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Schubert calculus and representations of the general linear group

Author(s): E. Mukhin; V. Tarasov; A. Varchenko
Journal: J. Amer. Math. Soc. 22 (2009), 909-940.
MSC (2000): Primary 17B67, 14P05, 82B23
Posted: 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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Additional Information:

E. Mukhin
Affiliation: Department of Mathematical Sciences, Indiana University, Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202-3216
Email: mukhin@math.iupui.edu

V. Tarasov
Affiliation: St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Peters- burg, 191023, Russia
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
Email: anv@email.unc.edu

DOI: 10.1090/S0894-0347-09-00640-7
PII: S 0894-0347(09)00640-7
Received by editor(s): January 11, 2008
Posted: 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 of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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