Showing papers on "Symmetric group published in 1979"

Closures of conjugacy classes of matrices are normal

Abstract: Hanspeter Kraft 1 and Claudio Procesi 2 Sonderforschungsbereich Theoretische Mathematik, Universit~it Bonn D-5300 Bonn, Federal Republic of Germany 2 Istituto di Matematica, Universitfi di Roma, I-Rome, Italy O. Introduction 0.1. The purpose of this paper is to prove the following theorem: Let A be an n “ n matrix over an algebraically closed field K of characteristic zero, C A the conjugacy class of A and C A its (Zariski-) closure. Theorem. C a is normal, Cohen-Macaulay with rational singularities. tf a variety X with a G-action (G reductive) is the closure of an orbit (9 and dim(X--. (9)< dim X-2, it is a crucial question for the geometry of X to decide whether the singularity (in X \ (9) is normal. In fact the normality of X allows to identify the ring K[X] of regular functions on X with the functions on the orbit (9 and so, by Frobenius reciprocity, to analyse K [X] as a representation of G (cf. [11], [1]). In our case this is closely related to the "multiplicity conjecture" of Dixmier; we refer the reader to the paper [1] for a detailed description of this connection and some applications. A different proof of this theorem will appear in [23]. 0.2. The theorem has also another interesting application, shown to us by Th. Vust, in the spirit of the classical theory of Schur. If U is a finite dimensional vector space one has the classical relation between the action of GL(U) and of the symmetric group ~,, on the tensor space U | If we restrict to the subgroup

Homology stability for the general linear group

Abstract: This thesis studies the homology stability problem for general linear groups over Euclidean rings and over subrings of the field of rational numbers Affine linear groups, acting on affine space rather than linear space, are also considered In order to get stability results one establishes that certain posets of ordered unimodular sequences of vectors are Cohen-Macaulay The ideas are first illustrated by reproving the Nakaoka stability theorem on homology stability for symmetric groups

The center of the ring of 3×3 generic matrices

Abstract: Following Procesi, the center of the division ring of generic matrices over a field F is described as the fixed field of the symmetric group acting on a purely transcendental extension of F. For 3×3 matrices, the center is shown to be purely transcendental over F. In characteristic zero this is equivalent to saying that the field of simultaneous rational invariants of 3×3 matrices over F is a purely transcendental extension field of F.

The Galois unsolvability of the sextic equation of anisotropic elasticity

Abstract: By an approximate numerical application of Galois theory it is proved that the sextic equation of anisotropic elasticity for cubic symmetry is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions and that its group is the full symmetric group. This implies the same unsolvability for tetragonal, orthorhombic, monoclinic, and triclinic symmetry. A separate investigation proves the same unsolvability for trigonal symmetry. Special cases of cubic symmetry which might have solvable equations are examined. Directions restricted to {111} or {112} planes give unsolvable equations, in contrast to {100} and {110} planes. Three additional classes of elastic constants which give solvable equations are found but only two limiting cases are physically possible. An extensive survey suggests that any further special elastic constants are rather unlikely.

The Permutation group in physics and chemistry

Abstract: 1. Counting Isomers and Such.- 2. Application of the Permutction Group to the Stereoisomer Generation for Computer Assisted Structure Elucidation.- 3. Applications of the Permutation Group in Dynamic Stereochemistry.- 4. The Spin Double Groups of Molecular Symmetry Groups.- 5. Relationship Between the Feasable Group and the Point Group of a Rigid Molecule.- 6. Some Suggestions Concerning a Geometric Definition of the Symmetry Group of Non-Rigid Molecules.- 7. Symmetry and Thermodynamics from Structured Molecules to Liquid Drops.- 8. Representations of the Symmetric Group as Special Cases of the Boson Polynomials in U(n).- 9. The Permutation Group and the Coupling of n Spin- 1/2 Angular Momenta.- 10. The Permutation Group in Atomic Structure.- 11. Double Cosets and the Evaluation of Matrix Elements.- 12. Properties of Double Cosets with Applications to Theoretical Chemistry.- 13. The Chirality Algebra some comments concerning mathematical aspects of the Ruch/Schoenhofer chirality theory.

Special symmetries of jm factors and j symbols

Abstract: It has been suggested that the existence of one-dimensional irreps of a group leads to symmetries in the Racah algebra of the group. The familiar 2jm symbol arises as a special case of these symmetries where the one-dimensional irrep is the identity irrep. The authors derive the general result and give examples for the symmetric groups and for the point groups. These examples show that these new symmetries are more complicated than the previous suggestions imply.

Automorphic Group Representations: A New Proof of Blattner's Theorem

Abstract: To each of these ways of obtaining 51 correspond automorphisms of 5(. The construction that will interest us is the fourth one. The automorphisms in question arise naturally in the following way. The vectors in E generate 51 as a von Neumann algebra; the full orthogonal group O(E) of E acts on E, hence on 5t as automorphisms. The normal subgroup G2 of O(E) which acts as inner automorphisms of 51 is of particular interest. The group G2 was determined by Blattner [2]. The normal subgroup Gi of O(E) which acts as inner automorphisms of the CAR algebra is also of interest, and was determined by Shale and Stinespring [21]. The purpose of this article is to give new, simpler proofs of these two results. We quote a result of the first-named author according to which each element of a proper normal subgroup of 0(E) is a compact perturbation either of the identity operator / or of —/. We prove directly that Gx and G2 are proper normal subgroups of O(E) and then exploit the spectral theory of compact operators. Although our proofs are new, they make liberal use of original ideas of Blattner, Shale and Stinespring. It is illuminating to prove the two basic results simultaneously. Our proofs are naive and constructive: corresponding to the spectral decomposition of an orthogonal operator in G2, the implementing unitary operator in 51 is explicitly constructed as an infinite product. Let ^ denote the CAR algebra. Since the centres of 51 and # comprise scalar multiples of the identity, each implementing unitary operator is determined up to such a scalar. Consequently we have projective unitary representations of Gx and G2 as follows: G2

