K polynomials and matroids
This talk is my about 2009 UMn PhD thesis problem which I have finally solved, only 10 years after the fact. The set-up was to take n vectors, tensor them together in n! ways, and consider their span as a representation of the symmetric group Sn. The problem was to see if the character of this representation could be determined knowing only the matroid of the n starting vectors. In this talk I present an affirmative solution to this problem, including an explicit generating function that computes the character from the matroid. The long overdue solution took a lengthy detour through representation theory, algebraic geometry and, fusing the two, equivariant K-theory. I will keep the technical aspects to a minimum, and focus on explaining how this problem in combinatorics led to a tractable geometric problem.