Representation Theoretical Methods in the Theory of Special Functions

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The proposed work involves computer explorations and transfer of information across several mathematical boundaries, from Representation Theory to the Theory Special Functions and then to Combinatorics. The connection between Representation Theory and the Theory of Special functions is provided by a process that goes back to Frobenius. Combinatorics plays a role in that certain important integers such as dimensions and multiplicities are obtained by counting tableaux, paths, trees, etc. These connections provide an invaluable vehicle of discovery, since results and mechanisms which may be obvious in one of these areas often translate into highly non trivial and unexpected facts in one of the other areas. The power of the combinatorial viewpoint should be easily understood from the truism `` A picture is worth a thousand words''. Combinatorial interpretations translate mathematical information be it algebraic, analytical, logical or otherwise into visual information. The proposed activities involve computation of Kronecker coefficients, determination of Hilbert series, constructions of explicit basic sets of rings of invariants, factorization of certain algebras as free modules over rings of invariants, decompositions of graded representations into their irreducible constituents. These are all activities that have a bearing in several branches of Mathematics and Physics. All the proposed research lies in areas which are particularly suitable to computer experimentation. Experience shows that, in this setting, even students with limited background can experience the joy of non trivial discovery. Kronecker coefficients, which are integers yielding the multiplicities of irreducibles in tensor products of representations, are difficult to compute directly from their original definition. New methodology that is being developped by the proposer in collaboration with A. Goupil and M. Zabrocki obtains polynomial generating functions of these coefficients, from which the coefficients themselves can be easily extracted. Constructing these Kronecker ``polynomials`` and seeking for their combinatorial interpretation will be one of the activities to be carried out under this grant. Hilbert series of graded vector spaces are generating functions of dimensions of the successive homogeneous components of these vector spaces. The proposed reseach will develop algorithms for the calculation of Hilbert series of rings of invariants. Invariants are polynomials which remain unchanged under the action of certain groups of matrices. Hilbert series are computed primarily to obtain information useful in the construction of basic sets of invariants. The proposer in collaboration with N. Wallach explored and expanded a way to obtain Hilbert series by constant term algorithms. Subsequent collaboration by the proposer with G. Xin succeeded in refining these algorithms to the extent that certain Hilbert series that hiterto required hours of computer time were recently obtained in a few seconds. The acquisition of explicit formulas and when not available the development of efficient algorithms yielding mathematical constructs are the primary activities that will be carried out under this grant. Significant success in these endeavours should be beneficial to other researchers in the applied sciences.