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

在这两年中，我们研究了在群为SU^*(2n)的情况下，对P.TRapa所明确描述的“实Lie群的广义Robinson-Schensted对应”提供某种组合解释的可能性。这是2n个点上的所谓的Brauer图(也可以解释为无定点对合)和列长都是偶数的2n个盒子的标准Young图面之间的双射。一种可能的方法是，利用经典的Robinson-Schensted对应给出相同集合之间的双射的事实，在Brauer图上找到一个变换，当与经典的Robinson-Schensted对应合成时，给出TRapa的双射。另一种可能的方法是使用一些组合量来刻画TRapa的双射，在相同的意义上，经典的Robinson-Schensted对应可以由偏序集的某些不变量来刻画，这是由于Greene和…更多的克莱特曼。在第一年，我们得到了第二个方向的一个猜想。通过与国内外不同研究人员的互动，我们仍在追寻这个猜想，我们收到了几个有趣的建议。一个是寻找进一步的可能性，将我们的几何解释扩展到由Pak和Postnikov设计的Updown Robinson-Schensted对应的Knuth版本。还有另一种建议，在某种程度上与之相关，即具有维度差距的各种N-稳定旗帜的不可约分解尚不清楚。鉴于这些建议，似乎也有必要通过推广到Knuth版本以及分段线性版本来找到组合解释的新方法。另一方面，我们继续支持T.Roby关于构造双射的研究，给出了与Sp(2n，R)的Weil表示的张量幂分解相对应的形式特征恒等式。这在n=2的情况下得出了一个结论。

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