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Transactions of the American Mathematical Society

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Galois groups of Schubert problems of lines are at least alternating

Authors: Christopher J. Brooks, Abraham Martín del Campo and Frank Sottile
Journal: Trans. Amer. Math. Soc. 367 (2015), 4183-4206
MSC (2010): Primary 14N15, 05E15
Published electronically: November 24, 2014
MathSciNet review: 3324924
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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.

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Additional Information

Christopher J. Brooks
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090

Abraham Martín del Campo
Affiliation: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
ORCID: 0000-0003-0369-0652

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
MR Author ID: 355336
ORCID: 0000-0003-0087-7120

Keywords: Galois groups, Schubert calculus, Kostka numbers, enumerative geometry
Received by editor(s): May 14, 2013
Published electronically: November 24, 2014
Additional Notes: This research was supported in part by NSF grant DMS-915211 and the Institut Mittag-Leffler
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.