Full Flag Variety Research Articles

On Landau–Ginzburg models for quadrics and flat sections of Dubrovin connections

This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau–Ginzburg model (Xˇcan,Wq). Here Xˇcan is the complement of an anticanonical divisor in a Langlands dual quadric. The superpotential Wq is a regular function on Xˇcan and is written in terms of coordinates which are naturally identified with a cohomology basis of the original quadric. This superpotential is shown to extend the earlier Landau–Ginzburg model of Givental, and to be isomorphic to the Lie-theoretic mirror introduced in [36]. We also introduce a Laurent polynomial superpotential which is the restriction of Wq to a particular torus in Xˇcan. Together with results from [31] for odd quadrics, we obtain a combinatorial model for the Laurent polynomial superpotential in terms of a quiver, in the vein of those introduced in the 1990's by Givental for type A full flag varieties. These Laurent polynomial superpotentials form a single series, despite the fact that our mirrors of even quadrics are defined on dual quadrics, while the mirror to an odd quadric is naturally defined on a projective space. Finally, we express flat sections of the (dual) Dubrovin connection in a natural way in terms of oscillating integrals associated to (Xˇcan,Wq) and compute explicitly a particular flat section.

Intersection theory of the Peterson variety and certain singularities of Schubert varieties

Precup recently proved that intersections with Schubert cells pave regular nilpotent Hessenberg varieties. We use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explicitly identify some of the coefficients of Schubert classes in this expansion, answering a problem of independent interest in Schubert calculus. We also identify some singular points in a certain family of Schubert varieties in general Lie type.

Schubert Calculus and the Homology of the Peterson Variety

We use the tight correlation between the geometry of the Peterson variety and the combinatorics the symmetric group to prove that homology of the Peterson variety injects into the homology of the flag variety. Our proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety, and shows that these intersections are proper and transverse.

Projected Richardson varieties and affine Schubert varieties

Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$.

KO-rings of full flag varieties

We present type-independent computations of the K O \mathrm {KO} -groups of full flag varieties, i.e. of quotient spaces G / T G/T of compact Lie groups by their maximal tori. Our main tool is the identification of the Witt ring, a quotient of the K O \mathrm {KO} -ring, of these varieties with the Tate cohomology of their complex K \mathrm {K} -ring. The computations show that the Witt ring is an exterior algebra whose generators are determined by representations of G G .

Oddification of the Cohomology of Type A Springer Varieties

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top degree component of the odd cohomology of Springer varieties is identified with the corresponding Specht module of the Hecke algebra at q=-1.

Schubert calculus of Richardson varieties stable under spherical Levi subgroups

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.

An example of orthogonal triple flag variety of finite type

Let G be the split special orthogonal group of degree 2n+1 over a field F of charF≠2. Then we describe G-orbits on the triple flag varieties G/P×G/P×G/P and G/P×G/P×G/B with respect to the diagonal action of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is a Borel subgroup. As by-products, we also describe GLn-orbits on G/B, Q2n-orbits on the full flag variety of GL2n where Q2n is the fixed-point subgroup in Sp2n of a nonzero vector in F2n and 1×Sp2n-orbits on the full flag variety of GL2n+1. In the same way, we can also solve the same problem for SO2n where the maximal parabolic subgroup P is of the shape (n,n).

Richardson Varieties have Kawamata Log Terminal Singularities

Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities.

K-orbits on [formula omitted] and Schubert constants for pairs of signed shuffles in types C and D

We give positive descriptions for certain Schubert structure constants cu,vw for the full flag variety in Lie types C and D. This is accomplished by first observing that a number of the K=GL(n,C)-orbit closures on these flag varieties coincide with Richardson varieties, and then applying a theorem of M. Brion on the decomposition of such an orbit closure in the Schubert basis in terms of paths in the weak order graph.

A Mirror Symmetric Solution to the Quantum Toda Lattice

We give a representation-theoretic proof of a conjecture from Rietsch (Adv Math 217:2401–2442, 2008) providing integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SLn/B in a uniform way to arbitrary type, and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive and totally negative critical point of the ‘superpotential’ in every mirror fiber.

