J. Van der Jeugt

CONVOLUTIONS FOR ORTHOGONAL POLYNOMIALS FROM LIE AND QUANTUM ALGEBRA REPRESENTATIONS

Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra

Irreducible representations of the exceptional Lie superalgebras D(2,1;α)

{\frak{su}}(1,1)

On the composition factors of Kac modules for the Lie superalgebras sl(m/n)

and the Clebsch--Gordan decomposition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the quantized universal enveloping algebra for

A Determinantal Formula for Supersymmetric Schur Polynomials

{\frak{su}}(1,1)

Coupling coefficients for Lie algebra representations and addition formulas for special functions

, convolution identities for the Al-Salam and Chihara polynomials and the Askey--Wilson polynomials are derived by using the Clebsch--Gordan and Racah coefficients. For the quantized universal enveloping algebra for

Finite oscillator models: the Hahn oscillator

{\frak{su}}(2)

Finite‐ and infinite‐dimensional representations of the orthosymplectic superalgebra OSP(3,2)

, q-Racah polynomials are interpreted as Clebsch--Gordan coefficients, and the linearization coefficients for a two-parameter family of Askey--Wilson polynomials are derived.

Invariance groups of transformations of basic hypergeometric series

The shift operator technique is used to give a complete analysis of all finite‐ and infinite‐dimensional irreducible representations of the exceptional Lie superalgebrasD(2,1;α). For all cases, the star or grade star conditions for the algebra are investigated. Among the finite‐dimensional representations there are no star and only a few grade star representations, but an infinite class of infinite‐dimensional star representations is found. Explicit expressions are given for the ‘‘doublet’’ representation of D(2,1;α). The one missing label problem D(2,1;α)→su(2)+su(2)+su(2) is discussed in detail and solved explicitly.

Quantum communication through a spin chain with interaction determined by a Jacobi matrix

In the classification of finite‐dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so‐called Kac module V(Λ) with highest weight Λ, which was shown to be simple if Λ is a typical highest weight. If Λ is an atypical highest weight, the Kac module is indecomposable and the simple module V(Λ) can be identified with a quotient module of V(Λ). In the present paper the problem of determining the composition factors of the Kac modules for the Lie superalgebra sl(m/n) is considered. An algorithm is given to determine all these composition factors, and conversely, an algorithm is given to determine all the Kac modules containing a given simple module as a composition factor. The two algorithms are presented in the form of conjectures, and illustrated by means of detailed examples. Strong evidence in support of the conjectures is provided. The combinatorial way in which the two algorithms are intertwined is both surprising and interesting, and is a convincing argument in favor of the solution to the composition factor problem presented here.

Realizations of su(1,1) and Uq(su(1,1)) and generating functions for orthogonal polynomials

We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for sλ(x/y). This new expression gives rise to a determinantal formula for sλ(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchys double alternant with Vandermondes determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.