Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 046, 15 pages      arXiv:1710.06534      https://doi.org/10.3842/SIGMA.2018.046

Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus

Kang Lu
Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA

Received November 27, 2017, in final form May 07, 2018; Published online May 14, 2018

Abstract
The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.

Key words: real Schubert calculus; self-dual spaces; Bethe ansatz; Gaudin model.

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References

1. Borisov L., Mukhin E., Self-self-dual spaces of polynomials, Adv. Math. 190 (2005), 77-118, math.QA/0308128.
2. Eremenko A., Gabrielov A., Degrees of real Wronski maps, Discrete Comput. Geom. 28 (2002), 331-347, math.AG/0108133.
3. Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 27-62, hep-th/9402022.
4. Frobenius F., Über die Charaktere der symmetrischen Gruppe, König. Preuss. Akad. Wiss. Berlin (1900), 516-534, reprinted in Gessamelte Abhandlungen, Vol. 3, Springer-Verlag, Berlin - New York, 1968, 148-166.
5. Hein N., Hillar C.J., Sottile F., Lower bounds in real Schubert calculus, S\ ao Paulo J. Math. Sci. 7 (2013), 33-58, arXiv:1308.4381.
6. Hein N., Sottile F., Beyond the Shapiro conjecture and Eremenko-Gabrielov lower bounds, in Lower Bounds Experimental Project, available at http://www.math.tamu.edu/~secant/lowerBounds/lowerBounds.php.
7. Hein N., Sottile F., Zelenko I., A congruence modulo four in real Schubert calculus, J. Reine Angew. Math. 714 (2016), 151-174, arXiv:1211.7160.
8. Lu K., Mukhin E., On the Gaudin model of type ${\rm G}_2$, Commun. Contemp. Math., to appear, arXiv:1711.02511.
9. Lu K., Mukhin E., Varchenko A., On the Gaudin model associated to Lie algebras of classical types, J. Math. Phys. 57 (2016), 101703, 23 pages, arXiv:1512.08524.
10. Lu K., Mukhin E., Varchenko A., Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$, Pure Appl. Math. Q., to appear, arXiv:1705.02048.
11. Molev A., Feigin-Frenkel center in types $B$, $C$ and $D$, Invent. Math. 191 (2013), 1-34, arXiv:1105.2341.
12. Molev A., Mukhin E., Eigenvalues of Bethe vectors in the Gaudin model, Theoret. and Math. Phys. 192 (2017), 1258-1281, arXiv:1506.01884.
13. Mukhin E., Tarasov V., Lower bounds for numbers of real solutions in problems of Schubert calculus, Acta Math. 217 (2016), 177-193, arXiv:1404.7194.
14. Mukhin E., Tarasov V., Varchenko A., The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. 170 (2009), 863-881, math.AG/0512299.
15. Mukhin E., Tarasov V., Varchenko A., Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), 909-940, arXiv:0711.4079.
16. Mukhin E., Varchenko A., Critical points of master functions and flag varieties, Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017.
17. Pontrjagin L., Hermitian operators in spaces with indefinite metric, Bull. Acad. Sci. URSS. Sér. Math. 8 (1944), 243-280.
18. Rybnikov L., A proof of the Gaudin Bethe Ansatz conjecture, arXiv:1608.04625.
19. Soprunova E., Sottile F., Lower bounds for real solutions to sparse polynomial systems, Adv. Math. 204 (2006), 116-151, math.AG/0409504.
20. Sottile F., Frontiers of reality in Schubert calculus, Bull. Amer. Math. Soc. 47 (2010), 31-71, arXiv:0907.1847.