Gabriel showed (among other things) that given an orientation on the $A_n$ Dynkin diagram, and a labeling of the vertices $Q$ by vector spaces $V_q$, the group $\prod_q GL(V_q)$ acts on the representation space $\prod_e \mathrm{Hom}(V_t,V_h)$ with finitely many orbits.

In the case that the orientation is all one direction, Zelevinskii showed how to understand these orbit closures, or "quiver cycles", in terms of Schubert varieties. Ezra Miller, Mark Shimozono, and I used this to give several positive formulae for the multigraded Hilbert polynomials of these cycles.

Ryan Kinser and Jenna Rajchgot modified Zelevinskii's map to deal with the case of arbitrary orientation. I'll explain how Kinser, Rajchgot and I use that to generalize three of the four [KMS] formulae.