An inequality of Kostka numbers and Galois groups of Schubert problems

2012/05/27 by Christopher J. Brooks, Brooks, Christopher J., Abraham Martin del Campo +3
Mathematics · #05E15 #14N15 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO #msc:05E15 #msc:14N15

paper · pdf · doi:10.48550/arxiv.1205.5972

Extended abstract for FPSAC 2012

arxiv created 2012/05/27 · arxiv updated 2012/05/29

Abstract

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 sl₂(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.

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