Extension-orthogonal components of nilpotent varieties

Abstract

Let Q be a Dynkin quiver, and let P(Q) be the corresponding preprojective algebra. Let I be a set of pairwise different indecomposable irreducible components of varieties of P(Q)-modules such that generically there are no extensions between these components. We show that the number of elements in I is at most the number of positive roots of Q. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.