Extension-orthogonal components of preprojective varieties

Abstract

Let Q Q be a Dynkin quiver, and let Λ \Lambda be the corresponding preprojective algebra. Let C = { C i ∣ i ∈ I } {\mathcal C} = \{ C_i \mid i \in I \} be a set of pairwise different indecomposable irreducible components of varieties of Λ \Lambda -modules such that generically there are no extensions between C i C_i and C j C_j for all i , j i,j . We show that the number of elements in C {\mathcal C} is at most the number of positive roots of Q Q . Furthermore, we give a module-theoretic interpretation of Leclerc’s counterexample to a conjecture of Berenstein and Zelevinsky.