共查询到20条相似文献,搜索用时 546 毫秒
1.
Julianna S. Tymoczko 《Selecta Mathematica, New Series》2007,13(2):353-367
Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology,
geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent
Hessenberg varieties for all classical types, generalizing results of De Concini–Lusztig–Procesi and Kostant. This paving
is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving
and their dimensions are identified by combinatorial conditions on roots. We use the paving to prove these Hessenberg varieties
have no odd-dimensional homology.
相似文献
2.
Cristian Lenart Frank Sottile 《Proceedings of the American Mathematical Society》2003,131(11):3319-3328
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.
3.
Izzet Coskun 《Inventiones Mathematicae》2009,176(2):325-395
This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties.
As a consequence, we obtain a positive, geometric rule for expressing the structure constants of the cohomology of two-step
flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for computing the structure constants
of the small quantum cohomology of Grassmannians. We also obtain a positive, geometric rule for computing the classes of subvarieties
of Grassmannians that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. These
rules have numerous applications to geometry, representation theory and the theory of symmetric functions.
Mathematics Subject Classification (2000) Primary 14M15, 14N35, 32M10 相似文献
4.
The recursive nature of cominuscule Schubert calculus 总被引:1,自引:0,他引:1
The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities. 相似文献
5.
Edward Richmond 《Journal of Algebraic Combinatorics》2009,30(1):1-17
Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag
varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller”
flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain
deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure
constant can be written in terms of structure constants coming from induced Grassmannians. 相似文献
6.
Izzet Coskun 《Advances in Mathematics》2011,(4):2441
This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles. 相似文献
7.
Martha Precup 《Selecta Mathematica, New Series》2013,19(4):903-922
In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of $\mathfrak{gl }_n(\mathbb C )$ are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology. 相似文献
8.
We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic
subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the
classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the
product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer
coefficients, in terms of special Schubert class generators and relations. 相似文献
9.
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. 相似文献
10.
We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus. 相似文献
11.
We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces,
to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in
the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them
as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding
Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on
its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive
degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology
of its associated complex discriminant variety.
Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000 相似文献
12.
In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag
varieties for rank 2 semisimple algebraic groups. 相似文献
13.
Kazhdan-Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras.
We give a lower bound for the values Px,w(1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a
subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds
geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple
group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action.
This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties. 相似文献
14.
Izzet Coskun 《Mathematische Annalen》2010,346(2):419-447
We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology
of partial flag varieties F(k
1, ..., k
r
; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed
from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to
the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing
these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals
a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties. 相似文献
15.
Seung Jin Lee 《Journal of Algebraic Combinatorics》2018,47(2):213-231
We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra. 相似文献
16.
《Indagationes Mathematicae》2021,32(6):1275-1289
We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara’s flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac–Moody algebras and give a new proof of the Kazhdan–Lusztig conjecture for blocks containing an antidominant element. 相似文献
17.
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic
tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes
Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection
cohomology stalks.
Received: 9 November 2000 / Published online: 24 September 2001 相似文献
18.
For each infinite series of the classical Lie groups of type B, C or D, we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov. 相似文献
19.
Harry Tamvakis 《Mathematische Zeitschrift》2011,268(1-2):355-370
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety ${\mathfrak X={\rm SO}_N/B}$ . We use these polynomials to describe the arithmetic Schubert calculus on ${\mathfrak X}$ . Moreover, we give a method to compute the natural arithmetic Chern numbers on ${\mathfrak X}$ , and show that they are all rational numbers. 相似文献
20.
Izzet Coskun 《Israel Journal of Mathematics》2014,200(1):85-126
A Schubert class σ in the cohomology of a homogeneous variety X is called rigid if the only projective subvarieties of X representing σ are Schubert varieties. A Schubert class σ is called multi rigid if the only projective subvarieties representing positive integral multiples of σ are unions of Schubert varieties. In this paper, we discuss the rigidity and multi rigidity of Schubert classes in orthogonal Grassmannians. For a large set of non-rigid classes, we provide explicit deformations of Schubert varieties using combinatorially defined varieties called restriction varieties. We characterize rigid and multi rigid Schubert classes of Grassmannian and quadric type. We also characterize all the rigid classes in OG(2, n) if n > 8. 相似文献