Permutation Action Research Articles

Quasigroup Homogeneous Spaces and Linear Representations

Using pseudoinverses of incidence matrices of finite quasigroups in partitions induced by left multiplications of subquasigroups, a quasigroup homogeneous space is defined as a set of Markov chain actions indexed by the quasigroup. A certain non-unital ring is afforded a linear representation by a quasigroup homogeneous space. If the quasigroup is a group, the linear representation is a factor in the usual linear representation of the group algebra afforded by the group homogeneous space. In the general case, the structure of the non-unital ring is analyzed in terms of the permutation action of the multiplication group of the quasigroup. The linear representation corestricts to the natural projection of the non-unital ring onto the quotient by its Jacobson radical.

The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines

The eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matrices, the proof gives a probabilistic interpretation to identities of Macdonald from symmetric function theory. For the case of upper triangular matrices over a finite field, connections between symmetric function theory and a probabilistic growth algorithm of Borodin and Kirillov emerge.

Flag-symmetry of the poset of shuffles and a local action of the symmetric group

We show that the posets of shuffles introduced by Greene in 1988 are flag symmetric, and we describe a permutation action of the symmetric group on the maximal chains which is local and yields a representation of the symmetric group whose character has Frobenius characteristic closely related to the flag symmetric function. A key tool is provided by a new labeling of the maximal chains of a poset of shuffles. This labeling and the structure of the orbits of maximal chains under the local action lead to combinatorial derivations of enumerative properties obtained originally by Greene. As a further consequence, a natural notion of type of shuffles emerges and the monoid of multiplicative functions on the poset of shuffles is described in terms of operations on power series. The main results concerning the flag symmetric function and the local action on the maximal chains of a poset of shuffles are obtained from new general results regarding chain labelings of posets.

Edge Sequences, Ribbon Tableaux, and an Action of Affine Permutations

An overview is provided of some of the basic facts concerning rim hook lattices and ribbon tableaux, using a representation of partitions by their edge sequences. An action is defined for the affine Coxeter group of type Ãr−1on ther-rim hook lattice, and thereby on the sets of standard and semistandardr-ribbon tableaux, and this action is related to the concept of chains inr-ribbon tableaux.

Computing Automatic Coset Systems and Subgroup Presentations

The concept of an automatic group can be generalized to a group that is automatic with respect to a specified subgroup. This means that there is a finite state automaton that recognizes a unique word in each coset of the subgroup, and others that essentially recognize the permutation action on these cosets induced by multiplying by a group generator. These automata make it possible to enumerate coset representatives as words in the generators, and to solve the generalized word problem for the subgroup efficiently. Algorithms to construct these automata have been described previously by Redfern. Here we describe improved versions, together with implementation details and some examples of successful calculations. A related algorithm to compute a finite presentation of the subgroup is also described.

Constructing Permutation Representations for Matrix Groups

New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image. The theory has been successfully tested on a representation of the sporadic simple groupLy, discovered by Lyons(1972). With noa prioriassumptions, we find a permutation representation of degree 9606125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a “point” was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2's running under MPI. The results of those parallel experiments are briefly reviewed.

Symmetric Generation of the Higman-Sims Group

In this paper we use the ideas developed in [Curtis, 1990] and [Curtis, 1993] to give a new elementary construction of the Higman-Sims group. The 176 point geometry found by Higman emerges naturally, complete with permutations on the 176 points plus 176 quadrics which generate HS:2. In addition, the permutation action of HS:2 on 100 points, with point stabilizer M22:2, is an easy by-product of this approach.

On the number of orbits of a group in two permutation actions

On the number of orbits of a group in two permutation actions

Symmetric Presentations II: The Janko Group J 1

In our paper [6] we exploited parallel behaviour of the Mathieu groups M12 and M24 which we described fully in [4, 5], to obtain presentations of those groups given implicitly in terms of certain highly symmetric generating sets. We made precise the concepts of symmetric generating set, symmetric presentation and primitive presentation, and indicated how these ideas can be generalized to other groups. In the present work, which is a continuation of [6], we turn the process around and seek groups which possess generating sets with certain specified symmetries. We shall find that the most straightforward ways of generalizing [6] immediately give rise to the two smallest Janko groups J1 and J2 Moreover the permutation actions of minimal degree of these two groups, on 266 and 100 letters respectively, emerge naturally and we are readily able to write down our symmetric generators in these representations. All assertions made can be easily checked on a computer, and we are again heavily indebted to the CAYLEY package. That said the final developments of J1 and J2 given here, and in the sequel to this paper, are entirely by hand and are complete in all detail.

