Schubert Varieties Research Articles

Characteristic Cycles of IH Sheaves of Simply Laced Minuscule Schubert Varieties Are Irreducible

Let G / P be a complex cominuscule flag manifold of type E 6 , E 7 . We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in G / P is irreducible. The proof utilizes an earlier algorithm by the same authors which calculates local Euler obstructions, then proceeds by direct computer calculation using Sage. This completes to the exceptional Lie types the characterization of irreducibility of IH sheaves of Schubert varieties in cominuscule G / P obtained by Boe and Fu. As a by-product, we also obtain that the Mather classes, and the Chern-Schwartz-MacPherson classes of Schubert cells in cominuscule G / P of type E 6 , E 7 , are strongly positive.

Spectrum of equivariant cohomology as a fixed point scheme

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology ring of $X$ is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.Comment: 57 pages, 7 figures. Comments are welcome

Singularities of local models

AbstractWe construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic 2 we construct local models, show reducedness of their special fiber, Cohen–Macaulayness and in equicharacteristic also (pseudo-)rationality. In mixed characteristic we conjecture their pseudo-rationality. This is based on the construction of parahoric group schemes over two dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.

Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite schubert cell

AbstractDale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\operatorname {\mathrm {Pet}}_n$ and the opposite Schubert cell associated with the identity element $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ . This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is given by Ciocan–Fontanine and Givental–Kim. In this paper, we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\operatorname {\mathrm {Hess}}(N,h)$ and $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ . In other words, we generalize the Peterson’s statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan–Fontanine and Givental–Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\operatorname {\mathrm {Hess}}(N,h_m)$ and $\Omega _e^\circ $ is the intersection of certain Schubert variety and $\Omega _e^\circ $ , where $h_m=(m,n,\ldots ,n)$ for $1<m<n$ . We also see that $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ is related with the cyclic quotient singularity.

Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers

AbstractLet G be a connected reductive group over an algebraically closed field k, and let $\operatorname {Fl}$ be the affine flag variety of G. For every regular semisimple element $\gamma $ of $G(k((t)))$ , the affine Springer fiber $\operatorname {Fl}_\gamma $ can be presented as a union of closed subvarieties $\operatorname {Fl}^{\leq w}_{\gamma }$ , defined as the intersection of $\operatorname {Fl}_{\gamma }$ with an affine Schubert variety $\operatorname {Fl}^{\leq w}$ .The main result of this paper asserts that if elements $w_1,\ldots ,w_n$ are sufficiently regular, then the natural map $H_i(\bigcup _{j=1}^n \operatorname {Fl}^{\leq w_j}_{\gamma })\to H_i(\operatorname {Fl}_{\gamma })$ is injective for every $i\in \mathbb Z$ . It plays an important role in our work [BV], where our result is used to construct good filtrations of $H_i(\operatorname {Fl}_{\gamma })$ . Along the way, we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.

A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let ${\mathcal {P}}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$ . Then our theorem implies $(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$ . As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

Generalized juggling patterns, quiver Grassmannians and affine flag varieties

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmannians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

On nonemptiness of Newton strata in the $B_{\mathrm{dR}}^{+}$-Grassmannian for $\mathrm{GL}_{n}$

We study the Newton stratification in the B_{\mathrm{dR}}^{+} -Grassmannian for \mathrm{GL}_{n} associated to an arbitrary (possibly nonbasic) element of B(\mathrm{GL}_{n}) . Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in B(\mathrm{GL}_{n}) , our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues–Fontaine curve.

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Let G be a linear semisimple algebraic group and B its Borel subgroup. Let T⊂B be the maximal torus. We study the inductive construction of Bott–Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory KT(G/B)[y], the right and left Demazure–Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on KT(G/B)[y] with respect to the Kirwan map. We show that the left and right twisted Demazure–Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.

Partial order on involutive permutations and double Schubert cells

As shown by Melnikov, the orbits of a Borel subgroup acting by conjugation on upper-triangular matrices with square zero are indexed by involutions in the symmetric group. The inclusion relation among the orbit closures defines a partial order on involutions. We observe that the same order on involutive permutations also arises while describing the inclusion order on [Formula: see text]-orbit closures in the direct product of two Grassmannians. We establish a geometric relation between these two settings.

Covexillary Schubert varieties and Kazhdan–Lusztig polynomials

Covexillary Schubert varieties and Kazhdan–Lusztig polynomials

Equivariant Schubert calculus and geometric Satake

Equivariant Schubert calculus and geometric Satake

Castelnuovo–Mumford regularity of matrix Schubert varieties

Castelnuovo–Mumford regularity of matrix Schubert varieties

Combinatorics of semi-toric degenerations of Schubert varieties in type C

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

On the Computation of the Cohomological Invariants of Bott–Samelson Resolutions of Schubert Varieties

Let X⊆G/B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\subseteq G/\\mathcal {B}$$\\end{document} be a Schubert variety in a flag manifold and let π:X~→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi : \ ilde{X} \\rightarrow X$$\\end{document} be a Bott–Samelson resolution of X. In this paper, we prove an effective version of the decomposition theorem for the derived pushforward Rπ∗QX~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R \\pi _{*} \\mathbb {Q}_{\ ilde{X}}$$\\end{document}. As a by-product, we obtain recursive procedure to extract Kazhdan–Lusztig polynomials from the polynomials introduced by Deodhar [7], which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan–Lusztig basis to the new one.

Following Schubert varieties under Feigin’s degeneration of the flag variety

We study the effect of Feigin’s flat degeneration of the type A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {A}$$\\end{document} flag variety on the defining ideals of its Schubert varieties. In particular, we describe two classes of Schubert varieties which stay irreducible under the degenerations and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some degenerate Schubert varieties (i.e. the vanishing sets of initial ideals of the ideals of Schubert varieties with respect to Feigin’s Gröbner degeneration) with Richardson varieties in higher rank partial flag varieties.

On singularity and normality of regular nilpotent Hessenberg varieties

On singularity and normality of regular nilpotent Hessenberg varieties

Tangent spaces and T-invariant curves of Schubert varieties

Tangent spaces and T-invariant curves of Schubert varieties

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem