The Grothendieck class of a quiver variety

A quiver variety is a general type of degeneracy locus which is obtained by putting arbitrary rank conditions on a sequence of vector bundles and bundle maps over a variety $X$. I will describe a formula for the structure sheaf of a quiver variety in the Grothendieck group $K^0(X)$ of vector bundles on $X$. This generalizes earlier joint work with Fulton on the cohomology class of quiver varieties. An important ingredient for the new formula is a generalization of the ring of symmetric functions which models the $K$-theory of Grassmannians.