Rouquier's conjecture and diagrammatic algebra

TL;DR

This paper proves Rouquier's conjecture linking decomposition numbers in category O of cyclotomic rational Cherednik algebras to Uglov's canonical basis, using new diagrammatic models that reveal rich algebraic structures.

Contribution

It develops two diagrammatic models for category O, providing explicit descriptions of representations and bases, and connecting these to canonical bases and braid group actions.

Findings

Proves Rouquier's conjecture on decomposition numbers

Constructs explicit diagrammatic models for category O

Establishes connections to canonical bases and braid group actions

Abstract

We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category $\mathcal{O}$; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category $\mathcal{O}$ in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.