Lower bounds for numbers of real solutions in problems of Schubert calculus. (English) Zbl 1365.14070

It is well known that the problem of finding the number of real solutions to algebraic systems is very difficult, and not many results are known. In this paper the authors address the counting of real points in intersections of Schubert varieties associated with osculating flags in the Grassmannian of \(n\)-dimensional planes in a \(d\)-dimensional space. These problems are parameterized by partitions \(\lambda^{(1)}, \dots, \lambda^{(k)}\) and \(\nu\) with at most \(n\) parts satisfying the condition \(|\nu|+ \sum_{i=1}^k|\lambda^{(i)}| =n(d-n)\), and distinct complex numbers \(z_1, \dots, z_k\).
In this parametrization,\(\lambda^{(1)}, \dots, \lambda^{(k)}\) and \(\nu\) are respectively paired with \(z_1, \dots, z_k\) and infinity.
Equivalently, they count \(n\)-dimensional real vector spaces of polynomials that have ramification points \(z_1, \dots, z_k\) with respective ramification conditions \(\lambda^{(1)}, \dots, \lambda^{(k)}\) and are spanned by polynomials of degrees \(d-i-\nu_{n+1-i},i=1,\dots, n\).
The same number is obtained by counting real monic monodromy-free Fuchsian differential operators with singular points \(z_1, \dots, z_k\) and infinity, exponents \({\lambda_{n}}^{(i)}, {\lambda_{n-1}}^{(i)} + 1,\dots, {\lambda_{1}}^{(i)}+ n-1\) at the \(z_i\)’s, \(i=1,\dots, k\), and exponents \(\nu_n + 1 - d,\nu_{n-1} + 2 - d, \dots,\nu_{1} + n - d \) at infinity.
The number of complex solutions to the above-mentioned algebraic systems is readily given by the Schubert calculus and equals the multiplicity of the irreducible Verma module \(L_{\mu}\) with highest weight \(\mu\).
Reviewer: Cenap Özel (Bolu)


14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
14P05 Real algebraic sets
Full Text: DOI arXiv


[1] Eremenko A., Gabrielov A.: The Wronski map and Grassmannians of real codimension 2 subspaces. Comput. Methods Funct. Theory, 1, 1-25 (2001) · Zbl 1052.14057 · doi:10.1007/BF03320973
[2] Eremenko A., Gabrielov A.: Degrees of real Wronski maps. Discrete Comput. Geom., 28, 331-347 (2002) · Zbl 1004.14011 · doi:10.1007/s00454-002-0735-x
[3] Eremenko A., Gabrielov A.: Pole placement static output feedback for generic linear systems. SIAM J. Control Optim., 41, 303-312 (2002) · Zbl 1031.93086 · doi:10.1137/S0363012901391913
[4] Eremenko A., Gabrielov A.: Rational functions with real critical points and the B. and M Shapiro conjecture in real enumerative geometry. Ann. of Math., 155, 105-129 (2002) · Zbl 0997.14015 · doi:10.2307/3062151
[5] Frobenius, F., Über die Charaktere der symmetrischen Gruppe. Sitzungberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, (1900), 516-534; reprinted in Gessamelte Abhandlungen, III, pp. 148-166, Springer Collected Works in Mathematics. Springer, Berlin-Heidelberg, 1968.
[6] Fulton, W., Young Tableaux. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997. · Zbl 1031.93086
[7] Hein N., Hillar C. J., Sottile F.: Lower bounds in real Schubert calculus. São Paulo J. Math. Sci., 7, 33-58 (2013) · Zbl 1351.14036 · doi:10.11606/issn.2316-9028.v7i1p33-58
[8] Hein, N. & Sottile, F., Beyond the Shapiro conjecture and Eremenko-Gabrielov lower bounds. Online database. Available at http://www.math.tamu.edu/ secant/lowerBounds/lowerBounds.php. · Zbl 0997.14015
[9] Hein N., Sottile F., Zelenko I.: A congruence modulo four in real Schubert calculus. J. Reine Angew. Math., 714, 151-174 (2016) · Zbl 1403.14088
[10] Mukhin, E., Tarasov, V. & Varchenko, A., Bethe eigenvectors of higher transfer matrices. J. Stat. Mech. Theory Exp., 8 (2006), P08002, 44 pp. · Zbl 1456.82301
[11] Mukhin, E., Tarasov, V. & Varchenko, A., Generating operator of XXX or Gaudin transfer matrices has quasi-exponential kernel. SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 060, 31 pp. · Zbl 1140.82015
[12] Mukhin E., Tarasov V., Varchenko A.: The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz. Ann. of Math., 170, 863-881 (2009) · Zbl 1213.14101 · doi:10.4007/annals.2009.170.863
[13] Mukhin E., Tarasov V., Varchenko A.: Schubert calculus and representations of the general linear group. J. Amer. Math. Soc., 22, 909-940 (2009) · Zbl 1205.17026 · doi:10.1090/S0894-0347-09-00640-7
[14] Pontryagin, L., Hermitian operators in spaces with indefinite metric. Izv. Akad. Nauk SSSR Ser. Math., 8 (1944), 243-280 (Russian). · Zbl 0061.26004
[15] Soprunova E., Sottile F.: Lower bounds for real solutions to sparse polynomial systems. Adv. Math., 204, 116-151 (2006) · Zbl 1102.14040 · doi:10.1016/j.aim.2005.05.016
[16] Sottile F.: Frontier of reality in Schubert calculus. Bull. Amer. Math. Soc., 47, 31-71 (2010) · Zbl 1197.14052 · doi:10.1090/S0273-0979-09-01276-2
[17] Talalaev, D., Quantization of the Gaudin system. Preprint, 2004. arXiv:hep-th/0404153 · Zbl 1111.82015
[18] White D.E.: Sign-balanced posets. J. Combin. Theory Ser. A, 95, 1-38 (2001) · Zbl 0982.05108 · doi:10.1006/jcta.2000.3146
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.