Cellularity of KLR and weighted KLRW algebras via crystals
Andrew Mathas, Daniel Tubbenhauer
TL;DR
This work establishes that finite-type weighted KLRW algebras and their cyclotomic quotients admit graded sandwich cellular structures with bases governed by crystal graphs, linking diagrammatic algebras to Lie-theoretic combinatorics. The authors develop a comprehensive framework, including definitions of wKLRW algebras, Q-polynomials, multilocal relations, and a robust notion of pulling strings and jumping dots, to produce explicit crystal-indexed cellular bases and to derive Morita equivalences with KLR algebras. They prove that simple modules correspond to crystal vertices and, in level-one cases, provide explicit bases for projectives and graded decomposition numbers; they also give criteria for semisimplicity, quasi-heredity, and indecomposability, all independent of the base field’s characteristic. The main technical engine is the crystal-theoretic realization of basis elements and their organization into sandwiched algebras, yielding a transparent, uniform method to study blocks and representation theory across finite types. The work also illuminates the interplay between crystal combinatorics and diagrammatic categorifications, with concrete running examples and SageMath computational support, and it delineates the boundaries where cellularity fails (notably outside certain cases in non-A types).
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.