Flag Manifolds Research Articles

The Critical Space for Orthogonally Invariant Varieties

Let q be a nondegenerate quadratic form on V. Let X ⊂ V be invariant for the action of a Lie group G contained in SO(V,q). For any f ∈ V consider the function df from X to mathbb C defined by df(x) = q(f − x). We show that the critical points of df lie in the subspace orthogonal to {mathfrak g}cdot f, that we call critical space. In particular any closest point to f in X lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.

Cohomology of generalized Dold spaces

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where X,S are arbitrary topological spaces. The resulting space P(S,X) will be called a generalized Dold space. When S and X are CW complexes satisfying certain natural requirements, we obtain a CW-structure on P(S,X). Under certain further hypotheses, we determine the mod 2 cohomology groups of P(S,X). We determine the Z2-cohomology algebra when X is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over P(S,X) associated to σ-conjugate complex bundles over X when the S is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

Schubert calculus from polyhedral parametrizations of Demazure crystals

One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this paper, we explicitly describe string parametrizations of opposite Demazure crystals, which give a natural generalization of reduced dual Kogan faces. We also relate reduced Kogan faces with Demazure crystals using the theory of mitosis operators, and apply these observations to develop the theory of Schubert calculus on symplectic Gelfand–Tsetlin polytopes.

On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime pge 5. Consider its perfectoid cover S^{text {ad}}(p^infty ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to S^{text {ad}}(p^infty ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to S^text {ad}(p^infty ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

Bn−1-bundles on the flag variety, I

In this paper, we consider the orbits of a Borel subgroup Bn−1 of Gn−1=GL(n−1) (respectively SO(n−1)) acting on the flag variety Bn of G=GL(n) (resp. SO(n)). The group Bn−1 acts on Bn with finitely many orbits, and we use the known description of Gn−1-orbits on B to study these orbits. In particular, we show that a Bn−1-orbit is a fibre bundle over a Bn−1-orbit in a generalized flag variety of Gn−1 with fibre a Bm−1-orbit on Bm for some m<n. We further use this fibre bundle structure to study the orbits inductively and describe the monoid action on the fibre bundle. In a sequel to this paper, we use these results to give a complete combinatorial model for these orbits and show how to understand the closure relations on these orbits in terms of the monoid action.

Extending torsors on the punctured Spec(A inf)

Abstract We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

Equivariant completions of affine spaces

Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства $\mathbb{C}^n$ в полные алгебраические многообразия $X$, для которых действие векторной группы $\mathbb{G}_a^n$ на $\mathbb{C}^n$ параллельными переносами продолжается до действия $\mathbb{G}_a^n$ на $X$. В первой части работы мы подробно изучаем соответствие Хассетта-Чинкеля, описывающее эквивариантные вложения $\mathbb{C}^n$ в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические многообразия и в многообразия Фано определeнных типов. Библиография: 109 названий.

Fano and Weak Fano Hessenberg Varieties

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 anticanonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.

Schubert calculus and intersection theory of flag manifolds

В 15-й проблеме Гильберта была поставлена задача строгого обоснования исчисления Шуберта; сложной частью такого обоснования является давняя проблема характеристик Шуберта. В ходе создания основ алгебраической геометрии Ван дер Варден и А. Вейль отнесли эту проблему к теории пересечений для многообразий флагов. В статье рассматриваются предыстория, содержание и решение проблемы характеристик. Нашими основными результатами являются единая формула для характеристик и систематическое описание колец пересечений многообразий флагов. Эффективность формулы и алгоритма проиллюстированы на явных примерах. Библиография: 71 название.

Schubert calculus and intersection theory of flag manifolds

Hilbert's 15th problem called for a rigorous foundation of Schubert calculus, of which a long-standing and challenging part is the Schubert problem of characteristics. In the course of securing a foundation for algebraic geometry, Van der Waerden and Weil attributed this problem to the intersection theory of flag manifolds. This article surveys the background, content, and solution of the problem of characteristics. Our main results are a unified formula for the characteristics and a systematic description of the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples. Bibliography: 71 titles.

Twisted quadratic foldings of root systems

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of typeE8E_8onto the subring of icosians of the quaternion algebra, which gives the root system of typeH4H_4.By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

Scalar Curvatures of Invariant Almost Hermitian Structures on Generalized Flag Manifolds

In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of K\"ahler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm C}$, where $s$ is Riemannian scalar curvature and $s_{\rm C}$ is the Chern scalar curvature.

Hilbert modules and complex analytic fibre spaces

We describe the ‘‘eigenbundle’’ (localization bundle) of Hilbert modules over bounded symmetric domains and show that the fibres are closely related to Kähler geometry of certain flag manifolds defined in terms of Jordan algebras. This gives an intriguing relationship between Hilbert space operators, coherent module sheaves and geometry of Jordan manifolds. The resulting complex-analytic fibre spaces are stratified of arbitrary depth r .

Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

Motivic Springer theory

We show that representations of convolution algebras such as Lustzig’s graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in type A and A˜ can be realized in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures.As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in type A˜ (with cyclic orientation) admit an affine paving.

UNIVERSAL GRAPH SCHUBERT VARIETIES

We consider the loci of invertible linear maps f : ℂn → (ℂn)* together with pairs of flags (E•, F•) in ℂn such that the various restrictions f : Fj → \( {E}_i^{\ast } \) 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 graph Schubert varieties.We consider similar loci where f is restricted to be symmetric or skew-symmetric. Their classes are now given by back-stable versions of the polynomials introduced by Wyser and Yong to represent classes of orbit closures for the orthogonal and symplectic groups acting on the type A flag variety. Using degeneracy locus formulas of Kazarian and of Anderson and Fulton, we obtain new Pfaffian formulas for these polynomials in the vexillary case. We also give a geometric interpretation of the involution Stanley symmetric functions of Hamaker, Marberg, and the author: they represent classes of involution graph Schubert varieties in isotropic Grassmannians.

Deformed sigma -models, Ricci flow and Toda field theories

It is shown that the Pohlmeyer map of a sigma -model with a toric two-dimensional target space naturally leads to the ‘sausage’ metric. We then elaborate the trigonometric deformation of the mathbb {CP}^{n-1}-model, proving that its T-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold sigma -models and Toda field theories.

Fundamental local equivalences in quantum geometric Langlands

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.

Flag manifolds over semifields

In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that the flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac-Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As a by-product, we prove a conjecture of Lusztig on the duality of totally nonnegative flag manifold of finite type.