Schubert calculus and representations of the general linear group

Authors:
E. Mukhin, V. Tarasov and A. Varchenko

Journal:
J. Amer. Math. Soc. **22** (2009), 909-940

MSC (2000):
Primary 17B67, 14P05, 82B23

Published electronically:
April 30, 2009

MathSciNet review:
2525775

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation -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.

**[B]**Prakash Belkale,*Invariant theory of 𝐺𝐿(𝑛) and intersection theory of Grassmannians*, Int. Math. Res. Not.**69**(2004), 3709–3721. MR**2099498**, 10.1155/S107379280414155X**[CG]**Vyjayanthi Chari and Jacob Greenstein,*Current algebras, highest weight categories and quivers*, Adv. Math.**216**(2007), no. 2, 811–840. MR**2351379**, 10.1016/j.aim.2007.06.006**[CL]**Vyjayanthi Chari and Sergei Loktev,*Weyl, Demazure and fusion modules for the current algebra of 𝔰𝔩ᵣ₊₁*, Adv. Math.**207**(2006), no. 2, 928–960. MR**2271991**, 10.1016/j.aim.2006.01.012**[CP]**Vyjayanthi Chari and Andrew Pressley,*Weyl modules for classical and quantum affine algebras*, Represent. Theory**5**(2001), 191–223 (electronic). MR**1850556**, 10.1090/S1088-4165-01-00115-7**[EG]**A. Eremenko and A. Gabrielov,*Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry*, Ann. of Math. (2)**155**(2002), no. 1, 105–129. MR**1888795**, 10.2307/3062151**[EH]**David Eisenbud and Joe Harris,*Limit linear series: basic theory*, Invent. Math.**85**(1986), no. 2, 337–371. MR**846932**, 10.1007/BF01389094**[F]**Edward Frenkel,*Affine algebras, Langlands duality and Bethe ansatz*, XIth International Congress of Mathematical Physics (Paris, 1994) Int. Press, Cambridge, MA, 1995, pp. 606–642. MR**1370720****[Fu]**William Fulton,*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620****[G]**Michel Gaudin,*La fonction d’onde de Bethe*, Collection du Commissariat à l’Énergie Atomique: Série Scientifique. [Collection of the Atomic Energy Commission: Science Series], Masson, Paris, 1983 (French). MR**693905****[GH]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR**1288523****[HP]**W. V. D. Hodge and D. Pedoe,*Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties*, Cambridge, at the University Press, 1952. MR**0048065****[HU]**Roger Howe and Tōru Umeda,*The Capelli identity, the double commutant theorem, and multiplicity-free actions*, Math. Ann.**290**(1991), no. 3, 565–619. MR**1116239**, 10.1007/BF01459261**[K]**Rinat Kedem,*Fusion products of 𝔰𝔩_{𝔑} symmetric power representations and Kostka polynomials*, Quantum theory and symmetries, World Sci. Publ., Hackensack, NJ, 2004, pp. 88–93. MR**2170716**, 10.1142/9789812702340_0010**[M]**I. G. Macdonald,*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144****[MNO]**A. Molev, M. Nazarov, and G. Ol′shanskiĭ,*Yangians and classical Lie algebras*, Uspekhi Mat. Nauk**51**(1996), no. 2(308), 27–104 (Russian); English transl., Russian Math. Surveys**51**(1996), no. 2, 205–282. MR**1401535**, 10.1070/RM1996v051n02ABEH002772**[MTV1]**E. Mukhin, V. Tarasov, and A. Varchenko,*Bethe eigenvectors of higher transfer matrices*, J. Stat. Mech. Theory Exp.**8**(2006), P08002, 44. MR**2249767****[MTV2]**E. Mukhin, V. Tarasov, A. Varchenko,*The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz*, Preprint math.AG/0512299 (2005), 1-18, to appear in Annals of Mathematics.**[MTV3]**E. Mukhin, V. Tarasov, A. Varchenko,*A generalization of the Capelli identity*, Preprint math/0610799 (2006), 1-14, to appear in Manin Festschrift.**[MTV4]**E. Mukhin, V. Tarasov, A. Varchenko,*Bethe algebra and algebra of functions on the space of differential operators of order two with polynomial solutions*, Selecta Mathematica (N.S.), ISSN 1022-1824 (Print) 1420-9020 (Online) DOI 10.1007/s00029-008-0056-x (2007), 1-24.**[MTV5]**E. Mukhin, V. Tarasov, A. Varchenko,*On reality property of Wronski maps*, arXiv:0710.5856 (2007), 1-20, to appear in Confluentes Mathematici.**[MV1]**E. Mukhin and A. Varchenko,*Critical points of master functions and flag varieties*, Commun. Contemp. Math.**6**(2004), no. 1, 111–163. MR**2048778**, 10.1142/S0219199704001288**[MV2]**Evgeny Mukhin and Alexander Varchenko,*Norm of a Bethe vector and the Hessian of the master function*, Compos. Math.**141**(2005), no. 4, 1012–1028. MR**2148192**, 10.1112/S0010437X05001569**[MV3]**E. Mukhin and A. Varchenko,*Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe ansatz conjecture*, Trans. Amer. Math. Soc.**359**(2007), no. 11, 5383–5418. MR**2327035**, 10.1090/S0002-9947-07-04217-1**[S]**Frank Sottile,*Rational curves on Grassmannians: systems theory, reality, and transversality*, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 9–42. MR**1837108**, 10.1090/conm/276/04509**[T]**D. Talalaev,*Quantization of the Gaudin System*, Preprint hep-th/0404153 (2004), 1-19.**[Tm]**Harry Tamvakis,*The connection between representation theory and Schubert calculus*, Enseign. Math. (2)**50**(2004), no. 3-4, 267–286. MR**2116717**

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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:
https://doi.org/10.1090/S0894-0347-09-00640-7

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

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.