|Date: Friday, September 28, 2012
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: An inequality of Kostka numbers and Galois groups of Schubert problems of lines
Abstract: Tensor decompositions of rational sl_2-modules and Schubert calculus on the Grassmannians of lines are both goverened 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 explaination of the recursion (due to Schubert) imply that all Schubert problems involving lines have Galois group that is at least aternating. 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 a trigonometric integral, which should be of independent interest. This integral formula reduces the inequaity of Kostka numbers to the positivity of a trigonometric integral, which is 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.
Speaker: Frank Sottile
Institution: Texas A&M