Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

By Jonathan Brundan, Alexander S. Kleshchev. 2008

363 citations recorded by the source corpus.

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b83ccaa60556a0974eb9910641756341e4930aab

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Citations in the corpus

Categorifications and cyclotomic rational double affine Hecke algebras (2013, background): “We will prove that the isomorphism αK in [48] (which differs from the one in [9]) restricts to an isomorphism αR : H s R,α „ ÑHs R,α.”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, neutral): “In [18] a remarkable connection between the representation theory of Hecke algebras and Lusztig’s canonical bases was established by showing that cyclotomic Hecke algebras are isomorphic to cyclotomic quotients of quiver Hecke algebras introduced in [46], [47], where also a connection between the re”
Homogeneous representations of Khovanov-Lauda algebras (2008, methodology): “In type A this can be explained using [BK].”
Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV (2013, background): “In [4, 35], Brundan-Kleshchev and Rouquier showed that the affine Hecke algebra of type A is isomorphic to the Khovanov-Lauda-Rouquier algebra of type A or of type A∞ up to a specialization and a localization.”
Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras (2011, methodology): “In [2], Brundan and Kleshchev proved this conjecture for type A∞ and A (1) n using the isomorphism between R and the cyclotomic Hecke algebra H which was constructed in [1].”
Crystals from categorified quantum groups (2009, background): “The more general conjecture was proven in finite and affine type A by Brundan and Kleshchev [BK08, BK09].”
Graded Specht modules (2009, methodology): “In this section we recall the definition of the cyclotomic Hecke algebras and make them into Z-graded algebras following [7].”
On category O for cyclotomic rational Cherednik algebras (2011, methodology): “Our proof is based on an isomorphism between the usual and degenerate cyclotomic Hecke algebra due to Brundan and Kleshchev [17].”
ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA (2013, methodology): “entations of Ts and the category Os. In particular, the category of graded modules over Ts is a graded lift of Os compatible with the graded lifts of the Hecke algebra deﬁned by Brundan and Kleshchev [BK09]. We should emphasize to the reader: this is the ﬁrst explicit description of the category Ofo”
Categorifications from planar diagrammatics (2010, background): “For instance, key results about simple and projective R(ν)-modules [26] follow via Kleshchev-Grojnowski constructions that amount to the representation-theoretical counterpart of the crystal graph structure.”
Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words (2009, neutral): “R T ] 1 0 Se p”