The case of equality in the Livingstone-Wagner Theorem

Let G be a permutation group acting on a set Ω of size n∈ℕ and let 1≤k<(n−1)/2. Livingstone and Wagner proved that the number of orbits of G on k-subsets of Ω is less than or equal to the number of orbits on (k+1)-subsets. We investigate the cases when equality occurs.     

Singularity of orbits in classical Lie algebras

We determine the maximum k such that the k-fold sum of some nontrivial, adjoint orbit in the Lie algebra of a classical, compact Lie group has measure zero. The orbits of minimal dimension are seen to be the extreme examples. We show that for this choice of k there is a central, continuous measure on the group such that ^k is singular to L ¹. For Lie groups other than type B _n or C ₃ this result is sharp.     

Cycle Action on Treelike Structures

The purpose of this paper is to study the action on cycles of several known classes of oligomorphic groups, that is, infinite permutation groups of countable degree having only finitely many orbits on k-sets for each k. The groups studied here are all related to trees and treelike relational structures. The sequence whose k-th term is the number of orbits in the action on k-cycles is called Parker sequence. It turns out that, if we are dealing with the automorphism group of a suitable relational structure, this sequence counts also the finite substructures admitting a cyclic automorphism; in calculating these sequences for various groups, we shall thus describe and enumerate such substructures.Di più dirò: ch'a gli alberi dà vita spirito uman che sente e che ragiona. Per prova sollo; io n'ho la voce udita che nel cor flebilmente anco mi suona.[I shall say more: the trees are given life by a human spirit that perceives and reasons. I know it by experience: I heard their voice and it still resounds faintly in my heart.]Torquato Tasso, Gerusalemme liberata, XIII, 49     

Block transitive Steiner systems with more than one point orbit

For all ‘reasonable’ finite t, k, and s, we construct a t‐(?₀, k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2‐(?0, 4, 1) design with a block‐transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial way, but is a by‐product of a new way of looking at a model‐theoretic construction of E. Hrushovski. © 2004 Wiley Periodicals, Inc.     

Cycle Action on Treelike Structures

The purpose of this paper is to study the action on cycles of several known classes of oligomorphic groups, that is, infinite permutation groups of countable degree having only finitely many orbits on k-sets for each k. The groups studied here are all related to trees and treelike relational structures. The sequence whose k-th term is the number of orbits in the action on k-cycles is called Parker sequence. It turns out that, if we are dealing with the automorphism group of a suitable relational structure, this sequence counts also the finite substructures admitting a cyclic automorphism; in calculating these sequences for various groups, we shall thus describe and enumerate such substructures.Di più dirò: ch’a gli alberi dà vita [I shall say more: the trees are given life spirito uman che sente e che ragiona. by a human spirit that perceives and reasons.Di più dirò: ch’a gli alberi dà vita spirito uman che sente e che ragiona. Per prova sollo; io n’ho la voce udita che nel cor flebilmente anco mi suona.[I shall say more: the trees are given life by a human spirit that perceives and reasons. I know it by experience: I heard their voice and it still resounds faintly in my heart.]Torquato Tasso, Gerusalemme liberata, XIII, 49     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-singular dichotomy for orbital measures of classical simple Lie algebras

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ ^k is singular to Lebesgue measure or μ ^k ∈ L ². The minimum k for which μ ^k ∈ L ² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.     

Action of general linear groups on grassmannians

Let A be a finite dimensional commutative semisimple algebra over a field k and let V be a finitely generated A—module. In previous work the author examined the action of the general linear group GL_A(V) on the Grassmannians of k—subspaces of V. The present paper examines the structure of the orbits in greater detail, in particular by working out the structure of the stabilizers in each of the cases when dim_k A≤3. From an algebraic point of view the most interesting situation occurs for A a cubic extension field of k     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.     

