Cellularity of KLR and wKLRW algebras via crystals

We prove that the weighted Khovanov–Lauda–Rouquier–Webster algebras of finite type, and their cyclotomic quotients, are sandwich cellular algebras. The sandwich cellular bases are explicitly described using crystal graphs. As a special case, this proves that the Khovanov–Lauda–Rouquier algebras of finite type are sandwich cellular. As one of our applications, we give explicit formulas for some graded decomposition numbers of the cyclotomic algebras in level one.