## Galois groups of Schubert problems via homotopy computation.(English)Zbl 1210.14064

Let $$G(k,n)$$ be the Grassmannian of $$k$$-planes in $$\mathbb C^n$$. A Schubert problem in $$G(k,n)$$ asks to determine the (number of) $$k$$-planes satisying some fixed incidence conditions, with respect to general fixed flags of subspaces of $$\mathbb C^n$$, such that this number is finite. Every incidence condition defines a Schubert variety, therefore the solution of a Schubert problem can be represented by a transverse intersection of Schubert varieties. In particular, by definition, a simple Schubert problem involves only two Schubert conditions of codimension higher than one, it enjoys the property of being a complete intersection.
To a given Schubert problem one naturally associates a Galois group; if it is the full symmetric group, it means that the problem has no underlying structures. The article under review tackles the problem of finding the Galois group of some simple Schubert problems, using the method of homotopy computation, recently developed in the new field of numerical algebraic geometry by Sommese and Wampler (see for instance [A. J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials. Arising in engineering and science. River Edge, NJ: World Scientific. (2005; Zbl 1091.65049)]). This method, initially conceived for applications of mathematics, has proved here to be very efficient also to solve purely theoretic problems.
As an example, the following numerical theorem is stated: The Galois group of the Schubert problem of $$3$$-planes in $$\mathbb C^8$$ meeting $$15$$ fixed $$5$$-planes non-trivially is the full symmetric group on $$6006$$ letters.
This kind of problems are intractable with the usual symbolic methods, so it appears that in the next future the numerical algorithms will be further developed. The article includes a precise description of the software used. The lines of future research in this field are also indicated.
The article is clearly written and very pleasant to read.

### MSC:

 14N15 Classical problems, Schubert calculus 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations

Zbl 1091.65049

### Software:

GAP; PHCpack; Bertini; polymake; HOM4PS
Full Text:

### References:

 [1] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Bertini: Software for numerical algebraic geometry, Available at http:// www.nd.edu/\~sommese/ bertini. · Zbl 1143.65344 [2] Daniel J. Bates, Andrew J. Peterson, Chrisand Sommese, and Charles W. Wampler, Numerical computation of the genus of an irreducible curve within an algebraic set, 2007. · Zbl 1211.14062 [3] 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. · Zbl 1132.14044 [4] 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. · Zbl 0701.93047 [5] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. · Zbl 0878.14034 [6] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.9, 2006. [7] Ewgenij Gawrilow and Michael Joswig, polymake: a framework for analyzing convex polytopes, Polytopes — combinatorics and computation (Oberwolfach, 1997) DMV Sem., vol. 29, Birkhäuser, Basel, 2000, pp. 43 – 73. · Zbl 0960.68182 [8] Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685 – 724. · Zbl 0433.14040 [9] Birkett Huber, Frank Sottile, and Bernd Sturmfels, Numerical Schubert calculus, J. Symbolic Comput. 26 (1998), no. 6, 767 – 788. Symbolic numeric algebra for polynomials. · Zbl 1064.14508 [10] Birkett Huber and Jan Verschelde, Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control, SIAM J. Control Optim. 38 (2000), no. 4, 1265 – 1287. · Zbl 0955.14038 [11] C. Jordan , Traité des substitutions, Gauthier-Villars, Paris, 1870. · JFM 02.0280.02 [12] R. Baker Kearfott and Zhaoyun Xing, An interval step control for continuation methods, SIAM J. Numer. Anal. 31 (1994), no. 3, 892 – 914. · Zbl 0809.65050 [13] Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287 – 297. · Zbl 0288.14014 [14] S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061 – 1082. · Zbl 0272.14016 [15] T. Lee, T.Y. Li, and C. Tsai, Hom4ps-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method, Available at http://www.math.msu.edu/\~li/Software.htm, 2007. · Zbl 1167.65366 [16] A. Leykin and F. Sottile, Galois groups of Schubert problems, 2007, www.math.tamu. edu/\~sottile/stories/Galois. [17] Anton Leykin and Jan Verschelde, Interfacing with the numerical homotopy algorithms in PHCpack, Mathematical software — ICMS 2006, Lecture Notes in Comput. Sci., vol. 4151, Springer, Berlin, 2006, pp. 354 – 360. · Zbl 1230.65061 [18] Anton Leykin, Jan Verschelde, and Yan Zhuang, Parallel homotopy algorithms to solve polynomial systems, Mathematical software — ICMS 2006, Lecture Notes in Comput. Sci., vol. 4151, Springer, Berlin, 2006, pp. 225 – 234. · Zbl 1230.65062 [19] T.-Y. Li, personal communication. [20] T. Y. Li, Tim Sauer, and J. A. Yorke, The cheater’s homotopy: an efficient procedure for solving systems of polynomial equations, SIAM J. Numer. Anal. 26 (1989), no. 5, 1241 – 1251. · Zbl 0689.65032 [21] 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. · Zbl 1111.14049 [22] H. Schubert, Anzahl-Bestimmungen für Lineare Räume, Acta Math. 8 (1886), no. 1, 97 – 118 (German). Beliebiger dimension. · JFM 18.0632.01 [23] -, Losüng des Charakteristiken-Problems für lineare Räume. Beliebiger Dimension, Mittheil. Math. Ges. Hamburg (1886), 135-155, (dated 1885). [24] Michael Shub and Steve Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459 – 501. · Zbl 0821.65035 [25] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Introduction to numerical algebraic geometry, Solving polynomial equations, Algorithms Comput. Math., vol. 14, Springer, Berlin, 2005, pp. 301 – 335. · Zbl 1152.14313 [26] 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. · Zbl 1091.65049 [27] Frank Sottile, Pieri’s formula via explicit rational equivalence, Canad. J. Math. 49 (1997), no. 6, 1281 – 1298. · Zbl 0933.14031 [28] Frank Sottile, Some real and unreal enumerative geometry for flag manifolds, Michigan Math. J. 48 (2000), 573 – 592. Dedicated to William Fulton on the occasion of his 60th birthday. · Zbl 1077.14562 [29] Frank Sottile, Elementary transversality in the Schubert calculus in any characteristic, Michigan Math. J. 51 (2003), no. 3, 651 – 666. · Zbl 1056.14072 [30] Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Springer, Berlin, 1977, pp. 217 – 251. Lecture Notes in Math., Vol. 579. [31] Ravi Vakil, A geometric Littlewood-Richardson rule, Ann. of Math. (2) 164 (2006), no. 2, 371 – 421. Appendix A written with A. Knutson. · Zbl 1163.05337 [32] Ravi Vakil, Schubert induction, Ann. of Math. (2) 164 (2006), no. 2, 489 – 512. · Zbl 1115.14043 [33] J. Verschelde, Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw. 25 (1999), no. 2, 251-276, Software available at http://www.math.uic.edu/\~jan. · Zbl 0961.65047
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