Representation Theoretical Methods in the Theory of Special Functions

拟议的工作涉及计算机探索和跨越几个数学边界的信息传输，从表示理论到特殊函数理论，然后到组合学。表象理论和特殊函数理论之间的联系可以追溯到弗罗贝尼乌斯的过程。组合数学的作用在于，某些重要的整数，如维度和多重性，是通过计算表格、路径、树等来获得的。这些联系提供了一个无价的发现工具，因为在这些领域中可能显而易见的结果和机制往往在其他领域中转化为非常不平凡和意想不到的事实。组合观点的力量应该很容易从“一幅画胜过千言万语”这句老生常谈中理解。组合解释将数学信息(无论是代数信息、分析信息、逻辑信息或其他信息)转换为视觉信息。所提出的活动包括Kronecker系数的计算，Hilbert级数的确定，不变量环的显式基本集的构造，某些代数作为不变量环上的自由模的因式分解，分次表示到它们的不可约分量的分解。这些都是与数学和物理的几个分支有关的活动。所有提出的研究都集中在特别适合计算机实验的领域。经验表明，在这种背景下，即使是背景有限的学生，也能体验到发现非同小可的喜悦。Kronecker系数是产生表示的张量积中不可约的重数的整数，很难直接从它们的原始定义计算出来。提出者与A.Goupil和M.Zabrocki合作开发的新方法获得了这些系数的多项式母函数，从中可以很容易地提取系数本身。构造这些Kronecker‘’多项式‘’并寻求它们的组合解释将是这笔赠款下要开展的活动之一。分次向量空间的Hilbert级数是这些向量空间的连续齐次分量的维度的生成函数。这项研究将开发计算不变量环的希尔伯特级数的算法。不变量是在某些矩阵组的作用下保持不变的多项式。计算希尔伯特级数主要是为了获得在构造基本不变量集时有用的信息。作者与N.Wallach合作，探索并扩展了一种用常数项算法求Hilbert级数的方法。随后，提出者与G.Xin合作，成功地改进了这些算法，以至于最近在几秒钟内就获得了某些希尔伯特级数，它需要几个小时的计算机时间。根据这项赠款，将开展的主要活动是获取显式公式，以及在没有公式的情况下开发产生数学结构的有效算法。这些努力的重大成功应该有利于应用科学领域的其他研究人员。

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.