classify our graph if it is strongly…
We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…