Degenerations of <Emphasis Type="Italic">k</Emphasis>[<Emphasis Type="Italic">t</Emphasis>]/(<Emphasis Type="Italic">t</Emphasis><Superscript>r</Superscript>)-Modules Giving Stratification of the Orbits

Let Λ be a finitely generated associative k-algebra where k is an algebraically closed field. For each natural number d, we have the variety of d-dimensional module structures on k^d given by the multiplication of the elements from a generating set of Λ. The general linear group Gl_d(k) acts on this variety by conjugation and the orbits under this action correspond to isomorphism classes of d-dimensional Λ-modules. For two d-dimensional Λ-modules M and N one says that M degenerates to N if the orbit corresponding to N is in the Zariski-closure of the orbit corresponding to M. Now in this situation the stabilizers of the elements in the orbit corresponding to N acts on the orbit corresponding to M. In this paper we characterize degenerations of k[t]/(t^r)-modules with the property that for each y in the orbit corresponding to N, there is an x_y in the orbit corresponding to M such that the orbit corresponding to M is the disjoint union of orbits of the x_y’s under the action of the stabilizer of y where y runs through the orbit corresponding to N. Presented by Idun ReitenMathematics Subject Classifications (2000) 14L30, 16G10.     

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds.     

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.     

A combinatorial approach to transitive extensions of generously unitransitive permutation groups

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H, Ω) on the k‐subsets of Ω, k=2, 3, 4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer‐generated data for the particular series of Hamming schemes H(m, 3), m⩾2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:369–391, 2010     

Closures of Orbits Under the Diagonal Action in the Wonderful Compactification of PGL(3)

In this article, we study the orbits under the diagonal action of a semisimple adjoint group G on its wonderful compactification X for the case G = PGL(3) and determine the closure relations between such orbits. Moreover, we show an example in the wonderful compactification of PSp(4) in which the closure of an orbit for the diagonal action consists of infinitely many orbits.     

Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

We determine the distribution of 3?(q + 1,k,λ) designs, with k ? {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ? 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3?(q + 1,k,λ) designs, with k ? {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.     

Apollonian circle packings: Number theory II. Spherical and hyperbolic packings

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in Part I (J. Number Theory 100, 1–45, 2003). Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup of the group of integral automorphs of the indefinite quaternary quadratic form . This subgroup, called the Apollonian group, acts on integer solutions . This paper gives a reduction theory for orbits of acting on integer solutions to valid for all integer k. It also classifies orbits for all k≡0 (mod 4) in terms of an extra parameter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit. Much of this work was done while the authors were at AT&T Labs-Research, whom the authors thank for support. N. Eriksson was also supported by an NDSEG fellowship and J.C. Lagarias by NSF grant DMS-0500555.     

A state space isomorphism theorem for minimal dynamical systems

Consider a discrete time dynamical systemx _k+1=f(x _k ) on a compact metric spaceM, wheref:MM is a continuous map. Leth:MB ^k be a continuous output function. Suppose that all of the positive orbits off are dense and that the system is observable. We prove that any output trajectory of the system determinesf andh andM up to a homeomorphism. IfM is a compact Abelian topological group andf is an ergodic translation, then any output trajectory determines the system up to a translation and a group isomorphism of the group.     

New coverings of t-sets with (t + 1)-sets

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C(v, k, t). In this article, a computer search for finding good such covering designs, leading to new upper bounds on C(v, k, t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the general framework of tabu search) is then used to find appropriate sets of orbits. A table of upper bounds on C(v, t + 1, t) for v 28 and t 8 is given, and the new covering designs are listed. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 217–226, 1999     

Cocycle Twisting of E(n)-Module Algebras and Applications to the Brauer Group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For any triangular structure R on E(n) we prove that the subgroup BM(k,E(n),R) of BQ(k,E(n)) arising from R is isomorphic to a direct product of BW(k), the Brauer-Wall group of the ground field k, and Sym_n(k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R on E(n) we construct a split short exact sequence having BM(k,E(n),R⁾ as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R). We finally describe how the image of the Hopf automorphism group inside BQ(k,E(n)) acts on Sym_n (k).