Related papers: The Chirotropical Grassmannian

The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of…

In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue…

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra…

In analogy with the origin of the additive structure of $K$-theory, we construct an $E_\infty$ structure on the matroid Grassmannian (the space of oriented matroids) for which the underlying binary operation is the direct sum of matroids.…

Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians…

The KP equation is a nonlinear dispersive wave equation which provides an excellent model for resonant interactions of shallow-water waves. It is well known that regular soliton solutions of the KP equation may be constructed from points in…

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto $X={\rm Gr}(N_c, n_f)$ is a circle compactification of the 3d $\mathcal{N}=2$…

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let \psi_j be the angles between L and M, let \phi_j be the angles between M and N, and let \theta_j be the angles between L and N. Consider…

We revisit the representation theory in type $A$used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that…

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has…

In this note I will explain how relative/log Gromov-Witten invariants of pairs $(X,D)$ with very ample smooth anticanonical divisor $D$ can be computed using algebro-combinatorial objects called scattering diagrams. The underlying principle…

Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in…

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition,…

The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We…

We establish an explicit bijection between the toric degenerations of the Grassmannian $\textrm{Gr}(2,n)$ arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to $\textrm{Gr}(2,n)$. We…

We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes,…

Recently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become…

We establish a derived geometric Satake equivalence for the real group $G_{\mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), to be called the Lorentzian Satake equivalence (resp. Octonionic Satake equivalence). By applying the real-symmetric…

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally nonnegative…

In the 1990s, Kita--Yoshida and Cho--Matsumoto introduced intersection forms on the twisted (co)homologies of hyperplane arrangement complements. We give a closed combinatorial formula for these intersection pairings. We show that these…