关于A-型模李代数与B-型李超代数的表示研究

发布时间：2018-02-22 02:11

本文关键词： KL理论特征标公式 KW猜想投射模 Cartan不变量　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文主要在sl-型模李代数和osp-型李超代数两个方面做了以下几件事情。在第一篇关于sl-型模李代数表示方面,我们在sl3的研究基础之上,给出了sln在p|n的情形下的所有正则幂零单模的特征标公式,维数等信息.因为它无法满足著名的三条假设(H1)-(H3),因此这是一系列(无限个)不满足上述三条假设的非平凡的模李代数表示的内容。同时也给出了其投射覆盖的维数,以及Cartan不变量。在Kac-Weisfeiler-Premet定理无法覆盖的情形首次给出了正则幂零表示的精确信息。另外作者从上述表示的Cartan不变量角度出发得到了一个复杂的组合恒等式。在第二篇关于osp-型李超代数表示方面,首先作者对于osp-型李超代数的KL理论之bar involution证明了它无法与其对应的Hecke代数在Fock space上的模结构兼容。进而得到无法将osp(2m+1|2n)的单模特征标公式的求解问题转化为一个组合问题。而对于gl(m|n)情形,正是因为Brundan将它的单模特征标的求解问题转化为组合问题,才有了后来的单模特征标的精确公式。其次是计算了osp(5|4)在复数域上的所有有限维单模的特征标公式以及相应的传递函子在有限维单模上的作用.同时也得到了非multiplicity free的现象,而这是不同于A-型李超代数表示的。众所周知,osp(5|4)的最高atypical度为2。而在此前公开的文献中,除了A-型李超代数,我们知道的单模特征标公式只有atypical度为1的,所以这个是第一个atypical度为2的完整的信息。
[Abstract]:In this paper, we mainly do the following things in two aspects: sl-type modular lie algebra and osp-type Li Chao algebra. In the first part, we study the representation of sl-type modular lie algebra on the basis of sl3. In this paper, the characteristic formulas of all regular nilpotent simple modules of sln in the case of p n are given. Because it can not satisfy the famous three hypotheses, this is a series of (infinite) representations of nontrivial modular lie algebras which do not satisfy the above three hypotheses. The dimensions of its projective coverage are also given. And Cartan invariants. The exact information of regular nilpotent representation is given for the first time when the Kac-Weisfeiler-Premet theorem cannot be covered. In addition, the author obtains a complex combinatorial identity from the Cartan invariant of the above representation. On the representation of Li Chao Algebra of Type osp-, First, the author proves that the bar involution of the KL theory of Li Chao algebra of type osp- type can not be compatible with the modular structure of the corresponding Hecke algebra on Fock space, and then obtains the solution of the single mode eigenvalue formula which cannot transform osp(2m 12 n) into a problem. And in the case of gl(m n, It is precisely because Brundan transforms its single-mode eigenvalue into a combinatorial problem. Then the exact formula of the single mode characteristic object is obtained. Secondly, the characteristic formula of all finite dimensional single modes on the complex field osp(5 4 and the action of the corresponding transitive functor on the finite dimensional single mode are calculated. At the same time, the phenomenon of non-#en1# free is also obtained. And this is different from the A- type Li Chao algebra. It is known that the highest atypical degree is 2. In the previous published literature, except for the A- type Li Chao algebra, we know that only the atypical degree is 1. So this is the first complete message of atypical degree 2.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O152.5

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