Brooks, Christopher J. and Campo, Abraham Mart\'ın, and Sottile, Frank - An inequality of Kostka numbers and Galois groups of Schubert problems

dmtcs:3099 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
An inequality of Kostka numbers and Galois groups of Schubert problems

Authors: Brooks, Christopher J. and Campo, Abraham Mart\'ın, and Sottile, Frank

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of $\mathfrak{sl}_2\mathbb{C}$ -modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.


Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
Section: Proceedings
Published on: January 1, 2012
Submitted on: January 31, 2017
Keywords: Kostka numbers, Galois groups, Schubert calculus, Schubert varieties,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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