Special Lagrangian fibrations on the flag variety F 3

We propose a construction of a Lagrangian torus fibration of the full flag variety in ℂ 3. In contrast to the classical fibration obtained from the Gelfand-Zeitlin system, the proposed fibration is special Lagrangian.

Maximal compatible splitting and diagonals of Kempf varieties

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group.

On Positivity in T-Equivariant K-Theory of Flag Varieties

We prove some general results on the T-equivariant K-theory K T (G/P) of the flag variety G/P, where G is a complex semisimple algebraic group, P is a parabolic subgroup, and T is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K T (G/P) for the product of two basis elements written in terms of the basis of K T (G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.

Poisson structures on affine spaces and flag varieties. II

The standard Poisson structures on the flag varieties $G/P$ of a complex reductive algebraic group $G$ are investigated. It is shown that the orbits of symplectic leaves in $G/P$ under a fixed maximal torus of $G$ are smooth irreducible locally closed subvarieties of $G/P$, isomorphic to intersections of dual Schubert cells in the full flag variety $G/B$ of $G$, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces, and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson–Dirac submanifolds. For a parabolic subgroup $P$ with abelian unipotent radical (in which case $G/P$ is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor $L$ of $P$ on $G/P$ are complete Poisson subvarieties which are quotients of $L$, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on $G/P$ vanishes at all special base points for the $L$-orbits on $G/P$ constructed by Richardson, Röhrle, and Steinberg.

Variation of Mumford quotients by torus actions on full flag varieties. II

This paper continues the study of the variation of the Mumford quotient by the action of a maximal torus on a flag variety in terms of its dependence on the projective embedding . In the case when is of type , the Picard group of the quotient is calculated under the assumption that the -linearization comes from the standard -linearization.Bibliography : 12 titles.

Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.

On Positivity in T-Equivariant K-Theory of Flag Varieties

We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K_T(G/P) for the product of two basis elements written in terms of the basis of K_T(G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.

Variation of Mumford quotients by torus actions on full flag varieties. I

We study the variation of the Mumford quotient by the action of a maximal torus on a flag variety as we change the projective embedding , where the -linearization is induced by the standard -linearization. To do this, we describe the linear spans of the supports of the semistable orbits. This enables us to calculate the rank of the Picard group of the quotient in the case when contains no simple components of type .

Global $F$-regularity of Schubert varieties with applications to $\mathcal {D}$-modules

A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F-regular if the section ring S(C) = 0n>o H?(X, Cn) of an ample line bundle C on X is strongly F-regular in the sense of H?chster and Huneke [9] (cf. Definition 1.1). This important notion was introduced by Karen Smith in [18]. In this paper we prove that Schubert varieties are globally F-regular. An im mediate consequence is that local rings of Schubert varieties are strongly F-regular and thereby F-rational. Another consequence is that local rings of varieties (like de terminantal varieties) that can be identified with open subsets of Schubert varieties (cf. [13]) are strongly F-regular Let X denote a flag variety and Y C X a Schubert variety over k. Then the local cohomology sheaves HJY(Ox) are equivariant (for the action of the Borel subgroup) and holonomic (in the sense of [4]) X>x-niodules. As an application of ir rationality of Schubert varieties we apply recent results of Blickle (cf. [2]) to prove that the simple objects in the category of equivariant and holonomic Px-modules are precisely the local cohomology sheaves Hy{Ox), where c is the codimension of Y in X. Using a local Grothendieck-Cousin complex from [11], we prove that the decomposition of the local cohomology modules with support in Bruhat cells is multiplicity free (see ?4.2). In characteristic zero the local cohomology modules with support in Bruhat cells correspond to dual Verma modules. In this setting the decomposition behavior and the simple T>x -modules arise from intersection cohomology complexes of Schubert varieties by the Riemann-Hilbert correspondence. If we pick the singular codi mension one Schubert variety Y in the full flag variety Z for SL4 in characteristic zero, computations in Kazhdan-Lusztig theory show that Hy(Oz) is not a simple ?>z-module (see ?4.1).