T-Divisible designs from imprimitive permutation groups

We study the close connection of some imprimitive permutation groups with t -divisible designs ( t -DD's). Some constructions of 2- and 3-DD's by affine and projective groups of dimension 1 over a local K -algebra A , considered both in their natural permutation action on the affine and projective line over A , are analysed in detail.

Construction of Graded Covariant GL ( m / n ) Modules Using Tableaux

Irreducible covariant tensor modules for the Lie supergroups GL(m/n) and the Lie superalgebras gl(m/n) and sl(m/n) are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on V⊗l. The isomorphism between this group of actions and the symmetric group Sl enables the graded generalization of the Young symmetrizers, and hence of the column relations and Garnir relations, to be made. Consequently, corresponding to each partition of l an irreducible GL(m/n) module may be obtained as a submodule of V⊗l. A basis for the module labeled by the partition λ is provided by GL(m/n)–standard tableaux of shape λ defined by Berele and Regev. The reduction of an arbitrary tableau to standard form is accomplished through the use of graded column relations and graded Garnir relations. The standardization procedure is algorithmic and allows matrix representations of the Lie superalgebras gl(m/n) and sl(m/n) to be constructed explicitly over the field of rational numbers. All the various steps of the standardization algorithm are exemplified, as well as the explicit construction of matrices representing particular elements of gl(m/n) and sl(m/n).

The Demazure–Tits subgroup of a simple Lie group

The Demazure–Tits subgroup of a simple Lie group G is the group of invariance of Clebsch–Gordan coefficients tables (assuming an appropriate choice of basis). The structure of the Demazure–Tits subgroups of An, Bn, Cn, Dn, and G2 is described. Orbits of the permutation action of the DT group in any irreducible finite-dimensional representation space of A2, C2, and G2 are decomposed into the sum of irreducible representations of the DT group.

Character correspondences induced by subgroups of operator groups

Let G and A be finite groups and suppose that A acts on G by automorphisms. Then A induces a permutation action on the set Irr(G) of irreducible characters of G. For x E Irr(G) and a E A, the character xa is defined by x”(g”) = x(g) for g E G. We let Irr,(G) = (x E Irr(G) 1 xa = x for all a E A). Assume further that (/ G /, 1 A 1) = 1. Then there is a “natural” one-to-one and onto map m(G, A): Irr,(G) + Irr(C,(A)). By “natural”, we mean that r(G, A) is completely determined by the action of A on G and is thus independent of any choices made in an algorithm used to compute the map. Since (I G 1, 1 A 1) = 1, the Odd-Order Th eorem implies that either A is solvable or / G / is odd. G. Glauberman [S] developed the correspondence for the case where A is solvable, and I. M. Isaacs [7] developed the correspondence when / G 1 is odd. The author [9] showed that the Glauberman correspondence and the Isaacs correspondence are equal when both are defined. We note E. C. Dade ([l], [2], and [3]), via his theory of Clifford systems, has done work inclusive of that of Isaacs [7]. We will follow Tsaacs methods. Let B < A. In this paper, we investigate the relationship between rr(G, A) and m(G, B). If B is subnormal in A, then much information is known about the relationship between m(G, A) and rr(G, B) (see Theorem 2.1(c)). Suppose B < A and C,(A) = C,(B). rt is then an easy consequence of the correspondence that Irr,(G) = Irr,(G). Th e main result of this paper will be to show that rr(G, A) = r(G, B) if G is solvable. The solvability of G is used heavily as we look at characters afforded A by chief sections of G. The more general question is unknown. In the event that G has odd order, we obtain a stronger result. For B < A, we have that C,(A) < C,(B). If / G 1 is odd and x E Irr,(G), then Theorem 2.14 shows that xm(G, A) is an irreducible constituent of the restriction of xrr(G, B) to C,(A). It is not known if this result holds when G is solvable of even order. All groups considered are finite.

Modeloids. I

If A is a set and  is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on  which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite ( ( ⩾ 5 ) \left ( { \geqslant \,\,5} \right ) ) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.

A Dual View of the Clifford Theory of Characters of Finite Groups, II

This paper continues the analysis of Clifford theory for the case of a finite group G, K a normal subgroup of G and G/K abelian which was developed in [7]. In [7] the permutation actions of G/K on the characters of K and of (G/K)^ on the characters of G were studied in relation to their effects on induction and restriction of group characters.