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

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Throughout the two years, we investigated the possibility of providing some combinatorial interpretation to the "generalized Robinson-Schensted correspondence for real Lie groups" in the case where the group is SU^*(2n), explicitly described by P. Trapa. This is a bijection between the so-called Brauer diagrams on 2n points (which can also be interpreted as the fixed-point-free involutions) and the standard Young tableaux with 2n boxes whose column lengths are all even. One possible approach is, utilizing the fact that the classical Robinson-Schensted correspondence gives a bijection between the same sets, to find a transformation on Brauer diagrams which, when composed with the classical Robinson-Schensted correspondence, gives Trapa's bijection. Another possible approach is to characterize Trapa's bijection using some combinatorial quantities, in the same sense that the classical Robinson-Schensted correspondence can be characterized by certain invariants of posets due to Greene and … More Kleitman. In the first year, we obtained a conjecture in the second direction. We are still in pursuit of this conjecture Through interaction with various researchers, both domestic and overseas, we have received several interesting suggestions. One is to look for further possibility of extending our geometric interpretation to the Knuth version of the updown Robinson-Schensted correspondence devised by Pak and Postnikov. There has been another suggestion, somewhat related, that the irreducible decomposition of the variety of N-stable flags with gaps in dimensions is not yet clear. In view of these suggestions, it looks also necessary to find new approaches to combinatorial interpretations through generalization to the Knuth version, possibly as well as piecewise linear version. On the other hand, we continued to support T. Roby's research on constructing bijections giving the formal character identities corresponding to the decomposition of the tensor powers of the Weil representation of Sp(2n, R). This has come to a conclusion in the case n = 2. Less