Representation theory and combinatorics of classical groups, quantum groups and Hecke algebras

在以前的一个项目中，我们使用交替的双线性形式、旗帜和幂零矩阵解释了由Stanley首先发现的Updown表和Brauer图之间的Robinson-Schensted类型对应。在本课题中，我们得到了一个更深入的结果，即我们还证明了由所有三元组组成的簇的不可约性：一个幂零矩阵，一个标志和一个非退化的交替双线性形式，使得后两者由第一个无穷小固定。这使得我们的结果与斯坦伯格对原始Ronbinson-Schensted对应的解释更接近。对于其他Robinson-Schensted型对应的类似几何解释尚待研究.在构造组合描述SP(2n，C)的Weil表示的张量幂分解的一组表和Robinson-Schensted型对应的过程中，我们通过将专门化同态的使用推广到某些普适特征标的无穷和，给出了组合处理的基础.我们帮助T.Roby得到了张量幂相对于秩大的稳定区域的结果，以及他几乎完成的任何秩次为2的幂的情况。进一步得到的结果有：K.Koike关于经典群的特征的结果；S.Okada关于次求和公式和菱形平铺的结果；T.Kobayashi关于约化群的某些无穷维表示的无重性和离散可分解性的结果。

In a previous project, we gave an interpretation of a Robinson-Schensted-type correspondence between updown tableaux and Brauer diagrams, first discovered by Stanley, using alternating bilinear forms, flags and nilpotent matrices. In the current project, we obtained a more thorough result, namely we also showed the irreducibility of the variety formed by all triples : a nilpotent matrix, a flag and a nondegenerate alternating bilinear form such that the latter two are infinitesimally fixed by the first. This gives a closer parallelism of our result with Steinberg's interpretation of the original Ronbinson-Schensted correspondence. Similar geometric interpretations of other Robinson-Schensted-type correspondences are yet to be intestigated.In persuing the construction of a set of tableaux and a Robinson-Schensted-type correspondence which would combinatorially describe the decomposition of the tensor powers of the Weil representation of sp (2n, C), we gave a basis of combinatorial treatment by extending the use of specialization homomorphisms to certain infinite sums of universal characters. We assisted T.Roby with obtaining results for the stable region where the tensor power is large relative to the rank, and also for the case of any power with rank 2, which he nearly completed. The method is yet to be generalized to the case of any power with any rank.Further obtained were : results on characters of the classical groups by K.Koike ; results concerning the minor summation formulas and rhombus tilings by S.Okada ; results on multiplicity-free and discrete decomposabilities of certain infinite-dimensional representations of reductive groups by T.Kobayashi.