Positivity of quiver cycles via deformation

Let Hom(V) be the set of quivers V₀ -> V₁ -> ... -> V_n. A quiver cycle is a subset O_r of Hom(V) where the ranks of the composite maps V_i -> V_j are bounded above by specified integers r=(r_ij) for i < j . Our goal is to compute the equivariant cohomology class [O_r]. As a special case one obtains Fulton's universal Schubert polynomials. Buch and Fulton expressed [O_r] in terms of Schur functions, and conjectured a combinatorial formula for the coefficients. In particular, they conjectured that the coefficients, which directly generalize the Littlewood-Richardson coefficients, are positive.

In this ongoing project, we construct a flat family whose general fiber is isomorphic to O_r, and whose special fiber has components that are direct products of matrix Schubert varieties. This proves that [O_r] is a sum of products of Stanley symmetric functions (stable double Schubert polynomials) where each summand is indexed by a list w of permutations. Our formula is obviously positive for geometric reasons and immediately implies the positivity of the Buch-Fulton formula. We conjecture that the special fiber is generically reduced, so that each list of permutations w occurs with multiplicity 1. We propose a simple nonrecursive combinatorial characterization of which lists w appear.

This is joint work with Allan Knutson and Ezra Miller.