Annals of Combinatorics 4 (2000) 413-432
Unitary Group Theory and the Discovery of the Factorial Schur Functions
James D. Louck
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Received June 4, 1998
AMS Subject Classification: 20C35, 20G05, 20H20, 0SA10, 47N50
Abstract. The unitary irreducible representations of the unitary group U(n) are obtained from a more general set of polynomials that satisfy the multiplication rule for representations for arbitrary indeterminates. These polynomials are the familiar boson polynomials that appeared in earlier work, but are now presented from a new viewpoint. The main results are (1) the proof that these same polynomials provide a basis for all U(n) irreducible tensor operators when the commuting indeterminates are replaced by non-commuting fundamental unit tensor operators, and (2) the construction of all sets of unit tensor operators whose matrix elements give U(n) Clebsch-Gordan coefficients that possess the null space required by the Littlewood-Richardson numbers. Recent advances in the combinatorial interpretations of some of these results are pointed out. An outline is given of how the factorial Schur functions were discovered during the investigation of the properties of certain polynomials that characterize the null space of U(3) tensor operators.
Keywords: group representations, polynomial bases, tensor operators, factorial Schur functions, multivariable hypergeometric functions
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