We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over a scheme $X$ equipped with a flag of subbundles and a further structure determined by $K$. We describe how our equivariant formulas can be interpreted as formulas for such loci in the Chern classes of the various bundles on $X$.
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formu…
Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of homogeneous vector bundles. We prove the Abelian/non-Abelian Correspondence in this context: this allows us to compute genus zero Gromov--Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, disproving a conjecture of Craw. In the Appendices, which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum p…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic descri…
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove…
We give two formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric degeneracy loci. We apply them in enumerative geometry, explore their algebraic combinatorics, and discuss K theory generalizations.
We characterize the Sp_{2n} orbits in the flag variety for SL_{2n} with rationally smooth closure via a pattern avoidance criterion, also showing that the singular and rationally singular loci of such orbit closures coincide.
We consider the action of a semisimple subgroup $\hat G$ of a semisimple complex group $G$ on the flag variety $X=G/B$, and the linearizations of this action by line bundles $\mathcal L$ on $X$. The main result is an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a combinatorial formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat G$-ample cone, and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension $q$ form a convex polyhedral c…
Formulas for Lagrangian and orthogonal degeneracy loci; the Q-polynomials approach
设V是赋非退化辛或正交形式的概型X上的向量丛。设G是V的极大迷向子丛的参数化的Grassmannian丛，本文的主要目的是给出G中的轨迹类的公式，这些轨迹类的参数化子丛与V的迷向子丛的固定旗的连续成员相交，维数大于1，2，3，…. .我们给出的公式是G上重言丛的Q-多项式和旗成员的旗S-多项式的二次表达式。这些公式全球化的拉格朗日和正交退化轨迹。它们对于X中两个极大迷向子丛相交维数超过给定数的点的轨迹有特别好的形式。（几年前，哈里斯提出了一个寻找这些轨迹公式的问题。）在这种情况下，我们的公式是应用于E和F的Q多项式的简单二次表达式。本文推广了[P.Pragacz，Schur S-和Q-多项式的代数几何应用，SLN 1478（1991），130-191]中给出的公式。其应用之一是计算Pryms中Brill-Noether基因座的类别，见[De Concini，P. Pragacz，On the class of Brill-Noether loci for Prym varieties，Math.Ann.302（1995），687-697]。
We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtai…
We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible 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 Schubert varieties with Richardson varieties in higher rank partial flag varieties.
The Witt ring of a complex flag variety describes the interesting -- i.e. torsion -- part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides full flag varieties, projective spaces, and other varieties whose Witt rings were previously known, this class contains many flag varieties of exceptional types. Complete results are obtained for flag varieties of types G2 and F4. The results also extend to flag varieties over other algebrai…
We compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at up to two marked points. When the Brill-Noether number $ρ$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $ρ=1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-Martín-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degen…
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 play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.
In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag vari…
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
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 group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal ste…
Degeneracy loci polynomials for quiver representations generalize several important polynomials in algebraic combinatorics. In this paper we give a nonconventional generating sequence description of these polynomials, when the quiver is of Dynkin type.
Given a generic map between flagged vector bundles on a Cohen-Macaulay variety, we construct maximal Cohen-Macaulay modules with linear resolutions supported on the Schubert-type degeneracy loci. The linear resolution is provided by the Schubert complex, which is the main tool introduced and studied in this paper. These complexes extend the Schubert functors of Kraśkiewicz and Pragacz, and were motivated by the fact that Schur complexes resolve maximal Cohen-Macaulay modules supported on determinantal varieties. The resulting formula in K-theory provides a "linear approximation" of the structu…
The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V; or, equivalently, the closure of an orbit of the group GL(V) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V; this is equivalent to describing the closure of a GL(V)-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our…
We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.
In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.
We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are…
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.
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 Grassmanians 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, …
These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V.Kiritchenko, V.Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics…
We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The strongest results are obtained when working over a Riemann surface. Weaker partial results are obtained for adiabatic polarisations over a base of arbitrary dimension. We develop the notion of relative K-stability which embeds the idea of working over a base variety into the theory of K-stability. Natural constructions in filtered algebras equip the collecti…
The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties. We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerati…
Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the $K$-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study th…
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. Identifying an invertible linear map with its graph viewed as a point in a Grassmannian, we show that the closures of these loci have cohomology classes represented by the back-stable Schubert polynomials of Lam, Lee, and Shimozono. As a special case, we recover the result of Knutson, Lam, and Speyer that Stanley symmetric functions represent the classes of gr…
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …
We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety a…
Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anti-canonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.
In our previous work arXiv:1305.3543, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schutzenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of arXiv:0908.3628, which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elu…
In this paper we express the class of the structure sheaves of the closures of Deligne--Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles.
Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to …
We generalize the Tutte polynomial of a matroid to a morphism of matroids via the K-theory of flag varieties. We introduce two different generalizations, and demonstrate that each has its own merits, where the trade-off is between the ease of combinatorics and geometry. One generalization recovers the Las Vergnas Tutte polynomial of a morphism of matroids, which admits a corank-nullity formula and a deletion-contraction recursion. The other generalization does not, but better reflects the geometry of flag varieties.
A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $π$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $\mathbb{CP}^3$ of $S^4$. Deformations of twistor fibres project to real surfaces in $\mathbb{CP}^2$, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest t…
In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case. T…
Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called _collapsings_. We give a simple formula for the G-equivariant cohomology class on V, or_multidegree_, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is 1, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, w…
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains a…
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.
The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's "$d$-complete posets" to extend the Thomas-Yong rule from minu…
This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.
Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach). These equations may be used to compute Galois and monodromy groups of intersections of…
