In the general case, quiver cycles are of the form of orbit closures of $GL\cdot V_{\vec{r}}$, where $GL= \prod_{i=0}^n GL_{r_i}$ is the possible changes of basis on all of the vector spaces on each of the vertices and $V_{\vec{r}}$ is any representation of the quiver with fixed dimension vector $\vec{r}$. In the equioriented A_{n} case, these are well understood by Zelevinsky and Lakshmibai-Magyar by showing them isomorphic to open sets in Schubert varieties. Bobinski and Zwara claim to reduce the non-equioriented case to the equioriented case, but I don't see how they are doing that.

In the introduction to ``Normality of Orbit Closures for Dynkin Quivers" (manuscripta math. 2001), Bobinski and Zwara say that they will generalize the result that equioriented A_{n} quivers have the same singularities as Schubert varieties to non-equioriented A_{n} quivers. They claim that they will do this by reducing the non-equioriented case to the equioriented case. So far, so good. But then, they say that this result follows from the proposition that they will prove, which I don't see has to do with the theorem at all.

The proposition is about a Dynkin quiver, Q, of type A_{p+q+1} with p arrows in one direction and q arrows in the other and Q' an equioriented Dynkin quiver of type A_{p+2q+1}, their respective path algebras B=kQ and A=kQ', and respective Auslander-Reiten quivers &Gamma_{B} and &Gamma_{A} over the category of finite dimensional left modules over A and B. The proposition says ``Let A=kQ' and B=kQ be the path algebras of quivers Q' and Q, respectively, where Q and Q' are Dynkin quivers of type A. Assume there exists a full embedding of translation quivers $F: \Gamma_B \to \Gamma_A$. Then there exists a hom-controlled exact functor $\mathcal{F}: \text{mod }B \to \text{mod }A$."

Can anyone tell me how (or if) their results translate into a result that tells me a recipe for constructing a Kazhdan-Lustzig variety from my non-equioriented quiver? (By K-L variety, I mean a Schubert variety intersect an opposite Bruhat cell.) Alternately, is there a way to see which particular sub-variety of the representation variety of equioriented A_{p+2q+1} I get out of this theorem and how that is (maybe a GIT quotient away from) a Kazhdan-Lustzig variety?

Thanks,

Anna