论文标题:完全分次代数和有限群特征标环的一些性质
Some Properties of Fully Graded Algebras and Character Rings of Finite Groups
论文作者
论文导师 樊恽,论文学位 博士,论文专业 基础数学
论文单位 武汉大学,点击次数 121,论文页数 86页File Size2556k
2005-04-01论文网 http://www.lw23.com/lunwen_668629972/
fully graded algebras; primitive idempotents; divisor; prime spectrum; connected components; 7r-regular conjugacy classes; irreducible characters; induction theorems; minimal family of subgroups
本文主要研究完全分次代数的Fong定理,有限群特征标环通过系数扩张之后素谱的连通分支的个数以及系数扩张之后的特征标环的诱导定理。全文共分为四个部分。前言部分介绍了三个主题的研究背景,研究思想,研究现状并概述了研究结果。第一章中,我们研究了剩余域是代数闭域的离散赋值环上的完全分次代数;给出了一个具体的寻找单位元的分解的方法,使得这些分解的长度以某个常数为界;接着我们得到了完全分次代数的Fong定理;在第一章的最后,我们讨论了交换局部环上的1-成分同构于矩阵代数直和的完全分次代数的结构。第二章中,我们研究了有限群的特征标环通过代数数域的某些子环的系数扩张后所得的交换环的素谱的连通分支的个数。最后,在本文的第三章,我们研究了有限群的特征标环通过复数域的某些子环进行系数扩张之后的特征标的诱导定理。 我们始终设G是一个有限群。第一章中,我们首先讨论的是剩余域是代数闭域的离散赋值环O上的完全分次代数,其中我们证明了若这样的完全分次代数A的1-成分A_1是同型的,那么这个完全分次代数实际上是G在A_1上的交叉积。借助这一引理,我们得到了任意完全分次代数的单位元的一个分解方法,使得在任何x∈G处这样的分解的长度在都以某个常数为界;接着我们利用交叉积中已有的结果,并应用一系列交换图,得到了关于有限p-可解群上的完全分次的O-代数的Fong定理,我们所得的结果与交叉积上的结果完全类似:设G是一个有限p-可解群,H是它的一个Hall p′-子群,A是一个完全G-分次的O-代数,那么,A的任何一个本原幂等元,都可以由A的H-部分A_H的某个本原幂等元共轭得到;并且我们还给出了A_H的本原幂等元在A中依然是本原幂等元的充分必要条件。在第一章的最后,设O是一个交换的局部环,我们研究了1-成分B_1同构于O上的矩阵代数的直和的完全G-分次代数B的结构,此时,G在B_1的各个矩阵直和因子的下标集合上有一个自然的作用,我们证明了当这个作用是正则的时候,B同构于O上的一个矩阵代数;而若这个作用是半正则的,那么B同构于O上矩阵代数的直和。 在第二章中,我们设有限群G的指数为e_G,ω为e_G-次本原单位根,Z是
This dissertation focuses on the fully graded algebras over a complete discrete valuation ring with an algebraically closed residue field of characteristic p; the number of the connected components of the prime spectrum of some kinds of extension rings of the character rings of finite groups; and the generalized Brauer induction theorem. It is composed of four parts. In the preface, we first introduce the background and main ideas of the present research, and then we list all of our main results of each chapter. In the first chapter, we discuss the fully graded algebras over discrete valuation ring with an algebraically closed residue field of characteristic p. Firstly, we obtain a method to decompose the identity element such that the length of the decomposition is bounded by a constant; secondly, we get the so-called Fong theorem for the fully graded algebras over finite p-solvable groups; thirdly, we study the construction of some special kinds of fully graded algebras over commutative local rings. We describe the connected components of the prime spectrum of some kinds of extension rings of the character rings of some special kinds of finite group in Chapter 2. The last chapter is concerned with the induction theorem for the extension rings of the character rings of finite groups.Let G be a finite group. In the first chapter, we consider the fully G-graded algebras over a complete valuation ring with an algebraically closed residue field of characteristic p. In the first place, we prove that when the 1-component A_1 of the fully G- graded algebra A is isotypic , A is a crossed product of G over A\. By using this lemma , we reach a way to decompose the identity element of the fully G-graded algebra at any x ∈ G with the length bounded by a constant; for a finite p-solvable group G, we get the Fong theorem for the fully G-graded algebra A over a complete valuation ring with an algebraically closed residue field of characteristic p. The theorem tells us that for any Hall p"-subgroup H of the finite p-solvable group G, any primitive idempotent in A is conjugate to an primitive idempotent in the H-part A_H of A; we also show a necessary and sufficient condition for a primitive idempotentof AH still being primitive in A. In the second place, let O be a commutative local ring. We think about the fully G-graded algebras B with 1-component isomorphic to a direct sum of full matrix algebras over O ; here G has a natural action on the indecomposable direct summand of B\. We prove that when the action is regular, B itself is isomorphic to a full matrix algebras over O; and that when the action is semi-regular, B is isomorphic to a direct sum of some full matrix algebrs over O.Let G be a finite group with exponent ea and w be a primitive e^-th root of unity. Suppose S is a subring of the algebraic number field containing the rational integer number ring Z, and n is a set of prime numbers defined below:7T = {p| p is a prime number such that p~* ￡ S}.Assuming further that G has a normal Hall-7r subgroup, denoting the character ring of G by R(G), in chapter 2 we prove that the number of connected components in Spec(5 z R(G)) equals the number of it -regular conjugacy classes in G.In the third chapter, G denotes a finite group and S expresses a subring of the complex field containing the ration integer number ring as subring. it is a set of prime numbers defined as follows:7T = {p\ p is a prime number such that p~* ￡ S[uj}}.where a; is a primitive e^-th root of unity, ea is the exponent of G and S[uj] is the extended ring of S generated by u. We still denote R{G) the character ring of G. We show that the minimal family of subgroups Ys of G such that the following mapS ?z Ind : ?HeYs Sis a surjection is equal to W(tt); W(tt) is the union of all the elementary p-subgroups, where p runs over all the prime numbers in w (when tt is an empty set, we set W(tx) be the family of all the cyclic subgroups of G).
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