Homomorphisms between standard modules over finite type KLR algebras

Khovanov-Lauda-Rouquier algebras of finite Lie type come with families of standard modules, which under the Khovanov-Lauda-Rouquier categorification correspond to PBW-bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and all non-zero endomorphisms of a standard module are injective. We obtain applications to extensions between standard modules and modular representation theory of KLR algebras.