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302
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 182 (38 self)
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To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h ∗ , then the algebras Hκ are certain ‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sninvariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the CalogeroMoser
Conjectures on the quotient ring by diagonal invariants
 J. ALGEBRAIC COMBIN
, 1994
"... We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[x1,...,xn,y1,...,yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1,.. ..."
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Cited by 98 (10 self)
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We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[x1,...,xn,y1,...,yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1,...,xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.
Yangians And Classical Lie Algebras
"... Introduction The term `Yangian' was introduced by V. G. Drinfeld to specify quantum groups related to rational solutions of the classical YangBaxter equation; see Belavin Drinfeld [BD1,BD2] for the description of these solutions. In Drinfeld [D1] for each simple finitedimensional Lie algebra a ..."
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Cited by 71 (16 self)
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Introduction The term `Yangian' was introduced by V. G. Drinfeld to specify quantum groups related to rational solutions of the classical YangBaxter equation; see Belavin Drinfeld [BD1,BD2] for the description of these solutions. In Drinfeld [D1] for each simple finitedimensional Lie algebra a, a certain Hopf algebra Y(a) was constructed so that Y(a) is a deformation of the universal enveloping algebra for the polynomial current Lie algebra a[x]. An alternative description of the algebra Y(a) was given in Drinfeld [D3]; see Theorem 1 therein. Prior to the intruduction of the Hopf algebra Y(a) in Drinfeld [D1], the algebra which may be called the Yangian for the reductive Lie algebra gl(N) and may be denoted by Y , was considered in the works of mathematical physicists from St.Petersburg; see for instance TakhtajanFaddeev [TF]. The latter algebra is a deformation of the universal enveloping algebra U . Representations of were studied in KulishReshetikhinSklya
Decomposition of quantics in sums of powers of linear forms
 Signal Processing
, 1996
"... Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering prob ..."
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Cited by 67 (20 self)
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Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the Eigen Value Decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms. Keywords. Tensors, Polynomials, Diagonalization, EVD, HighOrder Statistics, Cumulants. 1
Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
 Invent. Math
, 2001
"... Abstract. In an earlier paper [13], we showed that the Hilbert scheme of points in the plane Hn = Hilb n (C 2) can be identified with the Hilbert scheme of regular orbits C 2n //Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspond ..."
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Cited by 62 (2 self)
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Abstract. In an earlier paper [13], we showed that the Hilbert scheme of points in the plane Hn = Hilb n (C 2) can be identified with the Hilbert scheme of regular orbits C 2n //Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n + 1) n−1. 1.
Matrix models for circular ensembles
 Int. Math. Res. Not
"... Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e ..."
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Cited by 47 (6 self)
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Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e
Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
 Comm. Math. Phys
"... ABSTRACT. We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated po ..."
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Cited by 45 (14 self)
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ABSTRACT. We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haardistributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type. 1.
Minor Identities For QuasiDeterminants And Quantum Determinants
 COMM. MATH. PHYS
, 1994
"... We present several identities involving quasiminors of noncommutative generic matrices. These identities are specialized to quantum matrices, yielding qanalogues of various classical determinantal formulas. ..."
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Cited by 40 (4 self)
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We present several identities involving quasiminors of noncommutative generic matrices. These identities are specialized to quantum matrices, yielding qanalogues of various classical determinantal formulas.