Partial Flag Varieties Research Articles

Radon Transforms of Twisted D-Modules on Partial Flag Varieties

In this paper we study intertwining functors for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give equivalences of derived categories of twisted D-modules. This is a generalization of a result by Marastoni. We also show that these intertwining functors from dominant to antidominant direction are compatible with taking global sections.

On the [formula omitted]-vectors of Gelfand–Cetlin polytopes

A Gelfand–Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev et al. (2013).

Ulrich Schur bundles on flag varieties

In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(1,n−1;n), F(2,n−1;n), F(2,n−2;n), F(k,k+1;n) and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles.

Lifting preprojective algebras to orders and categorifying partial flag varieties

We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schr\"oer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)/[Ae] \cong \operatorname{Sub}\nolimits Q$ for an injective $B$-module $Q$ where $B := A/(e)$. These results generalize work by Jensen-King-Su concerning the cluster algebra structure of the Grassmannian $\operatorname{Gr}\nolimits_m(\mathbb{C}^n)$.

COMPARISON OF CANONICAL BASES FOR SCHUR AND UNIVERSAL ENVELOPING ALGEBRAS

We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In order to prove this result, we also explain the connections between categorifications of the Schur algebra which arise from parity sheaves on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's "flag category," which are of some independent interest.

Some results on equivariant contact geometry for partial flag varieties

We study equivariant contact structures on complex projective varieties arising as partial flag varieties [Formula: see text], where [Formula: see text] is a connected, simply-connected complex simple group of type ADE and [Formula: see text] is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby’s classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of [Formula: see text]-equivariant vector bundles on [Formula: see text]. A byproduct of our argument is a canonical, global description of the unique [Formula: see text]-invariant contact structure on the isotropic Grassmannian of 2-planes in [Formula: see text].

Total positivity, Schubert positivity, and geometric Satake

Let G be a simple, simply-connected complex algebraic group, and let X⊂G∨ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian GrG. Peterson furthermore connected X to the quantum cohomology rings of partial flag varieties G/P.In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković–Vilonen positivity obtained from the MV-cycles in GrG. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.

Ideals of bounded rank symmetric tensors are generated in bounded degree

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.

Standard Monomial Theory for Wonderful Varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.

COTANGENT BUNDLE TO THE GRASSMANN VARIETY

We show that there is an affine Schubert variety in the infinite-dimensional partial Flag variety (associated to the two-step parabolic subgroup of the Kac–Moody group , corresponding to omitting α 0; α d ) which is a natural compactification of the cotangent bundle to the Grassmann variety.

On equivariant quantum Schubert calculus for [formula omitted

We show a Z2-filtered algebraic structure and a “quantum to classical” principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety Fℓn1,⋯,nk;n+1 of Lie type A.

On the restriction of Zuckerman’s derived functor modules A q (λ) to reductive subgroups

In this article, we study the restriction of Zuckerman's derived functor $({\frak g},K)$-modules $A_{\frak q}}(\lambda)$ to ${\frak g}'$ for symmetric pairs of reductive Lie algebras $({\frak g},{\frak g}')$. When the restriction decomposes into irreducible $({\frak g}',K')$-modules, we give an upper bound for the branching law. In particular, we prove that each $({\frak g}',K')$-module occurring in the restriction is isomorphic to a submodule of $A_{\frak q}'}(\lambda')$ for a parabolic subalgebra ${\frak q}'$ of ${\frak g}'$, and determine their associated varieties. For the proof, we realize $A_{\frak q}}(\lambda)$ on complex partial flag varieties by using ${\mathcal D}$-modules.

Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence

We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0–1 polytopes. Moreover, we provide a decomposition of the marked poset into indecomposable marked posets, which respects this Minkowski sum decomposition for the marked chain polytopes.

Type A quiver loci and Schubert varieties

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 normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.

Partial flag varieties, stable envelopes, and weight functions

We consider the cotangent bundle T^*\mathfrak F_{\lambda} of a \mathrm {GL}_n partial flag variety, \lambda = (\lambda_1,\dots,\lambda_N), |\lambda|=\sum_i\lambda_i=n , and the torus T=(\mathbb C^\times)^{n+1} equivariant cohomology H^*_T(T^*\mathfrak F_{\lambda}) . In [9], a Yangian module structure was introduced on \bigoplus_{|\lambda|=n} H^*_T(T^*\mathfrak F_{\lambda}) . We identify this Yangian module structure with the Yangian module structure introduced in [5]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*\mathfrak F_{\lambda}) , described in [9], with the action of the dynamical Hamiltonians from [20], [10], [5]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [9]. The formula is in terms of the Yangian weight functions introduced in [19], c.f. [21], [22], in order to construct q-hypergeometric solutions of qKZ equations.

Existence of Affine Pavings for Varieties of Partial Flags Associated to Nilpotent Elements

The flag variety of a complex reductive linear algebraic group |$G$| is by definition the quotient |$G/B$| by a Borel subgroup. It can be regarded as the set of Borel subalgebras of |${\rm Lie}(G)$|⁠. Given a nilpotent element |$e$| in |${\rm Lie}(G)$|⁠, one calls Springer fiber the subvariety formed by the Borel subalgebras which |$e$| belongs to. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by De Concini, Lusztig, and Procesi asserts that when |$G$| is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In this paper, we study varieties generalizing the Springer fibers in two ways: being contained in a partial flag variety |$G/P$| (the quotient by a parabolic subgroup, instead of a Borel subgroup) and defined by a more general belonging condition (in terms of an ideal of |${\rm Lie}(P))$|⁠. These varieties arise, for instance, as fibers of resolutions of nilpotent orbit closures. The main result of the paper is a generalization of De Concini, Lusztig, and Procesi's theorem to this context.Communicated by Corrado de Concini

A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings

We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.

Trigonometric weight functions asK-theoretic stable envelope maps for the cotangent bundle of a flag variety

We consider the cotangent bundle $T^*F_\lambda$ of a $GL_n$ partial flag variety, $\lambda=(\lambda_1,...,\lambda_N)$, $|\lambda|=\sum_i\lambda_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_\lambda)$. We introduce K-theoretic stable envelope maps $\Stab_{\sigma}: \oplus_{|\lambda|=n} K_T((T^*F_\lambda)^T)\to\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$, where $\sigma\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_\lambda))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_\lambda))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_\lambda)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_\lambda))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_\lambda)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_\lambda)$ and with the help of the trigonometric weight functions introduced in [TV1, TV3] to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix 5 we describe the Bethe algebra of the XXZ model by generators and relations.

Pfaffian sum formula for the symplectic Grassmannians

We study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a sum of multi Schur-Pfaffians, whose entries are equivariantly modified special Schubert classes. Our result gives a proof to Wilson’s conjectural formula, which generalizes the Giambelli formula for the ordinary cohomology proved by Buch–Kresch–Tamvakis, given in terms of Young’s raising operators. Furthermore we show that the formula extends to a certain family of Schubert classes of the symplectic partial isotropic flag varieties.

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 $A$.