A presentation of the torus-equivariant quantum K-theory ring of flag manifolds of type A, Part II: quantum double Grothendieck polynomials

We obtain new families of (1, 2)‐symplectic invariant metrics on the full complex flag manifolds F(n). For n ≥ 5, we characterize n − 3 different n‐dimensional families of (1, 2)‐symplectic invariant metrics on F(n). Each of these families corresponds to a different class of nonintegrable invariant almost complex structures on F(n).

Double Schubert polynomials for the classical groups

Pieri-type multiplication formula for quantum Grothendieck polynomials

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are manifestly positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.

In recent work, Lusztig’s positive root vectors (with respect to a distinguished choice of reduced decomposition of the longest element of the Weyl group) were shown to give a quantum tangent space for every A-series Drinfeld–Jimbo full quantum flag manifold Oq(Fn). Moreover, the associated differential calculus Ωq(0,∙)(Fn) was shown to have classical dimension, giving a direct q-deformation of the classical anti-holomorphic Dolbeault complex of Fn. Here, we examine in detail the rank two case, namely the full quantum flag manifold of Oq(SU3). In particular, we examine the ∗-differential calculus associated with Ωq(0,∙)(F3) and its noncommutative complex geometry. We find that the number of almost-complex structures reduces from 8 (that is 2 to the power of the number of positive roots of sl3) to 4 (that is 2 to the power of the number of simple roots of sl3). Moreover, we show that each of these almost-complex structures is integrable, which is to say, each of them is a complex structure. Finally, we observe that, due to non-centrality of all the non-degenerate coinvariant 2-forms, none of these complex structures admits a left Oq(SU3)-covariant noncommutative Kähler structure.

Let G be a connected, compact Lie group, A (generalized) flag manifold for G is the quotient of G by the centralizer of a torus. Hermitian symmetric spaces (e.g. complex projective spaces and Grassmannians), which are of the form G/C(T) for a circle T ⊂ G, are flag manifolds. The generic examples, though, are flag manifolds of the form G/T for T a maximal torus. The name derives from the manifold U(n)/diagonals of flags C0 = V0 ⊂ V1 ⊂ V2 ⊂ … ⊂ Vn = ℂ n in complex n-space. Flag manifolds enjoy many favorable geometric properties. They can be realized as coadjoint orbits of G, and thus carry an invariant symplectic form. There is also a complex description of flag manifolds as quotients of the complex group Gℂ. The symplectic and complex structures merge nicely: flag manifolds are homogeneous Kahler. Using these two properties—homogeneity and the Kahler condition—we easily compute curvature formulas. For special metrics flag manifolds are Kahler-Einstein. In any metric the Ricci curvature is positive, from which we deduce a vanishing theorem in cohomology. The full flag manifold G/T plays an important role in the representation theory of G a la Borel-Weil-Bott, and in that story the vanishing theorem plays a crucial part.

The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3-space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modem solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of intersection points, counted with appropriate multiplicities. This reformulation of the problem, though one of the great achievements of algebraic geometry, is only part of a solution. It remains to give a concrete model for the cohomology ring that makes explicit computation with Schubert classes possible. As it happens, the cohomology rings of Grassmannians can be identified with quotients of a polynomial ring so that Schubert classes correspond to Schur functions. Intersection numbers such as we are considering then turn out to be Littlewood-Richardson coefficients. For example, the answer to our four-lines 4 problem is the coefficient of the Schur function s(2 2) in the product s(1) X or 2. For an extended treatment and history of the subject, see [10], [11], [18]. The identification of Schur functions as Schubert polynomials for Grassmannians is a consequence of a more general and now highly developed theory of Schubert polynomials for the flag varieties of the special linear groups SL(n, C). The starting point for this more general theory is a construction of

Harmonic maps and the related theory of minimal surfaces are variational problems of long standing in differential geometry. Many important advances have been made in understanding harmonic maps of Riemann surfaces into symmetric spaces. In particular, ""twistor methods"" construct some, and in certain cases all, such mappings from holomorphic data. These notes develop techniques applicable to more general homogeneous manifolds, in particular a very general twistor result is proved. When applied to flag manifolds, this wider viewpoint allows many of the previously unrelated twistor results for symmetric spaces to be brought into a unified framework. These methods also enable a classification of harmonic maps into full flag manifolds to be established, and new examples are constructed. The techniques used are mostly a blend of the theory of compact Lie groups and complex differential geometry. This book should be of interest to mathematicians with experience in differential geometry and to theoretical physicists.

We study homogeneous curves in generalized flag manifolds G/K with G_2-type t-roots, which are geodesics with respect to each G-invariant metric on G/K. These curves are called equigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with t-roots. Also, these spaces are a generalization of the exceptional full flag manifold G_2/T. We give a characterization for structural equigeodesics for flag manifolds with G_2-type t-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.

Noncommutative algebras related with Schubert calculus on Coxeter groups

For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce 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 corresponding 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. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov. Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

The classical Schubert cells on a flag manifold G/H give a cell decomposition for G/H whose Kronecker duals (known as Schubert classes) form an additive base for the integral cohomology H^{\ast}(G/H). We present a formula that expresses Steenrod mod-p operations on Schubert classes in G/H in terms of Cartan numbers of G.

Lagrangian submanifolds of the nearly Kähler full flag manifold [formula omitted

We investigate certain immersions of constant curvature from Riemann surfaces into flag manifolds equipped with invariant metrics, namely primitive lifts associated to pseudoholomorphic maps of surfaces into complex Grassmannians. We prove that a primitive immersion from the two-sphere into the full flag manifold which has constant curvature with respect to at least one invariant metric is unitarily equivalent to the primitive lift of a Veronese map, hence it has constant curvature with respect to all invariant metrics. We prove a partial generalization of this result to the case where the domain is a general simply connected Riemann surface. On the way, we consider the problem of finding the invariant metric on the flag manifold, under a certain normalization condition, that maximizes the induced area of the two-sphere by a given primitive immersion.