Permutation Representations Of Groups Research Articles

The Finite Primitive Permutation Groups Containing an Abelian Regular Subgroup

A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside in 1900. 2000 Mathematics Subject Classification 20B15, 20B30

The Permutation Group Theory and Electrons Interaction

The new mathematical formalism for the Green's functions of interacting electrons in crystals is constructed. It is based on the theory of Green's functions and permutation groups. We constructed a new object of permutation groups, which we call double permutation (DP). DP allows one to take into consideration the symmetry of the ground state as well as energy and momentum conservation in every virtual interaction. We developed the classification of double permutations and proved the theorem, which allows the selection of classes of associated double permutations (ADP). The Green's functions are constructed for series of ADP. We separate in the DP the convolving columns by replacing the initial interaction between the particles with the effective interaction. In convoluting the series for Green's functions, we use the methods developed for permutation groups schemes of Young–Yamanuti.

Association schemes and permutation groups

Every permutation group which is not 2-transitive acts on a non-trivial coherent configuration, but the question of which permutation groups G act on non-trivial association schemes (symmetric coherent configurations) is considerably more subtle. A closely related question is: when is there a unique minimal G-invariant association scheme? We examine these questions, and relate them to more familiar concepts of permutation group theory (such as generous transitivity) and association scheme theory (such as stratifiability). Our main results are the determination of all regular groups having a unique minimal association scheme, and a classification of groups with no non-trivial association scheme. The latter must be primitive, and are 2-homogeneous, or almost simple, or of diagonal type. The diagonal groups have some very interesting features, and we examine them further. Among other things we show that a diagonal group with non-abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more; and we characterise the quaternion group Q 8 as the unique non-abelian group T such that a diagonal group with eight factors T is generously transitive.

On partitioning the orbitals of a transitive permutation group

LetGGbe a permutation group on a setΩ\Omegawith a transitive normal subgroupMM. ThenGGacts on the setOrbl(M,Ω)\mathrm {Orbl}(M,\Omega )of nontrivialMM-orbitals in the natural way, and here we are interested in the case whereOrbl(M,Ω)\mathrm {Orbl}(M,\Omega )has a partitionP\mathcal Psuch thatGGacts transitively onP\mathcal P. The problem of characterising such tuples(M,G,Ω,P)(M,G,\Omega ,\mathcal P), called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where|P||\mathcal P|is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where|P|=2|\mathcal P|=2exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to theGG-actions onΩ\Omegaand onP\mathcal P, and gives some construction methods for TODs.

The non-abelian Born–Infeld action at order F6

To gain insight into the non-abelian Born–Infeld (NBI) action, we study coinciding D-branes wrapped on tori, and turn on magnetic fields on their worldvolume. We then compare predictions for the spectrum of open strings stretching between these D-branes, from perturbative string theory and from the effective NBI action. Under some plausible assumptions, we find corrections to the Str-prescription for the NBI action at order F 6 . In the process we give a way to classify terms in the NBI action that can be written in terms of field strengths only, in terms of permutation group theory.

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

On Subgroups of Prime Power Index

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

A labeling scheme for young tableaux spanning representations of permutation group S(N)

A labeling scheme for young tableaux spanning representations of permutation group S(N)

Set Ideals with Maximal Symmetry Group and Minimal Dynamical Systems

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <∣X∣, the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory. 1991 Mathematics Subject Classification 20B35, 20B27.

Dual symmetric inverse monoids and representation theory

AbstractThere is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

On Classifying All Full Factorisations and Multiple-Factorisations of the Finite Almost Simple Groups

We present some general results on factorisations of almost simple groups. These results are consequences of the classification of all maximal factorisations of almost simple groups (due to Liebeck, Saxl, and the second author), and are needed for applications to the theory of quasiprimitive permutation groups.

Reconstructibility of Graphs with a Large Branch: Permutation Group Theory and Graph Reconstruction

Let G be a connected graph having a pendant vertex v such that G-v has pendant vertices in at least two of its branches. Using techniques from permutation group theory, we prove that such a graph G is vertex reconstructible provided it has a branch of large order.

Spin Effects on Spur Kinetics. 4. Incorporating a Realistic Exchange Interaction

A long-standing controversy in the literature over the role of spin exchange in spur kinetics has been investigated for a four-radical spur using a modified Monte Carlo simulation technique which explicitly incorporates the quantum mechanics of the system. The dimensionality of the spin Hamiltonian is reduced using permutation group theory to improve the efficiency of the simulation program. Comparison of an exponential model of the exchange interaction with well-known approximations under a variety of conditions indicates that the “contact exchange” (CE) approximation of Brocklehurst and Highashimura is accurate to within 10% in most circumstances.

A graphical approach to permutation group representations for many-electron systems

A graphical approach to permutation group representations for many-electron systems

Images of commutator MAPS

The commutator map θα: G → G associated with the element α of the group G is defined by θα(g) = g −1α−1gα for all g ∊ G. In this paper we prove that if the image θα is a subgroup of the finite group G, then θα(G) is soluble. (This in fact generalises the well-known result which states that all finite groups that admit a fixed-point-free automorphism are soluble.) An application (that motivated this paper) to the theory of permutation groups is also given.

Applications of permutation group theory to Heisenberg spin-1/2 chain

Making use of permutation group technique, the eigenvalue problem of Heisenberg spin-1/2 chains can be solved. Especially the ground state of the antiferromagnetic chains can be easily written.

Random Permutations: Some Group-Theoretic Aspects

The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).

A NEW METHOD TO SOLVE THE S = l/2 HEISENBERG SPIN CHAIN

We present a new approach which is completely based on the permutation group theory to solve the eigenvalue problem of the S = 1/2 Heisenberg finite-chain with free-ends boundary. All of the eigenvalues and the corresponding eigenstates are obtained. The properties of the ground state and the low-lying excited states of the antiferromagnetic chain are examined.

Uniqueness of Parisi's scheme for replica symmetry breaking

Replica symmetry breaking in spin glass models is investigated using elements of the theory of permutation groups. It is shown how the various types of symmetry breaking give rise to special algebras and that Parisi's scheme may be uniquely characterized by two simple conditions on these algebras, namely transposition symmetry and simple extensibility. An alternative to the Parisi scheme is shown to be unacceptable.

Experimental tests of chirality algebra

Most chiral molecules can be dissected into a collection of ligands attached to an underlying skeleton. Application of permutation group theory and group representation theory to such a model can lead to chirality functions which can be used to approximate pseudoscalar measurements such as optical rotation or circular dichroism. Such chirality functions have been tested experimentally for the following skeletons: (1) The polarized triangle of phosphines and phosphine oxides; (2) the tetrahedron of methane derivatives; (3) the disphenoid of allene and 2, 2′-spirobiindane derivatives; (4) the polarized rectangle of [2, 2]-metacyclophanes; (5) the polarized pentagon of heterodisubstituted ferrocenes. The success of this method is fair to good for the polarized triangle, tetrahedron, and disphenoid skeletons but deteriorates rapidly for the polarized rectangle and polarized pentagon skeletons, in accord with the greater group-theoretical complexity of the latter skeletons.