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Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
"... This is the second of a series of four articles studying various generalisations of Khovanov’s diagram algebra. In this article we develop the general theory of Khovanov’s diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the ..."
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Cited by 103 (12 self)
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This is the second of a series of four articles studying various generalisations of Khovanov’s diagram algebra. In this article we develop the general theory of Khovanov’s diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the
Representations of rational Cherednik algebras
 CONTEMPORARY MATHEMATICS
"... This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with ..."
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Cited by 44 (4 self)
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This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with the representation theory of complex semisimple Lie algebras.
Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras
, 2008
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Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
, 2007
"... We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra va ..."
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Cited by 28 (2 self)
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We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the cfunction) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the afunction) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZfunctor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties,
On category O for the cyclotomic rational Cherednik algebras
, 2011
"... We study equivalences for category Op of the rational Cherednik algebras Hp of type Gℓ(n) = (µℓ) n ⋊ Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ Sℓ and an action of Sℓ on a nonempty Zariski open set of parameters p; a derived equivalence between Op and Op ′ whenever p and p ′ h ..."
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We study equivalences for category Op of the rational Cherednik algebras Hp of type Gℓ(n) = (µℓ) n ⋊ Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ Sℓ and an action of Sℓ on a nonempty Zariski open set of parameters p; a derived equivalence between Op and Op ′ whenever p and p ′ have integral difference; a highest weight equivalence between Op and a parabolic category O for the general linear group, under a nonrationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.
Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)
 IN PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY. ARXIV:MATH/0612733
"... The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r, p, n). As a first application, we give a selfcontained and elementary proof of the analog for the groups G( ..."
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The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r, p, n). As a first application, we give a selfcontained and elementary proof of the analog for the groups G(r, p, n), with r> 1, of Gordon’s theorem (previously Haiman’s conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p ̸ = r has been proved by Vale using a technique analogous to Gordon’s. Because of the combinatorial application to Haiman’s conjecture, the paper is logically selfcontained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon’s proof of his theorem.
Generalized Jack polynomials and the representation theory of rational Cherednik algebras
 Selecta Math. (N.S
"... Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1. ..."
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Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1.
Orthogonal functions generalizing Jack polynomials
 Trans. Amer. Math. Soc
"... Abstract. The rational Cherednik algebra H is a certain algebra of differentialreflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for H. This paper deals with the ..."
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Cited by 16 (9 self)
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Abstract. The rational Cherednik algebra H is a certain algebra of differentialreflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for H. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible Wmodules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on C n with values in the representation S λ. For G(1, 1, n) with λ = (n) these functions are the nonsymmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all welldefined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of H so that the rational Cherednik algebra for G(r, p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.