Schubert Varieties Research Articles

Springer fibers and Schubert points

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field

Let π be an irreducible, complex, smooth representation of GLn over a local non-archimedean (skew) field. Assuming π has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility of the parabolic induction of π⊗π to GL2n. The latter irreducibility property is the p-adic analogue of a special case of the notion of “real representations” introduced by Leclerc and studied recently by Kang–Kashiwara–Kim–Oh (in the context of KLR or quantum affine algebras). Our criterion is in terms of singularities of Schubert varieties of type A and admits a simple combinatorial description. It is also equivalent to a condition studied by Geiss–Leclerc–Schröer.

Chern–Schwartz–MacPherson classes of degeneracy loci

The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas---including residue generating sequences---for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (2-step) positivity properties for some relevant representations.

Weak Z-Property of the Absolute Order on Groups Generated by Sets Closed Under Taking Inverses

The Bruhat order is a well-studied partial order on Coxeter groups and Schubert varieties. Deodhar provided several characterizations of the Bruhat order, including the so-called “Z-property.” Another partial order on Coxeter groups that is relevant in combinatorics is the absolute order, which extends the notion of the classical noncrossing partitions to Coxeter groups. In this note, we prove a result that implies that the absolute order satisfies a weaker version of the Z-property.

EFFECTIVE DIVISORS ON BOTT–SAMELSON VARIETIES

We compute the cone of effective divisors on a Bott–Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain B-equivariant ℙ1 bundles. As an application, we compute the cone of effective codimension-two cycles on Bott–Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott–Samelson variety to contain a dense B-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott–Samelson variety is log Fano.

Diagrams and Essential Sets for Signed Permutations

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

Newton–Okounkov bodies for Bott–Samelson varieties and string polytopes for generalized Demazure modules

A Newton–Okounkov body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body inherits information about algebraic, geometric, and combinatorial properties of the original projective variety; for instance, Kaveh showed that string polytopes in the representation theory of algebraic groups are examples of Newton–Okounkov bodies for Schubert varieties. In this paper, we extend the notion of string polytopes for Demazure modules to generalized Demazure modules, and prove that the resulting generalized string polytopes are identical to the Newton–Okounkov bodies for Bott–Samelson varieties with respect to a specific valuation. As an application of this result, we show that these are indeed polytopes.

The Horn inequalities from a geometric point of view

We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale’s geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood–Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms.

Iitaka dimension for cycles

We define the Iitaka dimension of a numerical cycle class and develop its theory. We conjecture that the Iitaka dimension is integer valued and give some evidence in this direction. We focus on two cases of geometric interest: Schubert cycles on Grassmannians and cycles contracted by morphisms.

Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

AbstractWe study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.

Push-pull operators on the formal affine Demazure algebra and its dual

In the present paper we introduce and study the push pull operators on the formal affine Demazure algebra and its dual. As an application we provide a non-degenerate pairing on the dual of the formal affine Demazure algebra which serves as an algebraic counterpart of the natural pairing on the T-equivariant oriented cohomology of G/B induced by multiplication and push-forward to a point. This paper can be viewed as the next step towards the `algebraization program' for equivariant oriented cohomology theories started in arXiv:0905.1341 and continued in arXiv:1208.4114 and arXiv:1209.1676; the general idea being to match cohomology rings of algebraic varieties and elements of classical interest in them (such as classes of Schubert varieties) with algebraic and combinatorial objects that can be introduced in the spirit of [Demazure, Invariants sym\'etriques entiers des groupes de Weyl et torsion, Invent. Math. 21:287-301, 1973] and [Kostant, Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Advances in Math. 62:187-237, 1986].

Equivariant quantum cohomology of the odd symplectic Grassmannian

The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $$\mathrm {IG}$$ with two orbits, and $$\mathrm {IG}$$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $$\mathrm {IG}(k, 2n+2)$$ . We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $$\mathrm {IG}$$ , i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case $$k=2$$ , and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

Abstract We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.

Loop Models and K-Theory

This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems ($R$-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for $K$-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties.

Association Schemes on the Schubert Cells of a Grassmannian

Let $${\mathbb {F}}$$ be any field. The Grassmannian $${\mathrm {Gr}}(m,n)$$ is the set of m-dimensional subspaces in $${\mathbb {F}}^n$$ , and the general linear group $${\mathrm {GL}}_n({\mathbb {F}})$$ acts transitively on it. The Schubert cells of $${\mathrm {Gr}}(m,n)$$ are the orbits of the Borel subgroup $${\mathcal {B}} \subset {\mathrm {GL}}_n({\mathbb {F}})$$ on $${\mathrm {Gr}}(m,n)$$ . We consider the association scheme on each Schubert cell defined by the $${\mathcal {B}}$$ -action and show it is symmetric and it is the generalized wreath product of one-class association schemes, which was introduced by Bailey (Eur J Comb 27(3):428–435, 2006).

Cosmall Roots and Curve Neighborhoods

A previous result in a paper by Buch and Mihalcea relates the curve neighborhood of any Schubert variety to the curve neighborhood of a point and introduces the idea of P-cosmall roots. In our paper, we prove a conjecture given in that same paper about P-cosmall roots. We go on to provide explicit combinatorial conditions for a root to be P-cosmall.

Projected Gromov-Witten varieties in cominuscule spaces

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G / P X = G/P . When X X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3-point, genus zero) K K -theoretic Gromov-Witten invariants of X X are determined by projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer, and provides an alternative version of the ‘quantum equals classical’ theorem. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety.

Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus

The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.

Two Murnaghan-Nakayama Rules in Schubert Calculus

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan- Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming fromthe Chern character. Like the classical rule, both rules are multiplicity-free signed sums.

Parabolic subgroups and automorphism groups of Schubert varieties

Let G be a simple algebraic group of adjoint type over the field C of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and X(w) be the Schubert variety in G/B corresponding to w. In this article we show that given any parabolic subgroup P of G containing B properly, there is an element w∈W such that P is the connected component, containing the identity element of the group of all algebraic automorphisms of X(w).