对称群S5的表示环

本文利用有限群特征标理论计算了对称群S₅的所有不可约复表示的幂公式.根据求解幂公式过程中得到的S₅任意两个不可约表示张量积的分解情况,作者刻画了S₅上表示环r(S₅)及其若干结构性质,如极小生成元关系式表达、单位群、本原幂等元、行列式与Casimir数.     

三次对称群的一个特征性质

本给出三次对称群S1的一个特征性质：有限群G恰有两个非线性共轭类当且仅当G≈S2。     

对称群在面饰分类中的应用

近年来,中学课本和研究型学习的课程中,都涉及到一些平面图形的对称性问题.这一问题可划分为两大类,第一类:图形的对称变换有不动点,比如正方形的中心,等腰(非等边)三角形底边上的高等等.第二类:图形的对称变换没有不动点,在这种情况下,平移一定包含其中,而图形一定是无限的.这一类型最简单的情况是,平移仅沿某一固定直线进行,称为带饰;一般的情况是,平移可同时沿某两条相交直线进行,称为面饰.面饰的十七种图形在古埃及的装饰绘画中就已经出现,近三百年来,随着群论的逐步建立和完善,人们对这一问题进行了严格的理论证明.这篇文章是北京师范大学数学科学学院的本科毕业论文,郭佳意和董正林同学利用对称群的知识介绍了面饰的分类,给出了全部十七种面饰的生成元和定义关系,希望能够对中学老师和同学们有所启迪.     

正多边形对称群的子群

利用Lagrange定理和正多边形对称群的性质,首先对正多边形对称群的子群的性质进行了研究,其次讨论了正多边形对称群的子群的结构,由此完全确定了正多边形对称群的子群,最后应用所得结论求出了正六边形对称群的所有子群.     

对称群的一个特征性质

设G为有限群,∑_n为n次对称群,本文证明了:G≌∑_n当且仅当|G|=|∑_n|且Π_e(G)=Π_e(∑_n),此处Π_e(G)为G中元的阶的集合。     

无限对称群的正规子群

有限对称群Ｓｎ（ｎ≠４）非平凡的正规子群仅有一个交代群Ａｎ。无限集合Ｍ上的对称群ＳＭ则不是这样。本文的主要结果是：（１）确定了ＳＭ的全部正规子群，它们形成一个整序集；（２）ＳＭ正规子群的正规子群仍是ＳＭ的正规子群；（３）ＳＭ的正规子群是特征子群；（４）ＳＭ的正规子群（≠１）的自同构群与ＳＭ同构，ＳＭ是完全群。     

对称群在面饰分类中的应用(续)

（1）对称群的生成元只含有两个平移．此时生成关系唯一确定：tαtb=tbtα。 图示如下：     

D6等变非线性分歧问题的计算

本文研究D6对称群的分歧子群的分类,由此得到相应对称破缺分歧的计算方法,并应用到D6等变的Brusselator反应问题。     

群S4对称自治系统的Hopf分支

本文利用Ｌｙａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ方法对一类群Ｓ４对称的自治系统进行讨论,得到了Ｈｏｐｆ分支解的存在条件,研究了分支解的结构。     

Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape (, ), conjugate content and type . We also give lower and upper bounds for these numbers.     

Finite Group Schemes over Bases with Low Ramification

Let A be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category FF _A' of finite flat commutative group schemes over A withp-power order. When e= 1, Fontaine formulated the purely linear algebra notion of a finite Honda system over A and constructed an anti-equivalence of categories betweenineFF _A'> and the category of finite Honda systems over A when p> 2. We generalize this theory to the case e – 1.     

Applications of the Frobenius Formulas for the Characters of the Symmetric Group and the Hecke Algebras of Type A

We give a simple combinatorial proof of Ram's rule for computing the characters of the Hecke Algebra. We also establish a relationship between the characters of the Hecke algebra and the Kronecker product of two irreducible representations of the Symmetric Group which allows us to give new combinatorial interpretations to the Kronecker product of two Schur functions evaluated at a Schur function of hook shape or a two row shape. We also give a formula for the regular representation of the Hecke algebra.     

Purely Inseparable Extensions and Ramification Filtrations

In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.     

Symmetric complete intersections

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals and describe formulas for the graded characters of the corresponding quotient rings.     

Symmetric Group Decomposition Numbers for Some Three-Part Partitions

The decomposition numbers d _λμ for Specht modules S ^λ of partitions λ with three parts and whose third part is at most p ? 1 are obtained by induction and by using “node removal rules” developed in James and Williams (2000 James , G. D. , Williams , A. L. (2000). Decomposition numbers of symmetric groups by induction. J. Algebra 228:119–142. [CSA] [CROSSREF]  [Google Scholar]).     

Properties of Some Character Tables Related to the Symmetric Groups

We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S_n and their double covers. In particular, we give a simple computation, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table _RC of S_n with respect to an integer r 2. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke algebra. When r is prime we determine the Smith normal form of _RC. Taking r large yields the Smith normal form of the full character table of S_n. Analogous results are then given for spin characters.Partially supported by The Danish National Research Council.Partially supported by NSF grant #DMS-9988459.