Christopher J. Brooks ; Abraham Martín Campo ; Frank Sottile - 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) - https://doi.org/10.46298/dmtcs.3099
An inequality of Kostka numbers and Galois groups of Schubert problemsArticle

Authors: Christopher J. Brooks 1; Abraham Martín Campo 1; Frank Sottile 1

  • 1 Department of Mathematics [Austin]

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
Imported 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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