On a conjectured formula for quiver varieties

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In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be non-negative. We conjecture that each of these coefficients count the number of sequences of semistandard Young tableaux which satisfy certain conditions. In this paper I give a proof of this conjecture in the special case where the quiver variety can be described by at most four vector bundles. I also prove that the general conjecture follows from a simple combinatorial statement for which substantial computer verification has been obtained.