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582
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 177 (3 self)
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Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
The refined topological vertex
, 2009
"... We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the selfdual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a twoparameter (equivar ..."
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Cited by 90 (11 self)
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We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the selfdual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a twoparameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.
Advanced determinant calculus: a complement
 LINEAR ALGEBRA APPL
, 2005
"... This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular probl ..."
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Cited by 89 (8 self)
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This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems.
Kostka Polynomials and Energy Functions in Solvable Lattice Models
 Selecta Math. (N.S
"... The relation between the charge of LascouxSchuzenberger and the energy function in solvable lattice models is clarified. As an application, A.N.Kirillov's conjecture on the expression of the branching coefficient of c sl n =sl n as a limit of Kostka polynomials is proved. ..."
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Cited by 75 (7 self)
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The relation between the charge of LascouxSchuzenberger and the energy function in solvable lattice models is clarified. As an application, A.N.Kirillov's conjecture on the expression of the branching coefficient of c sl n =sl n as a limit of Kostka polynomials is proved.
Cherednik algebras and differential operators on quasiinvariants
 Duke Math. J
"... We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space h, and a parameter “c. ” We use it to show that, for integral values of “c, ” the algebra Hc is simple and Morita equivalent to D(h)#W, the cross product of W with the algeb ..."
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Cited by 67 (12 self)
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We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space h, and a parameter “c. ” We use it to show that, for integral values of “c, ” the algebra Hc is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, Qc, of quasiinvariant polynomials on h, such that C[h] W ⊂ Qc ⊂ C[h]. We prove that the algebra D(Qc) of differential operators on quasiinvariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Qc) W ⊂ D(Qc) of Winvariant operators turns out to be isomorphic to the spherical subalgebra eHce ⊂ Hc. We show that D(Qc) is generated, as an algebra, by Qc and its “Fourier dual ” Q ♭ c, and that D(Qc) is a rankone projective (Qc ⊗ Q ♭ c)module (via multiplicationaction on D(Qc) on opposite sides).
Matrix models, geometric engineering and elliptic genera
, 2008
"... We compute the prepotential of N = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kähler and complex moduli of T². We use three different methods to obtain this: matrix models, geometric engineering and ..."
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Cited by 67 (19 self)
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We compute the prepotential of N = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kähler and complex moduli of T². We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T². Geometric engineering involves considering Ftheory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R 4. We study the compactifications of N = 2 ∗ theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5branes wrapped on T² combines the Kähler and complex moduli of T² String theory has been rather successful in providing insights into the dynamics of supersymmetric
Rational Cherednik algebras and Hilbert schemes
"... Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also po ..."
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Cited by 58 (6 self)
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Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also possible to associate to a filtered Hc or Ucmodule M a coherent sheaf Φ(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities τ: Hilb(n) → h ⊕ h ∗ /W. For example, we prove: • If c = 1/n, so that Lc(triv) is the unique onedimensional simple Hcmodule, then Φ(eLc(triv)) ∼ = OZn, where Zn = τ −1 (0) is the punctual Hilbert scheme. • If c = 1/n + k for k ∈ N then, under a canonical filtration on the finite dimensional module Lc(triv), gr eLc(triv) has a natural bigraded structure which coincides with that on H 0 (Zn, L k), where L ∼ = O Hilb(n)(1); this confirms conjectures of Berest, Etingof and Ginzburg. • Under mild restrictions on c, the characteristic cycle of Φ(e∆c(µ)) equals ∑ λ Kµλ[Zλ], where Kµλ are Kostka numbers and the Zλ are (known) irreducible components of τ −1 (h/W). Contents
An algebraic characterization of the affine canonical basis
 Duke Math. J
, 1999
"... Abstract. The canonical basis for finite type quantized universal enveloping algebras was introduced in [L3]. The principal technique is the explicit construction (via the braid group action) of a lattice L over Z[q −1]. This allows the algebraic characterization of the canonical basis as a certain ..."
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Cited by 56 (15 self)
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Abstract. The canonical basis for finite type quantized universal enveloping algebras was introduced in [L3]. The principal technique is the explicit construction (via the braid group action) of a lattice L over Z[q −1]. This allows the algebraic characterization of the canonical basis as a certain barinvariant basis of L. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the “imaginary ” subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBWtype basis whose Z[q −1]span reduces to a “crystal ” basis at q = ∞, with the imaginary component given by the Schur functions. 0. Introduction. The canonical basis of the quantized universal enveloping algebra associated to a simple finitedimensional Lie algebra was introduced by Lusztig in [L3] via an elementary algebraic definition. The definition was characterized by three main components: 1) the basis was integral, 2) it was barinvariant, and 3) it spanned a