Isomorphic factorisations VI: Automorphisms

Abstract: An isomorphic factorisation of a graph G is a partition of its line set E(G) into isomorphic subgraphs called factor graphs. The question investigated here is how those automorphisms of a complete graph K p which preserve an isomorphic factorisation can act in permuting the factor graphs. The group of these permutations is called the symmetry group. It is clear that the symmetric group S2 is the only such factor group of degree 2. However, it is shown that all four permutation groups of degree 3 arise from isomorphic factorisations of complete graphs. For E1×S2 the smallest example requires 10 points. A natural conjecture is that every permutation group of degree d>2 is the symmetry group of an isomorphic factorisation of some complete graph. However no such representation is known for S6, and it is shown that the representations of S5 present certain irregularities.

Simple groups of finite order in the nineteenth century

Abstract: The history of simple groups starts with the work of Evariste Galois (1811-1832). Throughout the eighteenth century and on into the nineteenth century the all-consuming passion among algebraists was the determination of which polynomial equations could be solved by radicals. A polynomial equation of degree n, xn + alx"~1 + ... +an_lx + an = 0, where the coefficients af belong to a field F, is said to be solvable by radicals (or algebraically solvable) when it is possible to express the roots of the equation in terms of the coefficients using a finite number of algebraic operations addition, subtraction, multiplication, division, raising to powers and extraction of roots. Galois approached the problem of characterizing such equations by considering, as Lagrange had done before him, the notion of permutations of the roots of an equation. This in turn led to the concept of a group. Prior to Galois, Lagrange had worked with what is in effect the symmetric group in his studies of functions unchanged under all permutations of their variables, and GAUSS used essentially the cyclic group in the numbertheoretic setting of congruences. Galois, however, dealt not with special cases but recognized, without giving an explicit definition, a permutation group as a set of permutations having the closure property. Certain points should be clarified here. Galois was the first to use the term group in a technical sense, but he also used the word in its non-mathematical sense to refer to an arbitrary collection of objects. It is sometimes difficult to distinguish the mathematical from the colloquial meaning. Also meriting discussion is Galois' use of the terms "permutation" and "substitution". Whereas the contemporary definition of permutation is that of a one-to-one mapping of a set of objects onto itself, the word is also used more informally to denote the actual arrangement of the objects. Galois used the

Elementary abelian p -groups as automorphism groups of infinite groups. I

Abstract: If G is a group we will write Aut G for the group of all automorphisms of G and Inn G for the normal subgroup of all inner automorphisms of G. Many authors have studied the relationship between the structure of G and that of Aut G, in particular when the latter is finite. This paper is a further contribution to this study. The first results on groups whose automorphism groups are finite were published by Baer in a paper [2] in which he proved that a torsion group has finite automorphism group only if it is finite. Baer also proved that a group with only a finite number of endomorphisms is finite. In 1962 Alperin [1] characterized finitely generated groups with finitely many automorphisms as finite central extensions of cyclic groups. Nagrebeckil [9] discovered in 1972 the important result that in any group with finitely many automorphisms the elements of finite order form a finite subgroup. This of course generalizes Baer's original result. Robinson El0] has given another proof of Nagrebeckfi's Theorem as well as obtaining information on the primes dividing the order of the maximal torsion subgroup. He also characterized the center of a group whose automorphism group is finite and gave a general method for constructing examples. On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be several authors that torsion-free abelian groups with only one non-trivial automorphism the involution x F--~x~ are relatively common (de Groot [5], Fuchs [4], Corner [3]). However, Hallett and Hirsch have adopted a different approach, asking which finite groups can occur as the automorphism groups of torsion-free abelian groups. They have established the following definitive result [-7, 8]:

The monodromic Galois groups of the sextic equation of anisotropic elasticity

Abstract: It has previously been shown that the conventional algebraic Galois group of the sextic equation of anisotropic elasticity for cubic crystals is the symmetric group and the equation is therefore algebraically unsolvable in radicals. As an equation with four parameters it has also 15 monodromic Galois groups corresponding to different, relaxed, meanings of solvability in radicals. Three of these are appropriate, to solve explicitly for the functional dependence of the roots on the two directional parameters or on the two elastic parameters or on all four parameters. From the definition of a monodromic group as the root permutations induced by all complex circuits of the relevant parameters, it is shown numerically that these three monodromic groups must be either the alternating or the symmetric. The equation is therefore also unsolvable for these weaker and more appropriate meanings of solvability.

Hypergraphs, characteristic polynomials and permutation characters

Abstract: An (n m) hypergraph is a coupleH=(N E), where the vertex set N is {1,…n} and the edge set E is an m-element multiset of nonempty subsets of N. In this paper, we count nonisomorphic hypergraphs where isomorphism of hypergraphs is the natural extension of that of graphs. A main result is an explicit formula for the cycle index of the permutation representation of any permutation group P with object set N acting on the k-element subsets of N. By making a simple substitution in these cycle indices for P the symmetric group SN and k=1,…,n, we obtain generating functions which enumerate various types of hypergraphs. Using the technique developed, we extend Snapper's results on characteristic polynomials of permutation representations and group characters from the case where the group has odd order to the general case.