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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.
Combinatorics, Symmetric Functions, and Hilbert Schemes
 CDM 2002: Current Developments in Mathematics, Intl
, 2003
"... We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdon ..."
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Cited by 30 (0 self)
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We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdonald polynomials to the characters of doublygraded Sn modules. To make the treatment selfcontained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.
Cyclic sieving and noncrossing partitions for complex reflection groups
, 2007
"... We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups.
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Cited by 28 (2 self)
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We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups.
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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Cited by 20 (6 self)
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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.
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.
Quasi–invariants of complex reflection groups
"... We introduce quasiinvariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasiinvariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of some (singular) affine variety Xk. We extend the ma ..."
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Cited by 12 (0 self)
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We introduce quasiinvariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasiinvariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of some (singular) affine variety Xk. We extend the main results of [BEG] to this setting: in particular, we show that the variety Xk and the module Qk are CohenMacaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(C), where n = dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced in [DO] and is parallel to that of [BEG]. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam [DO]. Another result is a proof of a conjecture of Opdam [O2], concerning certain operations (KZ twists) on the set of irreducible representations of W.
Higher Trivariate Diagonal Harmonics via Generalized Tamari Posets
 Journal of Combinatorics
, 2012
"... Abstract. We consider the graded Snmodules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associa ..."
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Cited by 11 (5 self)
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Abstract. We consider the graded Snmodules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associated Hilbert series and graded Frobenius characteristics. This also leads to entirely new combinatorial formulas. Contents
On category O for the rational Cherednik algebra of G(m,1,n): the almost semisimple case
, 2008
"... We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dim ..."
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Cited by 11 (1 self)
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We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dimension (plus a mild genericity condition on the parameters). Our proof involves calculating the blocks of the ArikiKoike algebra in a special case.
Rational Cherednik algebras and diagonal coinvariants of G(m
 Department of Mathematics, University of Glasgow
"... Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational ..."
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Cited by 11 (2 self)
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Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational