Involution combinatorics, symmetric varieties, and Schubert polynomials
I will discuss an interesting partial order attached to the set of involutions in the symmetric group and describe certain intervals in this poset. This combinatorics has a geometric interpretation in terms of certain symmetric varieties, most specifically the variety that parametrizes smooth quadric hypersurfaces in projective space. Relating the combinatorial description of the intervals in the poset to certain equivariant cohomology calculations on these varieties yields identities that write certain sums of Schubert polynomials as products of linear factors. This is joint work with Mahir Can and Ben Wyser.