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.
In learning of Hilbert's (1893) solution of the fundamental problem of Invariant Theory, Paul Gordan is purported to have said This is Theology not Mathematics ". Gordan so expressed the change in the way of doing Mathematics heralded by Hilbert's results. Namely the abandonment of explicit constructions in favor of existence proofs. Ultimately this resulted into more than half a century of Theological Mathematics, rich in results but pauper in algorithms and numerical data. The advent of fast computers and powerful symbolic manipulation software has prompted a renewal of the constructive and algorithmic approach. The research carried out under this grant is quite suitable for this approach. It involves transfer of information from pure algebraic constructs, to explicit symmetric functions and ultimately to visual combinatorial objects such as tableaux, paths and trees. This results in enrichment of each of these areas 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. In particular the problems proposed can only be solved by advances in the theory of symmetric functions. Although this is not generally understood, symmetric functions are a computational device of wide applicability. Thus progress in the proposed research should widen the variety of tools available to physicists and engineers to obtain the hard data needed in their pursuit of applications of science. All the proposed research lies in areas of Mathematics that computers have transformed into experimental sciences. In fact it is not uncommon to witness theorems and conjectures literally jump out of the computer screen. In this setting even beginning students can experience the joy of non trivial discovery. This makes this type of research an ideal setting to convey to our new generations of researchers a deeper understanding of the wide range of possibilities offered by computer guided research.
