Partial Flag Manifolds Research Articles

Projections of Richardson varieties

Abstract. While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold G/P are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, Cohen–Macaulay, have rational resolutions, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem (Schwede, Kumar–Mehta) that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by Hague, whose proof we simplify somewhat.) One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an EL-shellable ball (Björner–Wachs). We prove that the projection of such a complex into the order complex of the Bruhat order on W / W P $W/W_P$ is again a shellable ball. This requires extensive analysis of “P-Bruhat order”, a generalization of the k-Bruhat order of Bergeron–Sottile. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Gröbner degeneration takes each projected Richardson variety to the Stanley–Reisner scheme of its corresponding ball.

Knot homology groups from instantons

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.

Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients

The Belkale-Kumar product on $H^*(G/P)$ is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case $G=GL_n$, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for $G/P$ a $(d-1)$-step flag manifold, each Belkale-Kumar structure constant is a product of $d\choose 2$ Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into $d-1$ factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward '04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always $1$; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension $1$ or $2$.

Chern numbers and the geometry of partial flag manifolds

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n = \mathit{SU}(n + 2)/S(U(n) × U(1) × U(1)) . For all n > 1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of ℂP^{n + 1} . The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because F_n is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that F_n is not geometrically formal.

Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T ⊂ G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad( G) c ≃ G/ Z( c), where c ∈ Lie( T), and Z( c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad( c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z( cc) B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.

GLSMs for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0, 2) models. Second, we review constructions of Calabi–Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSMs in which the Kähler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori–Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov’s homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSMs are precisely Quot and hyperquot schemes, as one would expect mathematically.

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in math.AG/0407254) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding ``abelianized'' quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.

A class of perverse sheaves on a partial flag manifold

We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G G defined over a finite field which are equivariant for the action of the group of rational points of G G . The definition of this class is similar to the definition of parabolic character sheaves.

Classification of Ding's Schubert Varieties: Finer Rook Equivalence

AbstractK. Ding studied a class of Schubert varieties Xƛ in type A partial flag manifolds, indexed by integer partitions ƛ and in bijection with dominant permutations. He observed that the Schubert cell structure of Xƛ is indexed by maximal rook placements on the Ferrers board Bƛ, and that the integral cohomology groups H*(Xƛ; ℤ), H*(Xμ; ℤ) are additively isomorphic exactly when the Ferrers boards Bƛ, Bμ satisfy the combinatorial condition of rook-equivalence.We classify the varieties Xƛ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.

Quantum cohomology of partial flag manifolds

We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties SL ⁡ ( n ) / P \operatorname {SL}(n)/P , including the quantum Pieri and quantum Giambelli formulas and the presentation.

Quantum cohomology of flag manifolds

We study two aspects of quantum Schubert calculus: a presentation of the (small) quantum cohomology ring of partial flag manifolds and a quantum Giambelli formula. Our proof gives a relationship between universal Schubert polynomials as defined by Fulton and quantum Schubert polynomials, as defined by Fomin, Gelfand, and Postnikov, and later extended by Ciocan-Fontanine. Intersection theory on hyperquot schemes is an essential element of the proof.

Cohomology of Smooth Schubert Varieties in Partial Flag Manifolds

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

Parabolic Kazhdan–Lusztig Polynomials and Schubert Varieties

We shall give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac–Moody Lie algebras in terms of parabolic Kazhdan–Lusztig polynomials introduced by Deodhar.

The geometry of the full Kostant–Toda lattice of [formula omitted

The full Kostant–Toda lattice is an integrable system whose geometry appears naturally in the setting of flag manifolds but is not easily apparent in the original phase space. Separatrices in the flows for two different families of integrals in the Toda lattice of sl(4, C ) appear in splittings of weight polytopes associated to partial flag manifolds of Sl(4, C ) . Each family of integrals gives rise to a C * -bundle of level set varieties with singular fibers. In a neighborhood of a separatrix, the monodromy is a single twist of the noncompact cycle around the cylinder, C * . Near another type of singularity, the cycle is twisted twice; this interchanges the source and sink of one of the Hamiltonian flows.

Mirror symmetry and toric degenerations of partial flag manifolds

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n1, . . . , nl, n) to a certain Gorenstein toric Fano variety P (n1, . . . , nl, n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n1, . . . , nl, n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n1, . . . , nl, n). Mathematisches Institut, Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany, email address: batyrev@bastau.mathematik.uni-tuebingen.de Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA email address: ciocan@math.okstate.edu Department of Mathematics, University of California Davis, Davis, CA 95616, USA email address: bumsig@math.ucdavis.edu FB 17, Mathematik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany, email address: straten@mathematik.uni-mainz.de

Total positivity in partial flag manifolds

The projective space of R n \mathbf {R}^{n} has a natural open subset: the set of lines spanned by vectors with all coordinates > 0 >0 . Such a subset can be defined more generally for any partial flag manifold of a split semisimple real algebraic group. The main result of the paper is that this subset can be defined by algebraic equalities and inequalities.

Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings

Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings Get access Bumsig Kim Bumsig Kim Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1995, Issue 1, 1995, Pages 1–16, https://doi.org/10.1155/S1073792895000018 Published: 01 January 1995 Article history Received: 11 November 1994 Published: 01 January 1995

Quantum cohomology of partial flag manifolds $$F_{n_1 } \ldots _{n_k }$$

We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.

The biflag manifold and the fixed points of a unipotent transformation on the flag manifold

We investigate the topology of the submanifold of the partial flag manifold consisting of those flags S 1 ⊂⋯⊂ S d in F n( F= R or C) that are fixed by a given unipotent A ϵ SL n ( F ) and for which the restrictions A∣ S i have prescribed Jordan type. It is shown that this submanifold has a compact nonsingular projective variety as a strong deformation retract. This variety, the biflag manifold, consists of rectangular arrays of subspaces which are nested along each row and column. It is proven that the biflag manifold has the homology type (mod 2 for F= R ) of a product of Grassmannians. A partition of the biflag manifold into a finite union of affine spaces is also given. In the special case when the biflag manifold is a flag manifold, this partition coincides with the Bruhat cell decomposition.