Flag Manifolds Research Articles

Chow–Witt rings and topology of flag varieties

AbstractThe paper computes the Witt‐sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch‐type theorem for Witt‐sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2‐torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.

AN EXAMPLE OF THE JANTZEN FILTRATION OF A D-MODULE

Abstract We compute the Jantzen filtration of a $\mathcal {D}$ -module on the flag variety of $\operatorname {\mathrm {SL}}_2(\mathbb {C})$ . At each step in the computation, we illustrate the $\mathfrak {sl}_2(\mathbb {C})$ -module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the $\mathcal {D}$ -module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson and Bernstein.

Generalized juggling patterns, quiver Grassmannians and affine flag varieties

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmannians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

Symplectic Grassmannians and cyclic quivers

AbstractThe goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

Curves on Brill–Noether special K3 surfaces

AbstractMukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface is equivalent to the Brill–Noether generality of smooth curves in the linear system , is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus , which provides a new and unified proof even in the genera where Mukai models exist.

Equigeodesics on Generalized Flag Manifolds with Five Isotropy Summands

Equigeodesics on Generalized Flag Manifolds with Five Isotropy Summands

The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag Varieties

Abstract Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.

The classical and quantum particle on a flag manifold

Abstract In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace–Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace–Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

Geometric Optimization of Restricted-Open and Complete Active Space Self-Consistent Field Wave Functions.

We explore Riemannian optimization methods for Restricted-Open-shell Hartree-Fock (ROHF) and Complete Active Space Self-Consistent Field (CASSCF) methods. After showing that ROHF and CASSCF can be reformulated as optimization problems on so-called "flag manifolds", we review Riemannian optimization basics and their application to these specific problems. We compare these methods to traditional ones and find robust convergence properties without fine-tuning of numerical parameters. Our study suggests that Riemannian optimization is a valuable addition to orbital optimization for ROHF and CASSCF, warranting further investigation.

Higher Tate traces of Chow motives

Abstract We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

Restrictions on Anosov subgroups of 𝑆𝑝(2𝑛,ℝ)

Let n ∈ N n\in \mathbb {N} and let Θ ⊂ { 1 , … , n } \Theta \subset \{1,\dots ,n\} be a nonempty subset. We prove that if Θ \Theta contains an odd integer, then any P Θ P_\Theta -Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) is virtually isomorphic to a free or surface group. On the other hand, if Θ \Theta does not contain any odd integers, then there exists a P Θ P_\Theta -Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) which is not virtually isomorphic to a free or surface group. We also exhibit new examples of maximally antipodal subsets of certain flag manifolds; these arise as limit sets of rank 1 1 subgroups.

Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties

Abstract Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G Q<G a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G / Q G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of H = SL 2 ⁡ ( R ) H=\operatorname{SL}_{2}(\mathbb{R}) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q G=Q , the type of stationary measures that 𝜇 admits depends strongly on the position of 𝐻 relative to 𝑄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss.

Stable envelopes for slices of the affine Grassmannian

The affine Grassmannian associated to a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}$$\\end{document} is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the G=PSL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}= \ extbf{PSL}_{2}$$\\end{document} case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

Cohomology-developed matrices: constructing families of weighing matrices and automorphism actions

The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call cohomology-developed matrices. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of quasiproducts, which is a generalization of the Kronecker product.

Tropical symplectic flag varieties: a Lie theoretic approach

Tropical symplectic flag varieties: a Lie theoretic approach

The Witt rings of many flag varieties are exterior algebras

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G 2 G_2 and F 4 F_4 . The results also extend to flag varieties over other algebraically closed fields.

Structure constants in equivariant oriented cohomology of flag varieties

We introduce generalized Demazure operators for the equivariant oriented cohomology of the flag variety, which have specializations to various Demazure operators and Demazure–Lusztig operators in both equivariant cohomology and equivariant K-theory. In the context of the geometric basis of the equivariant oriented cohomology given by certain Bott–Samelson classes, we use these operators to obtain formulas for the structure constants arising in different bases. Specializing to divided difference operators and Demazure operators in singular cohomology and K-theory, we recover the formulas for structure constants of Schubert classes obtained in Goldin and Knutson (Pure Appl Math Q 17(4):1345–1385, 2021). Two specific specializations result in formulas for the the structure constants for cohomological and K-theoretic stable bases as well; as a corollary we reproduce a formula for the structure constants of the Segre–Schwartz–MacPherson basis previously obtained by Su (Math Zeitschrift 298:193–213, 2021). Our methods involve the study of the formal affine Demazure algebra, providing a purely algebraic proof of these results.

Projective homogeneous varieties of Picard rank one in small characteristic

Projective homogeneous varieties of Picard rank one in small characteristic

Differential Calculi on Quantum Principal Bundles Over Projective Bases

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -the differential- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf–Galois extensions of algebras lift to Hopf–Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles Oq(SL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{SL}_2(\\mathbb {C}))$$\\end{document} and Oq(GL2(C))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_q(\ extrm{GL}_2(\\mathbb {C}))$$\\end{document} over the projective space P1(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{P}^1(\\mathbb {C})$$\\end{document} are discussed in detail.

Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special generalized flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special generalized flag manifold with a homogeneous bounded domain, (ii) a special generalized flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special generalized flag manifold. Our theorems strongly extend the results of Cheng and Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180], Cheng, Di Scala, and Yuan [Internat. J. Math. 28 (2017), p. 1750027], Loi and Mossa [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941], Loi and Mossa [Proc. Amer. Math. Soc. 151 (2023), pp. 3975–3984] and Umehara [Tokyo J. Math. 10 (1987), pp. 203–214].