Generalized flag variety
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Dynamics on flag manifolds: domains of proper discontinuity and cocompactness Open
For noncompact semisimple Lie groups $G$ we study the dynamics of the actions\nof their discrete subgroups $\\Gamma<G$ on the associated partial flag manifolds\n$G/P$. Our study is based on the observation that they exhibit also in higher\…
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Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety Open
Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\time…
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Degenerate Flag Varieties of Type A and C are Schubert Varieties Open
We show that any type A or C degenerate flag variety is, in fact, isomorphic to a Schubert variety in an appropriate partial flag manifold.
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Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds Open
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 …
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Integrable properties of σ-models with non-symmetric target spaces Open
It is well-known that sigma-models with symmetric target spaces are\nclassically integrable. At the example of the model with target space the flag\nmanifold U(3)/U(1)^3 -- a non-symmetric space -- we show that the introduction\nof torsion…
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Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem Open
Motivic Chern classes are elements in the K-theory of an algebraic variety $X$, depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic C…
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Loop structure on equivariant $K$-theory of semi-infinite flag manifolds Open
We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-grou…
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Schubert puzzles and integrability I: invariant trilinear forms Open
The puzzle rules for computing Schubert calculus on $d$-step flag manifolds, proven in [Knutson Tao 2003] for $1$-step, in [Buch Kresch Purbhoo Tamvakis 2016] for $2$-step, and conjectured in [Coskun Vakil 2009] for $3$-step, lead to vecto…
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Fano manifolds whose elementary contractions are smooth P^1-fibrations: a geometric characterization of flag varieties Open
ct. The present paper provides a geometric characterization of complete
\nflag varieties for semisimple algebraic groups. Namely, we show that if X is
\na Fano manifold whose elementary contractions are all P1-fibrations then X is
\nisomor…
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Flag Manifold Sigma Models and Nilpotent Orbits Open
In the present paper we study flag manifold sigma-models that admit a\nzero-curvature representation. It is shown that these models may be naturally\nconsidered as interacting (holomorphic and anti-holomorphic)\n$\\beta\\gamma$-systems. Be…
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Flag manifold sigma models: spin chains and integrable theories Open
This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in the description of continuum limits of spin ch…
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Two-step Homogeneous Geodesics in Homogeneous Spaces Open
We study geodesics of the form $\\gamma(t) = \\pi(\\exp(tX) \\exp(tY))$, $X, Y \\in \\mathfrak{g} = \\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\\pi \\colon G \\to G/K$ is the natural projection. These curves naturally gen…
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New homogeneous Einstein metrics on quaternionic Stiefel manifolds Open
We consider invariant Einstein metrics on the quaternionic Stiefel manifold V p ℍ n of all orthonormal p -frames in ℍ n . This manifold is diffeomorphic to the homogeneous space Sp( n )/Sp( n − p ) and its isotropy representation contains …
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Combinatorial models for Schubert polynomials Open
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other kno…
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Schubert presentation of the cohomology ring of flag manifolds Open
Let $G$ be a compact connected Lie group with a maximal torus $T$ . In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.
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Manifolds of Isospectral Matrices and Hessenberg Varieties Open
We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its…
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Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds Open
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of se…
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On quantum $K$-groups of partial flag manifolds Open
We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold in a way it respects the Schubert basis. This is a $K$-theoretic analogue of the parabolic version of Peterson'…
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Homogeneous geodesics and g.o. manifolds Open
Let $M$ be a homogeneous pseudo-Riemannian manifold, affine manifold, or Finsler space. A homogeneous geodesic is an orbit of a $1$-parameter group of isometries, respectively, of affine diffeomorphisms. A homogeneous manifold is called a …
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Flag Bott manifolds and the toric closure of a generic orbit associated to a generalized Bott manifold Open
To a direct sum of holomorphic line bundles, we can associate two fibrations,\nwhose fibers are, respectively, the corresponding full flag manifold and the\ncorresponding projective space. Iterating these procedures gives, respectively,\na…
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Quantum flag manifold $σ$-models and Hermitian Ricci flow Open
We show that flag manifold $σ$-models (including $\mathbb{CP}^{n-1}$, Grassmannian models as special cases) and their deformed versions may be cast in the form of gauged bosonic Thirring/Gross-Neveu-type systems. Quantum mechanically the g…
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Optimization on flag manifolds Open
A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix compu…
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Coxeter combinatorics and spherical Schubert geometry Open
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
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Maximal Newton Points and the Quantum Bruhat Graph Open
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
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Domains of discontinuity in oriented flag manifolds Open
We study actions of discrete subgroups Γ $\Gamma$ of semi-simple Lie groups G $G$ on associated oriented flag manifolds. These are quotients G / P $G/P$ , where the subgroup P $P$ lies between a parabolic subgroup and its identity componen…
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On the quantum K-ring of the flag manifold Open
We establish a finiteness property of the quantum K-ring of the complete flag manifold.
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The cohomology of compact Lie groups Open
Let $G$ be a compact Lie group with a maximal torus $T$. Based on a presentation of the integral cohomology ring $H^{\ast }(G/T)$ of the flag manifold $G/T$ in \cite{DZ1} we have presented in \cite{DZ2} an explicit and unified construction…
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Toward a classification of killing vector fields of constant length on pseudo-Riemannian normal homogeneous spaces Open
We give an almost complete classification of normal pseudo-Riemannian homogeneous spaces which admit a Killing vector field of costant length.
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Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module Open
The equivariant K -theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q -Heisenberg algebra. The action of the latt…
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Self-organizing mappings on the flag manifold with applications to hyper-spectral image data analysis Open
A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vec…