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68
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.
BCnsymmetric abelian functions
, 2006
"... We construct a family of BCnsymmetric biorthogonal abelian functions generalizing Koornwinder’s orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald’s conjectures. The construction is based on a direct construction for a special case generalizing Okoun ..."
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Cited by 44 (7 self)
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We construct a family of BCnsymmetric biorthogonal abelian functions generalizing Koornwinder’s orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald’s conjectures. The construction is based on a direct construction for a special case generalizing Okounkov’s interpolation polynomials. We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson’s summation and Bailey’s
Essays on the theory of elliptic hypergeometric functions
, 2008
"... We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An ..."
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Cited by 44 (10 self)
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We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.
An elementary approach to 6jsymbols (classical, quantum, rational, trigonometric, and elliptic)
, 2003
"... Elliptic 6jsymbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6jsymbols (or qRacah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6jsymbols, which i ..."
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Cited by 38 (2 self)
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Elliptic 6jsymbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6jsymbols (or qRacah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6jsymbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6jsymbols in terms of Sklyanin algebra representations.
Superconformal indices for N = 1 theories with multiple duals, Nucl
 Phys. B824
"... Following a recent work of Dolan and Osborn, we consider superconformal indices of four dimensional N = 1 supersymmetric field theories related by an electricmagnetic duality with the SP(2N) gauge group and fixed rank flavour groups. For the SP(2) (or SU(2)) case with 8 flavours, the electric theor ..."
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Cited by 34 (13 self)
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Following a recent work of Dolan and Osborn, we consider superconformal indices of four dimensional N = 1 supersymmetric field theories related by an electricmagnetic duality with the SP(2N) gauge group and fixed rank flavour groups. For the SP(2) (or SU(2)) case with 8 flavours, the electric theory has index described by an elliptic analogue of the Gauss hypergeometric function constructed earlier by the first author. Using the E7root system Weyl group transformations for this function, we build a number of dual magnetic theories. One of them was originally discovered by Seiberg, the second model was built by Intriligator and Pouliot, the third one was found by Csáki et al. We argue that there should be in total 72 theories dual to each other through the action of the coset group W(E7)/S8. For the general SP(2N), N> 1, gauge group, a similar multiple duality takes place for slightly more complicated flavour symmetry groups. Superconformal indices of the corresponding theories coincide due to the Rains identity for a multidimensional elliptic hypergeometric integral associated with the BCNroot system.
BCnsymmetric polynomials
, 2004
"... We consider two important families of BCnsymmetric polynomials, namely Okounkov’s interpolation polynomials and Koornwinder’s orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a n ..."
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Cited by 29 (7 self)
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We consider two important families of BCnsymmetric polynomials, namely Okounkov’s interpolation polynomials and Koornwinder’s orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate qanalogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald’s conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an
Periodic Schur processes and cylindric partitions
, 2006
"... Periodic Schur process is a generalization of the Schur process introduced in [OR1] (math.CO/0107056). We compute its correlation functions and their bulk scaling limits, and discuss several applications including asymptotic analysis of uniform measures on cylindric partitions, timedependent exte ..."
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Cited by 25 (4 self)
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Periodic Schur process is a generalization of the Schur process introduced in [OR1] (math.CO/0107056). We compute its correlation functions and their bulk scaling limits, and discuss several applications including asymptotic analysis of uniform measures on cylindric partitions, timedependent extensions of the discrete sine kernel, and bulk limit behavior of certain measures on partitions introduced in [NO] (hepth/0306238) in connection with supersymmetric gauge theories.
Elliptic determinant evaluations and the Macdonald identities for affine root systems
 Compositio Math
"... We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreduci ..."
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Cited by 22 (9 self)
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We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems. 1.