Related papers: Labelling Schubert intersections in the Grassmania…
Symmetries of combinatorial objects are known to complicate search algorithms, but such obstacles can often be removed by detecting symmetries early and discarding symmetric subproblems. Canonical labeling of combinatorial objects…
This paper aims to focus on Richardson varieties on symplectic groups, especially their combinatorial characterization and defining equations. Schubert varieties and opposite Schubert varieties have profound significance in the study of…
We describe a large-scale computational experiment to study structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions…
We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian Gr(n,d) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model…
Given a point z in P^1, let F(z) be the osculating flag to the rational normal curve at point z. The study of Schubert problems with respect to such flags F(z_1), F(z_2), ..., F(z_r) has been studied both classically and recently,…
We obtain new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical…
We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haiman's enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These…
In this paper we reduce the problem of counting the number of connected components in the intersection of two opposite open Schubert cells in the variety of real complete flags to a purely combinatorial question of counting the number of…
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of…
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order…
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of…
We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.
We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the F-rationality of matrix Schubert varieties. Although it is known…
We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Pl\"ucker embedding", this is a…
We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks.…
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary…
A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the…
We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving…
Peterson varieties are subvarieties of flag varieties and their (equivariant) cohomology rings are given by Fukukawa-Harada-Masuda in type A and soon later the author with Harada and Masuda gives an explicit presentation of the…
We classify all normal Schubert varieties in the affine Grassmannian of a semisimple group over an arbitrary field with special attention to small positive characteristic. The proof is elementary and relies on tangent space calculations for…