Hjalmar Rosengren

Elliptic hypergeometric series on root systems

Abstract We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A n , C n and D n . In the special cases of classical and q -series, our approach leads to new elementary proofs of the corresponding identities.

An Izergin--Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices

We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin-Korepin formula for the six-vertex model. As applications, we find dynamical (in the sense of the dynamical Yang-Baxter equation) generalizations of the enumeration and 2-enumeration of alternating sign matrices. The dynamical enumeration has a nice interpretation in terms of three-colourings of the square lattice.

Elliptic determinant evaluations and the Macdonald identities for affine root systems

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.

Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations

Abstract Using multiple q -integrals and a determinant evaluation, we establish a multivariable extension of Baileys nonterminating 1009 transformation. From this result, we deduce new multivariable terminating 10 φ 9 transformations, 8 φ 7 summations and other identities. We also use similar methods to derive new multivariable l 1 ψ 1 summations. Some of our results are extended to the case of elliptic hypergeometric series.

Multivariable orthogonal polynomials and coupling coefficients for discrete series representations

We study polynomials of several variables which occur as coupling coefficients for the analytic continuation of the holomorphic discrete series of SU(1,1). There are three types of such polynomials, one corresponding to each conjugacy class of one-parameter subgroups. They may be viewed as multivariable generalizations of Hahn, Jacobi, and continuous Hahn polynomials and include many orthogonal and biorthogonal families occurring in the literature. We give a simple and unified approach to these polynomials using the group theoretic interpretation. We prove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for the Heisenberg group, leading to multivariable generalizations of Krawtchouk and Hermite polynomials.

Sklyanin invariant integration

The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric series and integrals and with elliptic 6j-symbols.

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.

Elliptic U(2) Quantum Group and Elliptic Hypergeometric Series

We investigate an elliptic quantum group introduced by Felder and Varchenko, which is constructed from the R-matrix of the Andrews–Baxter–Forrester model, containing both spectral and dynamical parameter. We explicitly compute the matrix elements of certain corepresentations and obtain orthogonality relations for these elements. Using dynamical representations these orthogonality relations give discrete bi-orthogonality relations for terminating very-well-poised balanced elliptic hypergeometric series, previously obtained by Frenkel and Turaev and by Spiridonov and Zhedanov in different contexts.

A new pentagon identity for the tetrahedron index

A bstractRecently Kashaev, Luo and Vartanov, using the reduction from a fourdimensional superconformal index to a three-dimensional partition function, found a pentagon identity for a special combination of hyperbolic Gamma functions. Following their idea we have obtained a new pentagon identity for a certain combination of so-called tetrahedron indices arising from the equality of superconformal indices of dual three-dimensional

Karlsson–Minton type hypergeometric functions on the root system Cn

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