Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups

For a locally compact groupG and a groupB of topological automorphisms containing the inner automorphisms ofG and being relatively compact with respect to Birkhoff topology (that isG[FIA] _B,B I(G)) the spaceG _B of -orbits is a commutative hypergroup (=commutative convo inJewett's terminology) in a natural way asJewett has shown. Identifying the space of hypergroup characters ofG _B withE(G, B) (the extreme points ofB-invariant positive definite continuous functionsp withp (e)=1, endowed with the topology of compact convergence) we prove thatE(G, B) is a hypergroup, the hypergroup dual ofG _B.     

A Local Analysis of Imprimitive Symmetric Graphs

Let Г be a G-symmetric graph admitting a nontrivial G-invariant partition . Let Г be the quotient graph of Г with respect to . For each block B ∊ , the setwise stabiliser G_B of B in G induces natural actions on B and on the neighbourhood Г (B) of B in Г . Let G_(B) and G_[B] be respectively the kernels of these actions. In this paper we study certain “local actions" induced by G_(B) and G_[B], such as the action of G_[B] on B and the action of G_(B) on Г (B), and their influence on the structure of Г. Supported by a Discovery Project Grant (DP0558677) from the Australian Research Council and a Melbourne Early Career Researcher Grant from The University of Melbourne.     

Algorithms for Computing Characters for Symmetric Spaces

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice of the symmetric space in terms of the characters of the weight lattice of the group. These algorithms can easily be implemented in a computer algebra package. The root system of the symmetric space can be described as the image of the root system of the group under a projection π derived from an involution θ on . This implies that . Using these formulas for the characters of each of these lattices we show that in fact . A.G. Helminck is partially supported by N.S.F. Grant DMS-0532140.     

Numerically regular hereditary classes of combinatorial geometries

A hereditary class of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in is extremal if no proper extension of G of the same rank is in . The size function h(n) of is defined by h(n)=max {|G|: G and rank(G)=n}, where |G| is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in , |G|=h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n)–h(n–1): 1n<} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.Partially supported by a North Texas State University Faculty Research Grant.     

A condition for the nonsymmetric saddle point matrix being diagonalizable and having real and positive eigenvalues

This paper discusses the spectral properties of the nonsymmetric saddle point matrices of the form with A symmetric positive definite, B full rank, and C symmetric positive semidefinite. A new sufficient condition is obtained so that is diagonalizable with all its eigenvalues real and positive. This condition is weaker than that stated in the recent paper [J. Liesen, A note on the eigenvalues of saddle point matrices, Technical Report 10-2006, Institute of Mathematics, TU Berlin, 2006].     

Cyclic Ostrom Spreads

We call a transitive permutation group G stable if every non-trivial orbit B, of a point stabilizerG _O , is such that some element of G-G _O leaves invariant B {O}. We characterize all finite affine planes of order n that admit a collineation group G that acts, on the points of , as a stable 3/2-transitive permutation group of rank n+2 . The planes obtained are precisely what we have called cyclic Ostrom planes; these are translation planes whose associated spreads are obtained from the Desarguesian spread , of order n=p^r , by replacing a partial subspread of by another consisting of GF(p) subspaces that are unions of kern orbits. All André spreads, as well as many other spreads, are examples of cyclic Ostrom spreads.     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

Subfactors associated to compact Kac algebras

We construct inclusions of the form (B ₀P) ^G (B ₁P) ^G , whereG is a compact quantum group of Kac type acting on an inclusion of finite dimensional C^*-algebrasB ₀B ₁ and on aII ₁ factorP. Under suitable assumptions on the actions ofG, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.     

Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

Isoperimetric inequalities for lattices in semisimple lie groups of rank 2

LetG be a connected semisimple Lie group with finite center. Let be an irreducible non-uniform lattice inG. We show that if the real rank ofG is 2, then the Dehn (or filling) function of is exponential.     

