scholarly journals Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

2011 ◽  
Vol 22 (3) ◽  
pp. 447-447
Author(s):  
A. N. Kirillov ◽  
T. Maeno
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Quadratic Algebra ◽  
Flag Varieties

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 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 PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY

2019 ◽  
Vol 19 (6) ◽  
pp. 1889-1929
Author(s):  
Cristian Lenart ◽  
Kirill Zainoulline ◽  
Changlong Zhong
Keyword(s):  
Cohomology Ring ◽  
Flag Varieties ◽  
Special Cases ◽  
Oriented Cohomology ◽  
Schubert Classes ◽  
New Interpretation ◽  
The Moment ◽  
The Right ◽  
Relationship Of ◽  
The Relationship

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

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.

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