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164
Hecke algebras and Schur algebras of the symmetric group
, 1998
"... These notes give a fully self--contained introduction to the (modular) representation theory of the Iwahori--Hecke algebras and the q--Schur 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 self--contained introduction to the (modular) representation theory of the Iwahori--Hecke algebras and the q--Schur 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 Kac-Moody 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 Kac-Moody Lie algebra associated with the graph.
Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n)
- J. AMS
, 2002
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Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
- Invent. Math
"... Abstract. 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 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 (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. 1.
Canonical bases of higher-level q–deformed Fock spaces and Kazhdan-Lusztig polynomials
- in Physical combinatorics (Kyoto
, 1999
"... We define canonical bases of the higher-level q-deformed 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 higher-level q-deformed 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 Kazhdan-Lusztig 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 R--algebra 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 R--algebra 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 q-Deformed Fock Spaces
- Int. Math. Res. Notices
, 1996
"... We define a canonical basis of the q-deformed 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 q-analogues of decomposition numbers of the v-Schur algebras for v specia ..."
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Cited by 63 (12 self)
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We define a canonical basis of the q-deformed 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 q-analogues of decomposition numbers of the v-Schur algebras for v specialized to a nth root of unity. 1
q-Schur 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 q-Schur 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 q-Schur 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 semi-simplicity 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 q-Schur 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 base-change. 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 q-Schur 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 base-change. A key ingredient in our approachinvolves the theory of tilting modules for q-Schur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a q-Schur 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 irreducibleA-modulesL ( ) and the irreducibleR-modulesS. 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
