An inequality of Kostka numbers and Galois groups of Schubert problems of lines

Tensor decompositions of rational sl_2-modules and Schubert calculus on the
Grassmannians of lines are both governed by fillings of Young tableaux with 
two rows, which are certain Kostka numbers.  While these Kostka numbers
satisfy a simple recursion, there is no closed formula for them in general.

This details of this recursion have geometric consequences, for if the terms 
are always unequal (or if both equal 1), then a lemma of Vakil and a geometric
explanation of the recursion (due to Schubert) imply that all Schubert 
problems involving lines have Galois group that is at least alternating.  
Extensive computation suggested that this is the case.

The inequality is easy, in most cases.  To treat the remaining cases,
we discovered a formula for these Kostka numbers as trigonometric
integrals, which should be of independent interest.   This integral 
formula reduces the inequaity of Kostka numbers to the positivity of 
certain trigonometric integrals, which are established by estimation.  This shows 
that Galois groups of Schubert problems of lines are at least alternating.

My talk will describe this story with a emphasis on its combinatorial
aspects.  This is joint work with Christopher Brooks and Abraham
Martin del Campo, who were, respectively, an undergraduate and a
graduate student at the time of this research.