  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 formula to obtain
an integral formula for these Kostka numbers. This rewrites the inequality as
an integral, which we estimate to establish the inequality.
