Abstract
We interpret the equivariant cohomology \(H_{GL_n }^* \)(ℱ
λ
,ℂ) of a partial flag variety ℱ
λ
parametrizing chains of subspaces 0 = F
0 ⊂ F
1 ⊂ … ⊂ F
N
= ℂn, dimF
i
/F
i−1 = λ
i
, as the Bethe algebra 
of the 
-weight subspace 
of a 
[t]-module 
.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. F. Atiyah and R. Bott, “The moment map and equivariant cohomology,” Topology, 23:1 (1984), 1–28.
A. Braverman, D. Maulik, and A. Okounkov, Quantum cohomology of the Springer resolution, http://arxiv.org/abs/1001.0056.
V. Chari and S. Loktev, “Weyl, Demazure and fusion modules for the current algebra of sl r+1,” Adv. Math., 207:2 (2006), 928–960.
V. Chari and A. Pressley, “Weyl modules for classical and quantum affine algebras,” Represent. Theory (electronic), 5 (2001), 191–223.
A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, http://arxiv.org/abs/hep-th/0604128.
M. Gaudin, “Diagonalisation d’une classe d’Hamiltoniens de spin,” J. Physique, 37:10 (1976), 1089–1098.
M. Gaudin, La fonction d’onde de Bethe, Collection du Commissariat à l’Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
E. Mukhin, V. Tarasov, and A. Varchenko, “Bethe eigenvectors of higher transfer matrices,” J. Stat. Mech. Theory (electronic), No. 8 (2006), P08002.
E. Mukhin, V. Tarasov, and A. Varchenko, “Schubert calculus and representations of general linear group,” J. Amer. Math. Soc., 22:4 (2009), 909–940.
E. Mukhin, V. Tarasov, and A. Varchenko, “Spaces of quasi-exponentials and representations of glN,” J. Phys. A, 41:19, 194017 (2008).
E. Mukhin and A. Varchenko, “Critical points of master functions and flag varieties,” Commun. Contemp. Math., 6:1 (2004), 111–163.
A. Okounkov, “Quantum groups and quantum cohomology,” in: Lectures at the 15th Midrasha Mathematicae on Derived Categories of Algebro-Geometric Origin and Integrable Systems, December 19–24, Jerusalem, 2010.
R. Rimányi and A. Varchenko, Conformal blocks in the tensor product of vector representations and localization formulas, http://arxiv.org/abs/0911.3253.
R. Rimányi, V. Schechtman, and A. Varchenko, Conformal blocks and equivariant cohomology, http://arxiv.org/abs/1007.3155.
D. Talalaev, Quantization of the Gaudin system, http://arxiv.org/abs/hep-th/0404153.
A. Varchenko, “A Selberg integral type formula for an sl2 one-dimensional space of conformal blocks,” Mosc. Math. J., 10:2 (2010), 469–475.
E. Vasserot, “Représentations de groupes quantiques et permutations,” Ann. Sci. École Norm. Sup., 26:6 (1993), 747–773.
E. Vasserot, “Affine quantum groups and equivariant K-theory,” Transformation Groups, 3:3 (1998), 269–299.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 45, No. 4, pp. 16–31, 2011
Original Russian Text Copyright © by R. Rimányi, V. Schechtman, V. Tarasov, and A. Varchenko
To the memory of V. I. Arnold
The first author is supported in part by NSA grant CON:H98230-10-1-0171. The third author is supported in part by NSF grant DMS-0901616. The fourth author is supported in part by NSF grant DMS-0555327.
Rights and permissions
About this article
Cite this article
Rimányi, R., Schechtman, V., Tarasov, V. et al. Cohomology of a flag variety as a Bethe algebra. Funct Anal Its Appl 45, 252–264 (2011). https://doi.org/10.1007/s10688-011-0027-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-011-0027-4