正特征典型幂零轨道闭包正规性及BCD型高阶Schur-Weyl对偶

发布时间：2018-02-07 17:30

  本文关键词： 正特征 辛与正交幂零轨道闭包正规性 W-代数 Vust定理 仿射辫子代数 高阶Schur-Weyl对偶　出处：《华东师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本论文由两章构成.在第一章我们研究正特征代数闭域上正交与辛型幂零轨道闭包的正规性.我们证明不包含d与e型不可约极小退化的幂零轨道闭包是正规的.相反包含e型极小不可约退化的幂零轨道闭包不正规.这里,极小不可约退化是Hesselink在[Hes]中给出的,一共有8种参见表1.1.我们的结果是复数域上的结果[KP2,定理16.2(ii)]在正特征域上较弱一点的版本.我们采用的证明方法是[KP2]中的Kraft-Procesi论证,在正特征情形中需要更具体地实现一些过程.这一章的结果包含在已发表的论文[XS]中.作为幂零轨道闭包正规性的一个有趣应用,在第二章我们给出B,C和D型的Vust定理,然后用Vust定理研究高阶Schur-Weyl对偶.令G为复数域上的线性代数群,g=Lie(G)为其李代数,e ∈ g为一幂零元.若G = GL(V),Ge(?)G为幂零元e的稳定化子,Vust定理是说交换化代数EndGe,(V(?)d)由d次对称群的自然作用的像和所有形如{1(?)(i-1)(?)e(?)1(?)(d-i)|i=1,…,d}的线性变换生成.在本文第二章,我们把这个定理推广到G = O(V)和SP(V),此时我们需要限制条件幂零轨道闭包G·e是正规的.作为Vust定理的应用我们研究B,C和D型的高阶Schur-Weyl对偶,即建立有限W代数和退化仿射辫子代数之间的联系.第二章基于已发表的[LX].
[Abstract]:In chapter 1, we study the normality of nilpotent orbital closures of orthogonal and symplectic type on closed fields of positive characteristic algebras. We prove that nilpotent orbital closures without d and e type irreducible minimal degeneracy are normal. Instead, the closure of nilpotent orbits containing minimal irreducible degeneracy of type e is irregular. Minimal irreducible degeneracy is given by Hesselink in [Hes], and there are eight kinds of results given in [Hes]. Our result is a weaker version of the result [KP2, Theorem 16.2ii] on complex field in [Hes]. Our proof method is the Kraft-Procesi proof in [KP2]. In the case of positive features, some processes need to be realized more concretely. The results of this chapter are contained in the published paper [XS]. As an interesting application of nilpotent orbital closure normality, in chapter 2 we give Vust theorems of BG C and D type. Then the Vust theorem is used to study the duality of higher order Schur-Weyl. Let G be a linear algebraic group G over the complex field. Let G be its lie algebra G 鈭，

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