Rouquier's conjecture and diagrammatic algebra

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.

64 pages; numerous TikZ figures, PDF is preferable to DVI. v4: Revision in response to referee's report. Several proofs rewritten, examples and pictures added

References in corpus (7)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Knot invariants and higher representation theory
• Tensor product categorifications and the super Kazhdan-Lusztig conjecture
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• On graded presentations of Hecke algebras and their generalizations
• On generalized category $\mathcal{O}$ for a quiver variety
• Quiver Schur algebras and Koszul duality

Cited by in corpus (24)

• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Supports of simple modules in cyclotomic Cherednik categories O
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• On graded presentations of Hecke algebras and their generalizations
• The many graded cellular bases of Hecke algebras
• Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra
• Quiver Schur algebras and Koszul duality
• Geometric representations of graded and rational Cherednik algebras
• Cellularity for weighted KLRW algebras of types $B$, $A^{(2)}$, $D^{(2)}$
• Cellularity and subdivision of KLR and weighted KLRW algebras
• Harish-Chandra bimodules over rational Cherednik algebras
• Character formulas and Bernstein-Gelfand-Gelfand resolutions for Cherednik algebra modules
• Kleshchev's decomposition numbers for diagrammatic Cherednik algebras
• Skew cellularity of the Hecke algebras of type $G(\ell ,p,n)$
• Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$
• Graded cellular basis and Jucys-Murphy elements for generalized blob algebras
• On graded decomposition numbers of Cherednik algebras
• Generalized Mullineux involution and perverse equivalences
• Cellularity of endomorphism algebras of tilting objects
• Decomposable Specht modules indexed by bihooks
• An analogue of row removal for diagrammatic Cherednik algebras
• Cellularity of KLR and weighted KLRW algebras via crystals
• Morita equivalences between cyclotomic KLR algebras in types ${\mathtt{C}}_{\infty }$ and ${\mathtt{A}}_{\infty }$
• Empty runner removal theorem for Ariki-Koike algebras