  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
sl_2(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.
