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 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.

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 Berezin quantization on generalized flag manifolds

2009 ◽  
Vol 105 (1) ◽  
pp. 66 ◽  
Author(s):  
Benjamin Cahen
Keyword(s):  
Holomorphic Functions ◽  
Flag Manifold ◽  
Unitary Irreducible Representation ◽  
Flag Manifolds ◽  
Dense Open Subset ◽  
Generalized Flag Manifolds ◽  
Berezin Quantization ◽  
Generalized Flag Manifold ◽  
Simply Connected ◽  
Connected Lie Group

Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.

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.

INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

scholarly journals Differential operators on quantized flag manifolds at roots of unity, II

2014 ◽  
Vol 214 ◽  
pp. 1-52
Author(s):  
Toshiyuki Tanisaki
Keyword(s):  
Differential Operators ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Roots Of Unity ◽  
Central Character ◽  
Derived Equivalence ◽  
Bernstein Type ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Category Of Modules

AbstractWe formulate a Beilinson-Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twistedD-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

scholarly journals A geometric proof of an equivariant Pieri rule for flag manifolds

2019 ◽  
Vol 31 (3) ◽  
pp. 779-783
Author(s):  
Changzheng Li ◽  
Vijay Ravikumar ◽  
Frank Sottile ◽  
Mingzhi Yang
Keyword(s):  
Short Proof ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Algebraic Proof ◽  
Geometric Proof

Abstract We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.

A FIVE DIMENSIONAL MODEL OF EFFECTIVE GRAVITATIONAL AND FINE-STRUCTURE CONSTANTS

2002 ◽  
Vol 17 (29) ◽  
pp. 4317-4323 ◽  
Author(s):  
J. P. MBELEK ◽  
M. LACHIÈZE-REY
Keyword(s):  
Fine Structure ◽  
Scalar Field ◽  
Gravitational Constant ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Five Dimensional Model ◽  
Bulk Scalar Field ◽  
Structure Constants ◽  
Good Agreement ◽  
Kaluza Klein

It is shown that the coupling of the Kaluza-Klein (KK) internal scalar field both to an external stabilizing bulk scalar field and to the geomagnetic field may explain the observed dispersion in laboratory measurements of the (effective) gravitational constant. Except the PTB 95 value, the predictions are found in good agreement with all of the experimental data. The cosmological variation of the fine-structure constant is also addressed.

scholarly journals The cohomology ring of the sapphires that admit the Sol geometry

2018 ◽  
Vol 28 (03) ◽  
pp. 365-380 ◽  
Author(s):  
Daciberg Lima Gonçalves ◽  
Sérgio Tadao Martins
Keyword(s):  
Fundamental Group ◽  
Cohomology Ring ◽  
Free Resolution ◽  
Multiplicative Structure ◽  
Torus Bundle ◽  
Cohomology Rings ◽  
Cup Product ◽  
Diagonal Approximation

Let [Formula: see text] be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of [Formula: see text] over [Formula: see text] and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings [Formula: see text] for [Formula: see text] and [Formula: see text] for an odd prime [Formula: see text], and indicate how to compute the groups [Formula: see text] and the multiplicative structure given by the cup product for any system of coefficients [Formula: see text].

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.

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