scholarly journals Face functors for KLR algebras

2017 ◽  
Vol 21 (7) ◽  
pp. 106-131 ◽  
Author(s):  
Peter J. McNamara ◽  
Peter Tingley
Keyword(s):  
2020 ◽  
Vol 224 (11) ◽  
pp. 106410 ◽  
Author(s):  
Doeke Buursma ◽  
Alexander Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Type A ◽  

2017 ◽  
Vol 5 ◽  
Author(s):  
BEN WEBSTER

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.


2019 ◽  
Vol 100 (2) ◽  
pp. 447-469
Author(s):  
Peter J. McNamara
Keyword(s):  

2016 ◽  
Vol 369 (3-4) ◽  
pp. 1383-1433 ◽  
Author(s):  
Anton Evseev
Keyword(s):  

2018 ◽  
Vol 188 (2) ◽  
pp. 453-512 ◽  
Author(s):  
Anton Evseev ◽  
Alexander Kleshchev

Author(s):  
Doeke Buursma ◽  
Alexander Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Type A ◽  

2018 ◽  
Vol 371 (7) ◽  
pp. 4535-4583 ◽  
Author(s):  
Alexander Kleshchev ◽  
Robert Muth
Keyword(s):  

2016 ◽  
Vol 152 (8) ◽  
pp. 1648-1696 ◽  
Author(s):  
Peter Tingley ◽  
Ben Webster

We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.


2015 ◽  
Vol 422 ◽  
pp. 563-610 ◽  
Author(s):  
Ruslan Maksimau
Keyword(s):  

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