Showing papers on "Symmetric group published in 1980"

Polynomial representations of GLn

Abstract: Polynomial Representations of GLn(K): The Schur algebra.- Weights and Characters.- The modules D?,K.- The Carter-Lusztig modules V?,K.- Representation theory of the symmetric group.

Rings of Fricke characters and automorphism groups of free groups

Abstract: Let F n be the free group on n free generators g~(v= 1, , n) and let ~ , be its group of automorphisms Its quotient group (b* with respect to the inner automorphisms is the group of automorphism classes of F, For n = 2, it is easy to derive a presentation for 45 since ~* is simply the linear group GL(2, 77) of 2 by 2 matrices with integers as entries and determinant __ 1 For n>3 , finding a presentation becomes a difficult problem which was first solved by Nielsen [16] and later by McCool [14] who also proved important theorems about the presentations of subgroups of ~, However, not much is known about the structure of ~, and even less about that of ~b* According to a general theorem of Baumslag [-1], ~b, is residually finite It is already more difficult to prove the same result for ~* if n > 2 ; see Grossman [5] Since there exists a natural mapping of ~, onto GL(n, ~) with a kernel which we shall denote by K,, it seems to be natural to concentrate ones attention on the structure of K, since GL(n, Z) is a well investigated group for all n The quotient group of K, with respect to the inner automorphisms shall be called K* We know [11] that K, (and, therefore, K*) are finitely generated with explicitly known generators We also know that K, is residually torsion free nilpotent This follows immediately from the action of K n on the group ring of F, which is a graded ring [-10] in which the powers of the augmentation ideal provide the grading The action of K, on this ring then provides a faithful representation of K, in terms of upper triangular (infinite) matrices with integers as entries and with terms + 1 in the main diagonal But for K* it is not even known whether it is torsion free or not Nor is it known whether K, (and, therefore, K*) has a finite presentation Also, no finite dimensional matrix representation for ~b,, n > 2 or ~*, n > 3, are known and there is at least some support for the conjecture that none exist (It would be sufficient to show this for ~2; see [13]) However, it has been known for a long time ~4] that ~* acts as a group of automorphisms of a quotient ring Rn of a finitely generated (commutative) ring and it has been shown later [7] that this

Finite digraphs with given regular automorphism groups

Abstract: With five exceptions, every finite regular permutation group occurs as the automorphism group of a digraph.

On homomorphisms between Weyl modules and Specht modules

Abstract: The aim of this paper is to generalize a result of Carter and Lusztig(1) proving the existence of certain non-zero homomorphisms between Weyl modules for the special linear group SLn(K) over an infinite field of characteristic p, and between Specht modules for the symmetric group Sr over K.

Group Theory and the Interaction of Composite Nucleon Systems

Abstract: In the theory of nuclear structure and reactions, one often splits the full system into composite systems and studies the dynamics of these composite systems. In this report, I shall describe some methods of group theory which we have developed for dealing with these systems. We shall describe three types of groups and their application to composite particle theory. The symmetric group will be associated with exchange, orbital partitions and the supermultiplet scheme. The general linear group and its representations will be applied to exchange decompositions. The inhomogeneous symplectic transformations of classical phase space and their representations will be used to describe the kinematics and dynamics of composite particles.

The Galois group of a polynomial with two indeterminate coefficients

Abstract: Φ 0) is a polynomial in which two of the coefficients are indeterminates t, u and the remainder belong to a field F. We find the galois group of / over F(t, u). In particular, it is the full symmetric group Sn provided that (as is obviously necessary) /(X) Φ fχ(Xr) for any r > 1. The results are always valid if F has characteristic zero and hold under mild conditions involving the characteristic of F otherwise. Work of Uchida [10] and Smith [9] is extended even in the case of trinomials Xn + tXa + u on which they concentrated. 1* Introduction* Let F be any field and suppose that it has characteristic p, where p — 0 or is a prime. In [9], J. H. Smith, extending work of K. Uchida [10], proved that, if n and a are coprime positive integers with n > α, then the trinomial Xn + tXa + u, where t and u are independent indeterminates, has galois group Sn over F(t, u), a proviso being that, if p > 0, then p \ na(n — α). (Note, however, that this conveys no information whenever p — 2, for example.) Smith also conjectured that, subject to appropriate restriction involving the characteristic, the following holds. Let I be a subset (including 0) of the set {0, 1, , n — 1} having cardinality at least 2 and such that the members of / together with n are co-prime. Let T — {ti9 i e 1} be a set of indeterminate s. Then the polynomial Xn + ΣΓ^o1*^* has galois group Sn over F(T). In this paper, we shall confirm this conjecture under mild conditions involving p(>0), thereby extending even the range of validity of the trinomial theorem. In fact, we also relax the other assumptions. Specifically, we allow some of the tt to be fixed nonzero members of F and insist only that two members of T be indeterminates. Indeed, even if the co-prime condition is dispensed with, so that the galois group is definitely not SΛ, we can still describe

