Cellularity of KLR and weighted KLRW algebras via crystals

We prove that the weighted KLRW algebras of finite type, and their cyclotomic quotients, are cellular algebras. The cellular bases are explicitly described using crystal graphs. As a special case, this proves that the KLR algebras of finite type are cellular. As one application, we give explicit formulas for the graded decomposition numbers of the cyclotomic algebras in level one.