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Eigenvalues and eigenfunctions of the volume operator, associated with thesymmetric coupling of three SU(2) angular momentum operators, can be analyzedon the basis of a discrete Schroedinger-like equation which provides asemiclassical Hamiltonian picture of the evolution of a `quantum of space', asshown by the authors in a recent paper. Emphasis is given here to theformalization in terms of a quadratic symmetry algebra and its automorphismgroup. This view is related to the Askey scheme, the hierarchical structurewhich includes all hypergeometric polynomials of one (discrete or continuous)variable. Key tool for this comparative analysis is the duality operationdefined on the generators of the quadratic algebra and suitably extended to thevarious families of overlap functions (generalized recoupling coefficients).These families, recognized as lying at the top level of the Askey scheme, areclassified and a few limiting cases are addressed.

Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials