scholarly journals The equivariant cohomology ring of a cohomogeneity-one action

2019 ◽  
Vol 203 (1) ◽  
pp. 205-223
Author(s):  
Jeffrey D. Carlson ◽  
Oliver Goertsches ◽  
Chen He ◽  
Augustin-Liviu Mare
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

scholarly journals On the automorphism group of a symplectic half-flat 6-manifold

2019 ◽  
Vol 31 (1) ◽  
pp. 265-273
Author(s):  
Fabio Podestà ◽  
Alberto Raffero
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Group Action ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

A new equivariant cohomology ring

2019 ◽  
Vol 295 (3-4) ◽  
pp. 1163-1182 ◽  
Author(s):  
Bohui Chen ◽  
Cheng-Yong Du ◽  
Tiyao Li
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring

scholarly journals Double theta polynomials and equivariant Giambelli formulas

2015 ◽  
Vol 160 (2) ◽  
pp. 353-377 ◽  
Author(s):  
HARRY TAMVAKIS ◽  
ELIZABETH WILSON
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Generators And Relations ◽  
Raising Operators ◽  
Schubert Classes ◽  
Symplectic Grassmannians

AbstractWe use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

scholarly journals The equivariant cohomology ring of regular varieties

2004 ◽  
Vol 52 (1) ◽  
pp. 189-203 ◽  
Author(s):  
Michel Brion ◽  
James Carrell
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring

scholarly journals Poset Pinball, the Dimension Pair Algorithm, and Type A Regular Nilpotent Hessenberg Varieties

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-34 ◽  
Author(s):  
Darius Bayegan ◽  
Megumi Harada
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Type A ◽  
Successful Outcome ◽  
Combinatorial Game ◽  
Schubert Polynomials ◽  
Hessenberg Varieties ◽  
Hessenberg Variety ◽  
Springer Variety ◽  
Special Case

We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace X of a GKM space; Harada and Tymoczko also prove that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of X. First we define the dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type A regular nilpotent Hessenberg and any type A nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. The algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Insko. Second, in a special case of regular nilpotent Hessenberg varieties, we prove that our pinball outcome is poset-upper-triangular, and hence the corresponding classes form a HS1*(pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.

scholarly journals QUATERNIONIC KAHLER MANIFOLDS OF COHOMOGENEITY ONE

1999 ◽  
Vol 10 (05) ◽  
pp. 541-570 ◽  
Author(s):  
ANDREW DANCER ◽  
ANDREW SWANN
Keyword(s):  
Simple Group ◽  
Lie Algebras ◽  
Complex Structure ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action ◽  
Hyperkähler Manifolds ◽  
Adjoint Orbits ◽  
Quaternionic Kähler

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic Kähler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.

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