The structure of Banach algebras of bounded continuous functions on the open disk that contain H∞, the Hoffman algebra,and nontangential limits

LetH _B ^G be an algebra of bounded continuous functions on an open disk representable in the formH _B G, where andH _B is a closed subalgebra in C(D) consisting of the functions that have nontangential limits almost everywhere on {ie1023-06}, and these limits belong to the Douglas algebraB. In this paper we describe the spaceM(H _B ^G ) of maximal ideals of the algebraH _B ^G and prove thatM(H _B ^G ) =M(B) M(H _B ^G and prove thatM(H ₀ ^G ), whereH ₀ ^G is a closed ideal inG consisting of functions having nontangential limits equal to zero almost everywhere on {ie1023-12}. Moreover, it is established that on the disk. The Chang-Marshall theorem is generalized for the Banach algebrasH _B ^G . We also prove that for any Douglas algebraB, whereI B = {u } _B are inner functions such that on.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 924–931, July, 1993.     

-Linked planar graphs

A graph G is (m,n)-linked if for any two disjoint subsets R,BV(G) with |R|m and |B|n, G has two disjoint connected subgraphs containing R and B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if G is 4-connected and maximal.     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

Ramsey families which exclude a graph

For graphsA andB the relationA(B) _r ¹ means that for everyr-coloring of the vertices ofA there is a monochromatic copy ofB inA. Forb (G) is the family of graphs which do not embedG. A familyof graphs is Ramsey if for all graphsBthere is a graphAsuch thatA(B) _r ¹ . The only graphsG for which it is not known whether Forb (G) is Ramsey are graphs which have a cutpoint adjacent to every other vertex except one. In this paper we prove for a large subclass of those graphsG, that Forb (G) does not have the Ramsey property.This research has been supported in part by NSERC grant 69-1325.     

Almost Butler groups

Generalizing the notion of the almost free group we introduce almost Butler groups. An almost B ₂-group G of singular cardinality is a B ₂-group. Since almost B ₂-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that G is a B ₁-group. Some other results characterizing B ₂-groups within the classes of almost B ₁-groups and almost B ₂-groups are obtained. A theorem of [BR] stating that a group G of weakly compact cardinality having a -filtration consisting of pure B ₂-subgroup is a B ₂-group appears as a corollary.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Covering properties and Følner conditions for locally compact groups

Summary LetG be a locally compact group with left Haar measurem _G on the Borel sets IB(G) (generated by open subsets) and write |E|=m _G (E). Consider the following geometric conditions on the groupG.(FC If >0 and compact setKG are given, there is a compact setU with 0<|U|< and |x U U|/|U|< for allxK.(A) If >0 and compact setKG, which includes the unit, are given there is a compact setU with 0<|U|< and |K U U|/|U|<.HereA B=(A/B)(B/A) is the symmetric difference set; by regularity ofm _G it makes no difference if we allowU to be a Borel set. It is well known that (A)(FC) and it is known that validity of these conditions is intimately connected with amenability ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)(FC)(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies.(C) For at least one setK, with int(K)Ø and compact, there is an indexed family {x J}G such that {Kx } is a covering forG whose covering index at each pointg (the number of J withgKx ) is uniformly bounded throughoutG.     

An Asymptotic Independence Theorem for the Number of Matchings in Graphs

Let z(G) be the number of matchings (independent edge subsets) of a graph G. For a set M of edges and/or vertices, the ratio represents the probability that a randomly picked matching of G does not contain an edge or cover a vertex that is an element of M. We provide estimates for the quotient , depending on the sizes of the disjoint sets A and B, their distance and the maximum degree of the underlying graph G. It turns out that this ratio approaches 1 as the distance of A and B tends to ∞, provided that the size of A and B and the maximum degree are bounded, showing asymptotic independence. We also provide an application of this theorem to an asymptotic enumeration problem related to the dimer-monomer model from statistical physics. This material is based upon work supported by the German Research Foundation DFG under grant number 445 SUA-113/25/0-1 and the South African National Research Foundation under grant number 65972.