Related papers: Richardson varieties, projected Richardson varieti…
This article explores the relationship between Schubert varieties and equivariant embeddings, using the framework of homogeneous fiber bundles over flag varieties. We show that the homogenous fiber bundles obtained from…
We study positivity notions for the tropicalization of type A flag varieties and the flag Dressian. We focus on the hollow case, where we have one constituent of rank 1 and another of corank 1. We characterize the three different notions…
In this paper, we study invariant Poisson structures on homogeneous manifolds, which serve as a natural generalization of homogeneous symplectic manifolds previously explored in the literature. Our work begins by providing an algebraic…
We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gr\"obner bases. Each variety encodes a collection of matroids, whose flats…
This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that…
A stratified manifold has a Bruhat atlas on it if it can be covered with open charts such that each chart is stratified-isomorphic to an (opposite) Bruhat cell in a (usually Kac-Moody) flag manifold. In this paper, we construct an…
A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix…
We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian.…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive…
Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of…
We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants…
We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type $G_2$, verifying a conjecture of Rietsch.
This paper is concerned with the study of spaces of naturally defined cycles associated to SL(n,R)-flag domains. These are compact complex submanifolds in open orbits of real semisimple Lie groups in flag domains of their complexification.…
In this first of a series of articles on standard extension algebras we study standard perverse sheaves on varieties with $\mathbb{G}_m$-actions. Based on Braden's hyperbolic localisation, we describe their extension algebra geometrically…
Let G be a compact connected Lie group and H, the centralizer of a one-parameter subgroup in G. Combining the ideas of Bott-Samelson resulotions of Schubert varieties and the enumerative formula on a twisted products of 2-spheres obatained…
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry…
We show that braid varieties for any complex simple algebraic group $G$ are cluster varieties. This includes open Richardson varieties inside the flag variety $G/B$.
The $\Upsilon$ invariant is a concordance invariant defined by using knot Floer homology. F\"{o}ldv\'{a}ri gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\Upsilon$ invariant for balanced spatial…
We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the…