scholarly journals Face functors for KLR algebras

2017 ◽  
Vol 21 (7) ◽  
pp. 106-131 ◽  
Author(s):  
Peter J. McNamara ◽  
Peter Tingley
Keyword(s):  
Klr Algebras

scholarly journals Some extension algebras for standard modules over KLR algebras of type A

2020 ◽  
Vol 224 (11) ◽  
pp. 106410 ◽  
Author(s):  
Doeke Buursma ◽  
Alexander Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Type A ◽  
Klr Algebras

scholarly journals ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA

2017 ◽  
Vol 5 ◽  
Author(s):  
BEN WEBSTER
Keyword(s):  
Braid Group ◽  
Fock Space ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Intrinsic Interest ◽  
Cherednik Algebra ◽  
Klr Algebras ◽  
Derived Equivalences ◽  
Rational Cherednik Algebra ◽  
Decomposition Numbers

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.

scholarly journals Folding KLR algebras

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

scholarly journals RoCK blocks, wreath products and KLR algebras

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

scholarly journals Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

2018 ◽  
Vol 188 (2) ◽  
pp. 453-512 ◽  
Author(s):  
Anton Evseev ◽  
Alexander Kleshchev
Keyword(s):  
Symmetric Groups ◽  
Klr Algebras ◽  
Weyl Duality

scholarly journals Resolutions of standard modules over KLR algebras of type A

2020 ◽  
Author(s):  
Doeke Buursma ◽  
Alexander Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Type A ◽  
Klr Algebras

scholarly journals Affine zigzag algebras and imaginary strata for KLR algebras

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

scholarly journals Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

2016 ◽  
Vol 152 (8) ◽  
pp. 1648-1696 ◽  
Author(s):  
Peter Tingley ◽  
Ben Webster
Keyword(s):  
Lie Algebras ◽  
Finite Type ◽  
General Type ◽  
The Other ◽  
Explicit Description ◽  
Affine Grassmannian ◽  
Combinatorial Description ◽  
Extra Information ◽  
Klr Algebras ◽  
Affine Type

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.

scholarly journals Canonical basis, KLR algebras and parity sheaves

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