
Let Hom(V) be the set of quivers V_0 > V_1 > ... > 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
LittlewoodRichardson 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 BuchFulton 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.

