scholarly journals Schubert presentation of the cohomology ring of flag manifolds

2015 ◽  
Vol 18 (1) ◽  
pp. 489-506 ◽  
Author(s):  
Haibao Duan ◽  
Xuezhi Zhao
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Cohomology Ring ◽  
Minimal System ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Generators And Relations ◽  
Integral Cohomology ◽  
Minimal System Of Generators ◽  
Connected Lie Group

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.

scholarly journals Combinatorial integers(m,nj)and Schubert calculus in the integral cohomology ring of infinite smooth flag manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-55
Author(s):  
Cenap Özel ◽  
Erol Yilmaz
Keyword(s):  
Homogeneous Spaces ◽  
Cohomology Ring ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Cohomology Algebra ◽  
Loop Groups ◽  
Local Coefficient ◽  
Integral Cohomology ◽  
Ring Structures ◽  
Coefficient Ring

We discuss the calculation of integral cohomology ring ofLG/TandΩG. First we describe the root system and Weyl group ofLG, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures ofLG/TandΩGfor affine groupA^2. We introduce combinatorial integers(m,nj)which play a crucial role in our calculations and give some interesting identities among these integers. Last we calculate generators for ideals and rank of each module of graded integral cohomology algebra in the local coefficient ringℤ[1/2].

scholarly journals ALGORITHM FOR MULTIPLYING SCHUBERT CLASSES

2006 ◽  
Vol 16 (06) ◽  
pp. 1197-1210 ◽  
Author(s):  
HAIBAO DUAN ◽  
XUEZHI ZHAO
Keyword(s):  
Spectral Sequence ◽  
Lie Group ◽  
Cohomology Ring ◽  
Flag Manifold ◽  
Integral Cohomology ◽  
Sequence Method ◽  
Schubert Classes ◽  
Connected Lie Group

Based on the formula for multiplying Schubert classes obtained in [17], we develop an algorithm computing the product of two arbitrary Schubert classes in a flag manifold G/H, where G is a compact connected Lie group and H ⊂ G is the centralizer of a one-parameter subgroup in G. Since all Schubert classes on G/H constitute a basis for the integral cohomology H*(G/H), the algorithm gives also a method to compute the integral cohomology ring H*(G/H) independent of the classical spectral sequence method of Leray and Borel [32, 33, 8, 9].

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES

1991 ◽  
Vol 02 (01) ◽  
pp. 67-76
Author(s):  
KYUNG BAI LEE ◽  
FRANK RAYMOND
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Double Coset ◽  
Torus Action ◽  
Torus Actions ◽  
Coset Spaces ◽  
Aspherical Manifold ◽  
Aspherical Manifolds ◽  
Connected Lie Group

Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

scholarly journals Schubert Calculus on a Grassmann Algebra

2009 ◽  
Vol 52 (2) ◽  
pp. 200-212 ◽  
Author(s):  
Letterio Gatto ◽  
Taíse Santiago
Keyword(s):  
Cohomology Ring ◽  
Grassmann Algebra ◽  
Monic Polynomial ◽  
Free Module ◽  
Exterior Algebra ◽  
Schubert Calculus ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Small Quantum ◽  
Factorization Algebra

AbstractThe (classical, small quantum, equivariant) cohomology ring of the grassmannian G(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree n into the product of two monic polynomials, one of degree k.

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 Central lacunary sets for Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 30-45 ◽  
Author(s):  
A. H. Dooley
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Infinite Subset ◽  
Semisimple Lie Group ◽  
Infinite Sets ◽  
Connected Lie Group

AbstractIf G is a compact connected Lie group every infinite subset of Ĝ contains an infinite central Λ(p) set, for p < 2 + 2 rank G/(dim G - rank G). A subset R of Ĝ is of type central Λ(2) if and only if the associated set of characters on the maximal torus is of type Λ(2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of Su(2) contains such a set.

scholarly journals Divided difference operators in equivariant KK-theory

2014 ◽  
Vol 06 (02) ◽  
pp. 237-261
Author(s):  
Ho-Hon Leung
Keyword(s):  
Direct Summand ◽  
Lie Group ◽  
Maximal Torus ◽  
Divided Difference ◽  
Difference Operators ◽  
C Algebras ◽  
Work Done ◽  
Connected Lie Group

Let G be a compact connected Lie group with a maximal torus T. Let A, B be G-C*-algebras. We define certain divided difference operators on Kasparov's T-equivariant KK-group KKT(A, B) and show that KKG(A, B) is a direct summand of KKT(A, B). More precisely, a T-equivariant KK-class is G-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.

scholarly journals Weyl functions and the Ap condition on compact lie groups

1982 ◽  
Vol 33 (2) ◽  
pp. 185-192 ◽  
Author(s):  
Giancarlo Travaglini
Keyword(s):  
Trigonometric Polynomial ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Weyl Function ◽  
Compact Lie Groups ◽  
Simply Connected ◽  
Weyl Functions ◽  
Uniformly Bounded ◽  
Connected Lie Group

AbstractLet G be a compact, simple, simply connected Lie group. The Lp-norm of a central trigonometric polynomial reduces naturally to a weighted Lp-norm of a trigonometric polynomial on a maximal torus T. The weight is | Δ |2-p, where Δ is the usual Weyl function. If p ≥ 2, we prove that | Δ |2-p satisfies Muckenhoupt's Ap condition if and only if the Lp-norms of the irreducible characters of G are uniformly bounded.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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