The early proofs of Sylow's theorem

Abstract: The early history of Sylow's theorem is surprisingly unfamiliar. A recent bulky history of mathematics, for instance, recounts that "Having arrived at the abstract notion of a group, the mathematicians turned to proving theorems about abstract groups that were suggested by known results for concrete cases. Thus Frobenius proved Sylow's theorem for finite abstract groups."1 Now a moment's thought shows that this in itself is not much of an accomplishment : any finite group can be realized as a group of permutations (Cayley's theorem), and so Sylow's theorem for permutation groups implies the abstract result. The reader then wonders whether Frobenius somehow did not know Cayley's theorem. But a glance at the paper in question2 shows that Frobenius not only mentioned that result but had included it in one of his own earlier publications. Obviously more careful analysis is needed to grasp what FROBENIUS was actually doing. The specific purpose of this paper is precisely to analyze the various different proofs that were given for Sylow's theorem in the first fifteen years after its discovery. This is of interest in itself, since the theorem is still basic, and the major lines of argument now known all turn out to have been discovered in some form by that time. But also, by tracing this single theorem through a crucial period in the development of group theory, we will see more generally how the introduction of new ideas throws new light on the same result. Indeed, we almost have a case study in levels and use of abstraction. Some of the authors still thought of changes of variable in polynomials, others applied group-theoretic reasoning to permutation groups, and finally Frobenius himself made serious use of abstract groups. In addition to the advance, the continuity in this development will also be illuminated, since we will see techniques equivalent to the construction of coset actions and quotient groups used in proofs before these concepts were formulated explicitly.

Configuration interaction matrix elements. III. Spin functions relating the unitary and symmetric group approaches

Abstract: Spin functions that are compatible with orbital ordering and geminal antisymmetry conditions are investigated. It is shown that two widely used classes of spin functions, namely, the spin-bonded functions and Yamanouchi–Kotani (or, equivalently, Gelfand–Tsetlin) functions possess these properties. The relationship of the latter with Young–Yamanouchi spin functions is also outlined using graphical techniques of spin algebras. These techniques are also used to rederive the Hamiltonian matrix elements between spin-bonded functions and to show the relationship among the various schemes used in this case.

The endomorphism near-rings of the symmetric groups of degree at least five

Abstract: The near-ring distributively generated by the semigroup of all endomorphisms of Sn, the symmetric group of degree n, for n ≥ 5, is close to being the near-ring of all mappings from Sn to itself respecting the identity. In this paper, the structure of these near-rings is studied in detail. In particular, addition and multiplication rules for the elements given in canonical form are determined. A complete list of all right ideals, left ideals, right invariant and left invariant subgroups is given.

A first course in group theory

Abstract: 1 First Ideas.- 1.1 Introduction.- 1.2 The Definition of a Group.- 1.3 The General Associative Law.- 1.4 Further Examples of Groups.- 1.5 Aims.- Exercises 1.- 2 Multiplication Table, Generators, Relations, Isomorphism.- 2.1 Multiplication Table.- 2.2 Multiplication Table for the Dihedral Group D3.- 2.3 Order of an Element.- 2.4 The Symmetric Group Sn.- 2.5 Isomorphism n.- 2.6 Generators and Relations.- 2.7 All Possible Groups of Orders 1, 2, 3, 4.- 2.8 Some Results on Orders of Elements.- Exercises 2.- 3 Subgroups, Lagrange's Theorem, Cyclic Groups.- 3.1 Cosets and Lagrange's Theorem.- 3.2 Some Results on Subgroups.- 3.3 Generators.- 3.4 Products of Subsets of Groups.- 3.5 Cyclic Groups.- 3.6 Subgroups of S3.- Exercises 3.- 4 Factor Groups, Permutation Representations, Finite Point Groups.- 4.1 Normal Subgroups.- 4.2 Simplicity.- 4.3 Conjugacy.- 4.4 Conjugacy Classes.- 4.5 Homomorphisms.- 4.6 Permutation Representation of a Group.- 4.7 Subgroups of Factor Groups.- 4.8 Factor Groups of Factor Groups.- 4.9 Groups of Order p2, p prime.- 4.10 Symmetry and the Orthogonal Group.- 4.11 Classification of the Finite Rotation Groups.- 4.12 Examples of Finite Rotation Groups.- 4.13 Classification of Finite Point Groups of the Second Kind.- 4.14 Examples of Some of the Finite Point Groups of the Second Kind.- Exercises 4.- 5 Finitely Generated Abelian Groups.- 5.1 Introduction.- 5.2 Direct Sum.- 5.3 Free Abelian Groups.- 5.4 Structure Theorems for Finitely Generated Abelian Groups.- 5.5 Uniqueness.- 5.6 Possible Groups of Order p2.- Exercises 5.- 6 The Sylow Theorems.- 6.1 Introduction.- 6,2 Double Cosets.- 6.3 The Sylow Theorems.- 6.4 Applications of the Sylow Theorems.- Exercises 6.- 7 Groups of Orders 1 To 15.- 7.1 Introduction.- 7.2 Groups of Order 6.- 7.3 Groups of Order 7.- 7.4 Groups of Order 8.- 7.5 Groups of Order 9.- 7.6 Groups of Order 10.- 7.7 Groups of Order 11.- 7.8 Groups of Order 12.- 7.9 Groups of Order 13.- 7.10 Groups of Order 14.- 7.11 Groups of Order 15.- 7.12 Summary.- Exercises 7.- 8 Epilogue.- 8.1 Introduction.- 8.2 Construction of Finite Groups.- 8.3 Solvable and Nilpotent Groups.- 8.4 The Isomorphism Theorems.- 8.5 The Schreier-Jordan-Holder Theorem.- 8.6 Some Basic Results on Solvable Groups.- Exercises 8.- Miscellaneous Exercises.- Outline Solutions to the Exercises.- Exercises 1.- Exercises 2.- Exercises 3.- Exercises 4.- Exercises 5.- Exercises 6.- Exercises 7.- Exercises 8.- Miscellaneous Exercises.- Further Reading and References.- Intermediate.- Advanced.- Sources of Further Problems.- References.- Further Reading and References for Scientists.- General Reference.

