Applications of Symmetry and Group Theory for the Investigation of Molecular Vibrations

The application of symmetry and mathematical group theory is a powerful tool for investigating the vibrations of molecules. In this paper, we present an overview of the methods utilized. First we briefly discuss the quantum mechanical nature of vibrations and the experimental methods used. We then present the principal concepts for applying group theory to molecules. The symmetry operations which are used to comprise groups are described and then used to determine the point groups of molecules. The properties of character tables are presented and the method for obtaining a reducible representation for all the motions of a molecule is detailed. This can then be broken down to obtain the irreducible representation which contains the symmetry species of the individual vibrations. The determination of symmetry adapted linear combinations is outlined and the basis for spectroscopic selection rules is presented. The paper concludes by examining how matrix algebra along with symmetry concepts simplifies calculations with molecular force constants.     

特征标表各列零点个数不超过2个的有限群

继续考虑特征标的零点对有限群结构的影响, 并给出了特征标表中 每列至多有两个零点的有限群的分类,从而完成了特征标表中每列至多 $p$ ($p$是群的阶的最小素因子)个零点的有限群的完全分类.     

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

In this paper we characterise finite groups whose character tables fuse from the character table of a cyclic group. We also show a connection with a generalisation of the Camina pair condition introduced by Camina in [Ca].     

NON-ISOMORPHIC GROUPS WITH ISOMORPHICSPECTRAL TABLES AND BURNSIDE MATRICES

ＮＯＮ－ＩＳＯＭＯＲＰＨＩＣＧＲＯＵＰＳＷＩＴＨＩＳＯＭＯＲＰＨＩＣＳＰＥＣＴＲＡＬＴＡＢＬＥＳＡＮＤＢＵＲＮＳＩＤＥＭＡＴＲＩＣＥＳ￥Ｗ．ＫＩＭＭＥＲＬＥ；Ｋ．Ｗ．ＲＯＧＧＥＮＫＡＭＰ（ＭａｔｈｅｍａｔｉｓｃｈｅｓｉｎｓｔｉｔｕｔＢ，Ｕｎｉｖｅｒｓ...     

Reversible hypergroups

Using the properties of the characters of a finite group as a prototype, a new notion of a reversible hypergroupH is introduced which takes account of degrees and multiplicities. Many properties of finite abelian groups have analogues for reversible hypergroups. For example, the hypergroup algebra CH is defined by analogy with a group algebra. Using the representations of this algebra, reversible hypergroups are shown to be in one-to-one correspondence with abstract «character tables» (of which Brauer’s «pseudo groups» are examples). These possess the usual orthogonality relations, whether or not they are character tables of groups. Examples are given of reversible hypergroups which are not isomorphic with the set of characters of any finite group.     

Right division in groups, dedekind-frobenius group matrices, and ward quasigroups

The variety of quasigroups satisfying the identity (xy)(zy) = xz mirrors the variety of groups, and offers a new look at groups and their multiplication tables. Such quasigroups are constructed from a group using right division instead of multiplication. Their multiplication tables consist of circulant blocks which have additional symmetries and have a concise presentation. These tables are a reincarnation of the group matrices which Frobenius used to give the first account of group representation theory. Our results imply that every group matrix may be written as a block circulant matrix and that this result leads to partial diagonalization of group matrices, which are present in modern applied mathematics. We also discuss right division in loops with the antiautomorphic inverse property.     

Character tables of p-groups with derived subgroup of prime order I

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.     

Non-rigid parabolic geometries of Monge type

In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by gradings such that their −1 component contains a nonzero co-dimension 1 abelian subspace whose bracket with its complement is non-degenerate. We completely classify the simple Lie algebras with such gradings in terms of elementary properties of the defining set of simple roots. In addition we characterize those parabolic geometries of Monge type which are non-rigid in the sense that they have nonzero harmonic curvatures in positive weights. Standard models of all non-rigid parabolic geometries of Monge type are described by under-determined ODE systems. The full symmetry algebras for these under-determined ODE systems are explicitly calculated; surprisingly, these symmetries are all just prolonged point symmetries.     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

On the minimum number of blocks of a maximal partial spread in STS(v) and SQS(v)

We give some results on the minimum number of blocks which a maximal partial spread can have in an STS(v) or in an SQS(v).This work has been supported by GNSAGA (C.N.R) and MPI,1986.     

自同构群基柱为~3D_4(q)的2-(v,k,1)设计

2-(v,k,1)设计的存在性问题是组合设计理论中重要的问题,当这类设计具有一个有意义自同构群时,讨论其存在性是尤其令人感兴趣的.30年前,一个6人团队基本上完成了旗传递的2-(v,k,1)设计分类.此后,人们开始致力于研究区传递但非旗传递的2-(v,k,1)设计的分类课题.本文证明了自同构群基柱为~3D_4(q)的区传递及点本原非旗传递的2-(v,k,1)设计是不存在的.     

Recognizing the Automorphism Groups of Mathieu Groups Through Their Orders and Large Degrees of Their Irreducible Characters

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

Positively Curved Manifolds with Almost Maximal Symmetry Rank

The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n–1)/2] (n 6, 7).     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

Character tables of commutative hecke algebras associated with exceptional weyl groups

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups when they are commutative. In the case when Weyl groups are of classical type, they are already known in [D.1] and [D.2]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with exceptional Weyl groups.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

Extended centroid and central closure of semiprime normed algebras. a first approach

In this paper, we classify all the multiplicity-free permutation characters of sporadic simple groups and their automorphism groups. This project is an application of the group theory system GAP, its character table library, and its library of tables of marks.