Let Hom(V) be the set of quivers
V0 -> V1 -> ... -> Vn.
A quiver cycle is a subset Or of Hom(V)
where the ranks of the composite maps
Vi -> Vj are
bounded above by specified integers r=(rij)
for i < j .
Our goal is to compute the equivariant cohomology class
As a special case one obtains Fulton's universal Schubert polynomials.
Buch and Fulton expressed [Or] in
terms of Schur functions, and conjectured
a combinatorial formula for the coefficients.
In particular, they conjectured that the coefficients, which directly
Littlewood-Richardson coefficients, are positive.
In this ongoing
project, we construct
a flat family whose general fiber is isomorphic to Or,
and whose special fiber
has components that are direct products of matrix Schubert varieties.
This proves that
[Or] is a sum of products of Stanley symmetric functions
(stable double Schubert polynomials) where each summand is indexed by a
of permutations. Our formula is obviously positive for geometric reasons
implies the positivity of the Buch-Fulton formula. We conjecture that
the special fiber
is generically reduced, so that each list of permutations w
We propose a simple nonrecursive combinatorial characterization of which
This is joint work with Allan Knutson and Ezra Miller.