Cellularity of KLR and weighted KLRW algebras via crystals
Crystal combinatorics gives explicit sandwich cellular bases for finite-type weighted KLRW algebras.
Data
- Title: Cellularity of KLR and weighted KLRW algebras via crystals
- Authors: Andrew Mathas and Daniel Tubbenhauer
- Status: Commun. Am. Math. Soc. 6 (2026), 548–633. Last update: Sun, 2 Nov 2025 11:57:39 UTC
- Code / errata: GradedDimKLR, klr-and-crystal
- arXiv: https://arxiv.org/abs/2309.13867
Abstract
We prove that the weighted KLRW 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 KLR algebras of finite type are sandwich cellular. As one application, we give explicit formulas for the graded decomposition numbers of the cyclotomic algebras in level one.
What is the point?
KLR and KLRW algebras are central objects in modern categorification. They are powerful, but structurally complicated. This paper uses crystal graphs as a guide for writing down cellular bases, turning representation-theoretic combinatorics into explicit algebraic structure.
KLR and weighted KLRW algebras.
crystal graphs.
sandwich cellular bases and decomposition numbers.
A picture
An example of how the crystal combinatorics organizes the algebra.