On the structure of block ideals in group algebras of finite groups

Abstract: (1980). On the structure of block ideals in group algebras of finite groups. Communications in Algebra: Vol. 8, No. 19, pp. 1867-1872.

On the ramification theory of two-dimensional local fields

Abstract: Filtrations are defined on the group of a two-dimensional local field of characteristic and on the Galois group of its -extension. Results are proved which are analogous to the one-dimensional case (Proposition 2.4, Theorem 2.1).It is proved that, for an Artin-Schreier extension the reciprocity map carries the filtration on the group to the filtration on the group , with the Herbrand numbering. An example is given which shows that this is not true for an arbitrary -extension.Bibliography: 7 titles.

Generalized back coupling rules for the Racah algebra of Gln

Abstract: Two subgroup relations for double coset matrix elements (DCME) of the symmetric group SN are derived by considering the processes of subduction and induction. The duality of outer product coupling in SN and inner product coupling in Gln identifies these as generalized back coupling rules for the Racah algebra of Gln. An iterative procedure for evaluating the DCME, once a consistent phase convention is established, is given. As an example the Racah sum formula for the Clebsch–Gordan coefficients of SU2 is derived from consideration of SN coupling only.

Pauli Principle and Indirect Exchange Phenomena in Molecules and Solids

Abstract: All analyses of interactions in molecular and solid-state systems, however diverse they may be in method of approach and degree of sophistication, must conform to basic principles imposed by what is called the quantum-mechanical “symmetry” of the system. The concept of symmetry of a system characterized by a Hamilton operator H is embodied by the group of Hilbert-space operators which have the following two important properties: a) they preserve the absolute value of the scalar product, |(f, g)|, between any two vectors f, g in the Hilbert space. Let A be such an operator. Then, either (Af, Ag) = (f, g) or (Af, Ag) = (g, f) = (f, g)*. From this it follows that A must be unitary,or an anti-unitary operator, respectively. Further, b) any A must commute with H: AHf = HAf, for all f. We call them “symmetry operators” for the system. It can indeed be easily verified, and it is left as an exercise to the Reader, from the properties a) and b), that these operators form a group in the abstract sense of the word, the symmetry group of the system under consideration. We denote this group by GH.

Infinite soluble groups with no outer automorphisms

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On Permutation Cliques

Abstract: The symmetric group S n is a metric space with distance d(a, b) = |E(a −1 b)| where E(c) is the set of points moved by c ∈ S n . Let L be a given subset of {1, 2,…, n}, a permutation clique A = A(L, n) is any subset A ⊆ S n with d(a, b) ∈ L whenever a, b ∈ A, a ≠ b. We give a framework of new and known information on some special A = A(L, n): maximal, largest, largest subgroups of S n , subscheme of Hamming metric scheme, permutation geometry and some other problems related to this metric space. Some links with classical problems of classification of permutation groups and with extremal problems on finite sets are given.

The direct configuration interaction method for general multireference expansions: Symmetric group approach

Abstract: A computer implementation of the direct configuration interaction method formulated within the symmetric group approach is discussed. The formulation allows for an open-shell as well as for a multiconfigurational reference state. The number of all necessary formulas, derived by a computer for each integral type rather than for the individual integrals, is lower than in the currently existing techniques, including the unitary group approach. The logical structure of a general program for singly and doubly excited configurations is outlined. The efficiency of the symmetric group approach is demonstrated on a recently developed program, restricted to one reference state only.

Some Metrical Problems on Sn

Abstract: We present some problems related to right- (or bi-)invariant metrics d on the symmetric group of permutations S n . Characterizations and constructions of bi-invariant extremal (in the corresponding convex cone) metrics are given, esp. for n ⩽ 5. We also consider special subspaces of the metric space (S n , d): unit balls, sets with prescribed distances ( L -cliques), “hamiltonian” sets. Here we give (for proofs, see [3]) some results and problems arising by analogy with extremal set systems. Related problems of coding type (with Hamming metric) are considered in [1, 4].