On the construction of highly symmetric tight frames and complex polytopes

Many “highly symmetric” configurations of vectors in C^d${\mathbb{C}}^d$, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n   vectors for C^d${\mathbb{C}}^d$ (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.     

Homology decompositions for p-compact groups

We construct a homotopy theoretic setup for homology decompositions of classifying spaces of p-compact groups. This setup is then used to obtain a subgroup decomposition for p-compact groups which generalizes the subgroup decomposition with respect to p-stubborn subgroups for a compact Lie group constructed by Jackowski, McClure and Oliver.     

The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n$n$-polytopes with the smallest number of flags are found, for every rank n>1$n>1$. With a few small exceptions, the smallest regular n$n$-polytopes come from a family of ‘tight’ polytopes with 2⋅4ⁿ⁻¹$2\cdot 4^{n-1}$ flags, one for each n$n$, with Schläfli symbol {4∣4∣?∣4}$\{4\mid 4\mid ?\mid 4\}$. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.     

Integral geometry and geometric inequalities in hyperbolic space

Using results from integral geometry, we find inequalities involving mean curvature integrals of convex hypersurfaces in hyperbolic space. Such inequalities generalize the Minkowski formulas for euclidean convex sets.     

Good coverings for section spaces of fibre bundles

In this note, existence of good coverings for section spaces of smooth fibre bundles is shown by resorting to the Riemannian geometry of the L²-metric, via the construction of strongly convex neighbourhoods, in close analogy with the finite dimensional case.     

The configuration space of the n-arms machine in the Euclidean space

As a typical kind of mechanical linkage, we consider the n-arms machine in Rd. The machine consists of n 2-legs with equal length such that the initial point of each 2-leg is fixed to a circle, while all tips of the 2-legs are combined to a central joint. We determine the homotopy type of the configuration space of the n-arms machine.     

Gaussian rational quadrature formulas for ill-scaled integrands

A flexible treatment of Gaussian quadrature formulas based on rational functions is given to evaluate the integral , when f is meromorphic in a neighborhood V of the interval I and W(x) is an ill-scaled weight function. Some numerical tests illustrate the power of this approach in comparison with Gautschi’s method.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950s.     

The lattice of flats and its underlying flag matroid polytope

LetM be a matroid andF the collection of all linear orderings of bases ofM, orflags ofM. We define the flag matroid polytope Δ(F). We determine when two vertices of Δ(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats ofM and certain maximal faces of Δ(F). Supported in part by NSA grant MDA904-95-1-1056.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

General affine surface areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for L? affine surface areas are established.     

Cartan models and cellular decompositions of symmetric Riemannian spaces

In this paper, by making use of the Cartan models, we will construct cellular decompositions of some symmetric Riemannian spaces such as Sp(n)/U(n), U(n)/O(n), U(2n)/Sp(n), O(2n)/U(n), SU(n)/SO(n), SU(2n)/Sp(n), SO(2n)/U(n).     

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product red_? on Mred from a strongly invariant star product ? on M. The new questions which are addressed in this paper concern the existence of natural ^∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ^∗-algebra.We assume that ? is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ^∗-involution for red_? on the reduced space. Looking into the question whether the corresponding ^∗-involution is the complex conjugation (which is a ^∗-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C^∞(M)?λ?,?)-submodule and a right (C^∞(Mred)?λ?,red_?)-submodule of C^∞(C)?λ?; we define on it a C^∞(Mred)?λ?-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^∞(Mred)?λ? and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ^∗-representations of (C^∞(Mred)?λ?,red_?) on pre-Hilbert right D-modules to those of (C^∞(M)?λ?,?), for any auxiliary coefficient ^∗-algebra D over C?λ?.     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

Bounds for the cup-length of Poincaré spaces and their applications

Our main result offers a new (quite systematic) way of deriving bounds for the cup-length of Poincaré spaces over fields; we outline a general research program based on this result. For the oriented Grassmann manifolds, already a limited realization of the program leads, in many cases, to the exact values of the cup-length and to interesting information on the Lyusternik-Shnirel'man category.