Schubert calculus and representations of the general linear group

Mukhin, E.; Tarasov, V.; Varchenko, A.

doi:10.1090/S0894-0347-09-00640-7

Schubert calculus and representations of the general linear group
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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.

References

Prakash Belkale, Invariant theory of $\textrm {GL}(n)$ and intersection theory of Grassmannians, Int. Math. Res. Not. 69 (2004), 3709–3721. MR 2099498, DOI 10.1155/S107379280414155X
Vyjayanthi Chari and Jacob Greenstein, Current algebras, highest weight categories and quivers, Adv. Math. 216 (2007), no. 2, 811–840. MR 2351379, DOI 10.1016/j.aim.2007.06.006
Vyjayanthi Chari and Sergei Loktev, Weyl, Demazure and fusion modules for the current algebra of $\mathfrak {s}\mathfrak {l}_{r+1}$, Adv. Math. 207 (2006), no. 2, 928–960. MR 2271991, DOI 10.1016/j.aim.2006.01.012
Vyjayanthi Chari and Andrew Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223. MR 1850556, DOI 10.1090/S1088-4165-01-00115-7
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, DOI 10.2307/3062151
David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932, DOI 10.1007/BF01389094
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
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, DOI 10.1007/978-3-662-02421-8
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
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, DOI 10.1002/9781118032527
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 48065
Roger Howe and T\B{o}ru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619. MR 1116239, DOI 10.1007/BF01459261
Rinat Kedem, Fusion products of $\mathfrak {sl}_N$ symmetric power representations and Kostka polynomials, Quantum theory and symmetries, World Sci. Publ., Hackensack, NJ, 2004, pp. 88–93. MR 2170716, DOI 10.1142/9789812702340_{0}010
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
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, DOI 10.1070/RM1996v051n02ABEH002772
E. Mukhin, V. Tarasov, and A. Varchenko, Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 8 (2006), P08002, 44. MR 2249767, DOI 10.1088/1742-5468/2006/08/p08002

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

A generalization of the Capelli identity

Bethe algebra and algebra of functions on the space of differential operators of order two with polynomial solutions

On reality property of Wronski maps

E. Mukhin and A. Varchenko, Critical points of master functions and flag varieties, Commun. Contemp. Math. 6 (2004), no. 1, 111–163. MR 2048778, DOI 10.1142/S0219199704001288
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, DOI 10.1112/S0010437X05001569
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, DOI 10.1090/S0002-9947-07-04217-1
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, DOI 10.1090/conm/276/04509

Quantization of the Gaudin System

Harry Tamvakis, The connection between representation theory and Schubert calculus, Enseign. Math. (2) 50 (2004), no. 3-4, 267–286. MR 2116717