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

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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.