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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
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.
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