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164
Hecke algebras and Schur algebras of the symmetric group
, 1998
"... These notes give a fully selfcontained introduction to the (modular) representation theory of the IwahoriHecke algebras and the qSchur algebras of the symmetric groups. The central aim of this work is to give a concise, but complete, and an elegant, yet quick, treatment of the classificatio ..."
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Cited by 188 (19 self)
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These notes give a fully selfcontained introduction to the (modular) representation theory of the IwahoriHecke algebras and the qSchur algebras of the symmetric groups. The central aim of this work is to give a concise, but complete, and an elegant, yet quick, treatment of the classification of the simple modules and of the blocks of these two important classes of algebras.
A diagrammatic approach to categorification of quantum groups I
, 2009
"... To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph. ..."
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Cited by 182 (18 self)
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To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph.
KazhdanLusztig polynomials and character formulae for the Lie superalgebra gl(mn)
 J. AMS
, 2002
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Blocks of cyclotomic Hecke algebras and KhovanovLauda algebras
 Invent. Math
"... Abstract. We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (signmodified) cyclotomic KhovanovLauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The KhovanovLauda algebr ..."
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Cited by 83 (12 self)
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Abstract. We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (signmodified) cyclotomic KhovanovLauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The KhovanovLauda algebras are naturally graded, which allows us to exhibit a nontrivial Zgrading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic. 1.
Canonical bases of higherlevel q–deformed Fock spaces and KazhdanLusztig polynomials
 in Physical combinatorics (Kyoto
, 1999
"... We define canonical bases of the higherlevel qdeformed Fock space modules of the affine Lie algebra ̂ sln generalizing the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of certa ..."
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Cited by 71 (0 self)
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We define canonical bases of the higherlevel qdeformed Fock space modules of the affine Lie algebra ̂ sln generalizing the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of certain affine KazhdanLusztig polynomials. 1
The number of simple modules of the Hecke algebras of type G(r,1,n)
 n), Math. Zeitschrift
, 1998
"... Introduction Let n and r be integers with n 0 and r 1. Let R be a commutative ring with 1 and let q, Q 1 ; : : : ; Q r be elements of R with q invertible. The cyclotomic Hecke algebra H R;n = H R;n (q; fQ 1 ; : : : ; Q r g) of type G(r; 1; n) is the unital associative Ralgebra with generators T ..."
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Cited by 68 (13 self)
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Introduction Let n and r be integers with n 0 and r 1. Let R be a commutative ring with 1 and let q, Q 1 ; : : : ; Q r be elements of R with q invertible. The cyclotomic Hecke algebra H R;n = H R;n (q; fQ 1 ; : : : ; Q r g) of type G(r; 1; n) is the unital associative Ralgebra with generators T 0 ; T 1 ; : : : ; Tn\Gamma1 and relations (T 0 \Gamma Q 1 ) \Delta \Delta \Delta (T 0 \Gamma Q r ) = 0; T 0 T 1 T 0<F1
Canonical Bases of qDeformed Fock Spaces
 Int. Math. Res. Notices
, 1996
"... We define a canonical basis of the qdeformed Fock space representation of the affine Lie algebra b gl n . We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are qanalogues of decomposition numbers of the vSchur algebras for v specia ..."
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Cited by 63 (12 self)
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We define a canonical basis of the qdeformed Fock space representation of the affine Lie algebra b gl n . We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are qanalogues of decomposition numbers of the vSchur algebras for v specialized to a nth root of unity. 1
qSchur algebras and complex reflection groups
"... Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These r ..."
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Cited by 58 (2 self)
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Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These results follow from the unicity of certain highest weight categories covering Hecke algebras. We also provide a semisimplicity criterion for Hecke algebras of complex reflection groups. 1.
Quantum Weyl reciprocity and tilting modules
 Comm. Math. Phys
, 1998
"... Abstract. Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of qSchur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ring Z[q;q,1] of Laurent polynomials) and that it behaves well under basechange. A key ingre ..."
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Cited by 51 (18 self)
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Abstract. Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of qSchur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ring Z[q;q,1] of Laurent polynomials) and that it behaves well under basechange. A key ingredient in our approachinvolves the theory of tilting modules for qSchur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a qSchur algebra in all cases. In particular, in the most interesting situation, the Ringel dual identi es with a natural quotient algebra of the Hecke algebra. Weyl reciprocity refers to the connection between the representation theories of the general linear groupGLn(k) and the symmetric group Sr. LetV be a vector space (over a eldk) of dimensionn and form the tensor spaceV r. The natural (left) action ofGLn(k) onV r commutes with the (right) permutation action of Sr. LetA(resp.,R) be the algebra generated by the image ofGLn(k) (resp., Sr) in the algebra End(V r) of linear operators onV r. Classically [We], whenk = C, these algebras satisfy the double centralizer property (1)a)A = EndR(V r) andb)R = EndA(V r): Further, the set + (n;r) of partitions ofrinto at mostnnonzero parts indexes both the irreducibleAmodulesL ( ) and the irreducibleRmodulesS. TheL ( ) are the irreducible polynomial representations ofGLn(C) of homogeneous degreer, while theS are Specht modules for Sr. Weyl reciprocity also entails the decomposition (2)V r = M L ()S 2 + (n;r) of the tensor space into irreducible (A;R op)bimodules. Whenkhas positive characteristicp, property (1) remains true, but it is more di cult to establish; see [CL; (3.1)] for the equality (1a) and [dCP; (4.1)] or [D2; x2 Cor.] for (1b). (The latter is easy whennr.) The set + (n;r) still indexes