scholarly journals Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p) × U(n − p)

2010 ◽  
Vol 38 (4) ◽  
pp. 413-438 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Ioannis Chrysikos ◽  
Yusuke Sakane
Keyword(s):  
Einstein Metrics ◽  
Flag Manifold ◽  
Generalized Flag Manifold

scholarly journals INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS

2011 ◽  
Vol 90 (2) ◽  
pp. 237-251 ◽  
Author(s):  
ANDREAS ARVANITOYEORGOS ◽  
IOANNIS CHRYSIKOS
Keyword(s):  
Variational Approach ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Semisimple Lie Group ◽  
Generalized Flag Manifolds ◽  
Generalized Flag Manifold ◽  
Scalar Curvature Functional ◽  
Homogeneous Einstein Metrics ◽  
Adjoint Orbit

AbstractLet M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

scholarly journals New homogeneous Einstein metrics on quaternionic Stiefel manifolds

2018 ◽  
Vol 18 (4) ◽  
pp. 509-524 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Homogeneous Space ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Total Space ◽  
Stiefel Manifold ◽  
Isotropy Representation ◽  
Stiefel Manifolds ◽  
Generalized Flag Manifold ◽  
Generalized Wallach Space ◽  
Homogeneous Einstein Metrics

Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

scholarly journals HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS

2013 ◽  
Vol 24 (10) ◽  
pp. 1350077 ◽  
Author(s):  
ANDREAS ARVANITOYEORGOS ◽  
IOANNIS CHRYSIKOS ◽  
YUSUKE SAKANE
Keyword(s):  
Einstein Equation ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Polynomial Systems ◽  
Flag Manifolds ◽  
Isotropy Representation ◽  
Generalized Flag Manifolds ◽  
A New Technique ◽  
Kähler Einstein Metrics ◽  
Homogeneous Einstein Metrics

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

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

10.37236/563 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Allen Knutson ◽  
Kevin Purbhoo
Keyword(s):  
Richardson Number ◽  
Cohomology Ring ◽  
Structure Constant ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Combinatorial Formula ◽  
Cup Product ◽  
New Family ◽  
Generalized Flag Manifold ◽  
Structure Constants

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 $2$.

scholarly journals On the Finiteness of Quantum K-Theory of a Homogeneous Space

2020 ◽  
Author(s):  
David Anderson ◽  
Linda Chen ◽  
Hsian-Hua Tseng
Keyword(s):  
Flag Manifold ◽  
Quadratic Growth ◽  
Module Structure ◽  
Geometric Representation ◽  
Geometric Representation Theory ◽  
Shift Operators ◽  
The Core ◽  
Generalized Flag Manifold ◽  
K Theory ◽  
Quadratic Growth Condition

Abstract We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

scholarly journals Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

2016 ◽  
Vol 152 (12) ◽  
pp. 2603-2625 ◽  
Author(s):  
Paolo Aluffi ◽  
Leonardo C. Mihalcea
Keyword(s):  
Divided Difference ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Generalized Flag Manifold ◽  
Effective Combination ◽  
A Cell ◽  
Schubert Classes ◽  
Dimension One ◽  
Schubert Class

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.

scholarly journals A Characterization of the Quantum Cohomology Ring of G/B and Applications

2008 ◽  
Vol 60 (4) ◽  
pp. 875-891
Author(s):  
Augustin-Liviu Mare
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Main Idea ◽  
Flag Manifold ◽  
Schubert Calculus ◽  
Generators And Relations ◽  
Cup Product ◽  
Generalized Flag Manifold ◽  
Standard Monomials ◽  
Small Quantum

AbstractWe observe that the small quantum product of the generalized flag manifold G/B is a product operation ★ on H*(G/B) ⊗ ℝ[q1, . . . , ql] uniquely determined by the facts that it is a deformation of the cup product on H*(G/B); it is commutative, associative, and graded with respect to deg(qi ) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H*(G/B)⊗ℝ[q1, . . . , ql], ★) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B: the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL(n, ℂ). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum -module of G/B one can decode all information about the quantum cohomology of this space.

Sign in / Sign up

Export Citation Format

Share Document