Galois groups of Schubert problems of lines are at least alternating

Brooks, Christopher; Martín del Campo, Abraham; Sottile, Frank

doi:10.1090/S0002-9947-2014-06192-8

Galois groups of Schubert problems of lines are at least alternating
HTML articles powered by AMS MathViewer

by Christopher J. Brooks, Abraham Martín del Campo and Frank Sottile PDF

Trans. Amer. Math. Soc. 367 (2015), 4183-4206 Request permission

Abstract:

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.

References

Désiré André, Mémoire sur les combinaisons régulières et leurs applications, Ann. Sci. École Norm. Sup. (2) 5 (1876), 155–198 (French). MR 1508638
Sara Billey and Ravi Vakil, Intersections of Schubert varieties and other permutation array schemes, Algorithms in algebraic geometry, IMA Vol. Math. Appl., vol. 146, Springer, New York, 2008, pp. 21–54. MR 2397936, DOI 10.1007/978-0-387-75155-9_{3}
C. I. Byrnes, Pole assignment by output feedback, Three decades of mathematical system theory, Lect. Notes Control Inf. Sci., vol. 135, Springer, Berlin, 1989, pp. 31–78. MR 1025786, DOI 10.1007/BFb0008458
Christopher I. Byrnes and Peter K. Stevens, Global properties of the root-locus map, Feedback control of linear and nonlinear systems (Bielefeld/Rome, 1981) Lect. Notes Control Inf. Sci., vol. 39, Springer, Berlin, 1982, pp. 9–29. MR 837447, DOI 10.1007/BFb0006816
Q. Feng and G. Baini, Extensions and sharpenings of the noted Kober’s inequality, Jiāozuò Kuàngyè Xuéyuàn Xuébaò (Journal of Jiaozuo Mining Institute) 12 (1993), no. 4, 101–103, (Chinese).
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
Charles Hermite, Sur les fonctions algébriques, CR Acad. Sci. (Paris) 32 (1851), 458–461.
C. Jordan, Traité des substitutions, Gauthier-Villars, Paris, 1870.
Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
Steven L. Kleiman, Intersection theory and enumerative geometry: a decade in review, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 321–370. With the collaboration of Anders Thorup on §3. MR 927987
S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI 10.2307/2317421
Anton Leykin and Frank Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comp. 78 (2009), no. 267, 1749–1765. MR 2501073, DOI 10.1090/S0025-5718-09-02239-X
A. McD. Mercer, Ulrich Abel, and Donald Caccia, Problems and Solutions: Solutions of Elementary Problems: E2952, Amer. Math. Monthly 93 (1986), no. 7, 568–569. MR 1540921, DOI 10.2307/2323042
G.A. Miller, H.F. Blichfeldt, and L.E. Dickson, Theory and applications of finite groups, John Wiley, New York, 1916.
Jim Ruffo, Yuval Sivan, Evgenia Soprunova, and Frank Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds, Experiment. Math. 15 (2006), no. 2, 199–221. MR 2253007
H. Schubert, Die $n$-dimensionalen Verallgemeinerungen der fundamentalen Anzahlen unseres Raume, Math. Ann. 26 (1886), 26–51, (dated 1884).
Andrew J. Sommese and Charles W. Wampler II, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Arising in engineering and science. MR 2160078, DOI 10.1142/9789812567727
Frank Sottile, Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J. 87 (1997), no. 1, 59–85. MR 1440063, DOI 10.1215/S0012-7094-97-08703-2
Ravi Vakil, A geometric Littlewood-Richardson rule, Ann. of Math. (2) 164 (2006), no. 2, 371–421. Appendix A written with A. Knutson. MR 2247964, DOI 10.4007/annals.2006.164.371
Ravi Vakil, Schubert induction, Ann. of Math. (2) 164 (2006), no. 2, 489–512. MR 2247966, DOI 10.4007/annals.2006.164.489
J. Weber, Lehrbuch der algebra, Zweiter Band, Vieweg und Sohn, Braunschweig, 1896.