## Schubert calculus and representations of the general linear group

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- by E. Mukhin, V. Tarasov and A. Varchenko;
- J. Amer. Math. Soc.
**22**(2009), 909-940 - DOI: https://doi.org/10.1090/S0894-0347-09-00640-7
- Published electronically: April 30, 2009
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## Abstract:

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation $\mathfrak {gl}_N[t]$-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection.

In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.

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## Bibliographic Information

**E. Mukhin**- Affiliation: Department of Mathematical Sciences, Indiana University, Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 317134
- Email: mukhin@math.iupui.edu
**V. Tarasov**- Affiliation: St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Peters- burg, 191023, Russia
- MR Author ID: 191119
- Email: vt@math.iupui.edu, vt@pdmi.ras.ru
**A. Varchenko**- Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250
- MR Author ID: 190269
- Email: anv@email.unc.edu
- Received by editor(s): January 11, 2008
- Published electronically: April 30, 2009
- Additional Notes: The first author is supported in part by NSF grant DMS-0601005.

The second author is supported in part by RFFI grant 05-01-00922.

The third author is supported in part by NSF grant DMS-0555327 - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 909-940 - MSC (2000): Primary 17B67, 14P05, 82B23
- DOI: https://doi.org/10.1090/S0894-0347-09-00640-7
- MathSciNet review: 2525775