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

经典群、量子群和赫克代数的表示论和组合学

基本信息

批准号：
12640011
负责人：
TERADA Itaru
金额：
$ 2.24万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640011/
关键词：
combinatorics representation theory Young diagrams the Robinson-Schensted correspondence the Springer variety partially ordered sets involutions Brauer diagrams Springer varieyy Brauer diagram ロビンソン・シェンステッド対応 tableau Steinberg vari

项目摘要

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

在这两年中，我们研究了在群为SU^*（2n）的情况下，为“真实的李群的广义Robinson-Schensted对应”提供某种组合解释的可能性，这是P. Trapa明确描述的。这是在2n个点上的所谓布劳尔图（也可以被解释为无不动点对合）和具有2n个列长都是偶数的盒子的标准杨表格之间的双射。一种可能的方法是，利用经典的罗宾逊-申斯特对应给出相同集合之间的双射的事实，在布劳尔图上找到一个变换，当与经典的罗宾逊-申斯特对应组合时，给出特拉帕的双射。另一种可能的方法是使用一些组合量来表征Trapa的双射，在相同的意义上，经典的Robinson-Schensted对应可以由由于格林和 ...更多信息克莱特曼在第一年，我们得到了第二个方向的猜想。我们仍然在追求这个猜想通过与国内外各种研究人员的互动，我们收到了一些有趣的建议。一个是寻找进一步的可能性，将我们的几何解释扩展到克努特版本的上下罗宾逊-申斯特德对应设计的帕克和Postnikov。还有另一个建议，有点相关，即在维度上有间隙的各种N-稳定标志的不可约分解尚不清楚。鉴于这些建议，它看起来也有必要找到新的方法，通过推广到克努特版本的组合解释，可能以及分段线性版本。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表示的张量积幂的分解相对应的形式特征标恒等式。这在n = 2的情况下得到结论。少