Cellularity of KLR and weighted KLRW algebras via crystals

TL;DR

This paper proves that weighted KLRW algebras and their cyclotomic quotients of finite type are cellular algebras, providing explicit bases via crystal graphs and applications to graded decomposition numbers.

Contribution

It establishes the cellularity of weighted KLRW algebras and their cyclotomic quotients, with explicit cellular bases constructed using crystal graphs.

Findings

Weighted KLRW algebras of finite type are cellular.

Explicit cellular bases are described using crystal graphs.

Formulas for graded decomposition numbers in level one are provided.

Abstract

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.