Spherical 5-designs obtained from finite unitary groups

Let M M be a cusped hyperbolic 3 3 -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in P G L ( 2 , C ) \mathrm {PGL}(2,\mathbf {C}) (up to conjugation) is of complex dimension the number ν \nu of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν \nu peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M M , for which the peripheral representation is no more faithful. We work here in the framework of P G L ( n , C ) \mathrm {PGL}(n,\mathbf {C}) . The hyperbolic structure lifts, via the n n -dimensional irreducible representation, to a representation ρ g e o m \rho _{\mathrm {geom}} . We know from the work of Menal-Ferrer and Porti that the space of deformations of ρ geom \rho _{\textrm {geom}} has complex dimension ( n − 1 ) ν (n-1)\nu . We prove here that, unlike the P G L ( 2 ) \mathrm {PGL}(2) -case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1 \geq 1 .

Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2

LetV be a complex vector space and letG⊂GL(V) be a finite unitary reflection group. Letγ:V →V/G be the orbit map. We show that the image of the orbit stratification underγ is determined by a matrix of polynomials called the discriminant matrix, and that this stratification agrees with the stratification ofV/G obtained from logarithmic vector fields tangent to the discriminant hypersurface.

Let R be a quantum solvable algebra. It is proved that every prime ideal I of R which is stable with respect to the quantum adjoint action is completely prime, and Fract(R/I) is isomorphic to the skew field of fractions of an algebra of twisted polynomials. We study the correspondence between symplectic leaves and irreducible representations. A conjecture of De Concini-Kac--Procesi on the dimension of irreducible representations is proved for sufficiently large 1.

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∈ cd (G). We determine the graph Δ(G) for the finite simple groups of types A ℓ(q) and 2 A ℓ (q 2), that is, for the simple linear and unitary groups.

In !.heir fundamental paper IBH I j, J. Bruning and E. Heintze studied elliptic operators and their eigenvalues by means of the action of a compact Lie group of isometries conunuting with the operator. For a fixed irreducible representation of the group, they single out those eigenvalues in whose eigenspaces this representation occurs, and prO\'e general theorems, inter alia, on the asymptotic distribution of such eigenvalues. The aim of the present paper is to use this approach in a very ronrrete situation, namely for the eigenfunctions and eigenvalues of the Laplacian on dosed surfaces of revolution. Such surfaces are homeomorphic to the sphere or to the torus; their isomt'try group cont i\in~ 0(2). The reprt'sentation ohhis group on C2(M) induces a representation on each eigenspace E~ . We determine the form of the eigenfunctions belonging to irredudble components of this representation, and the of their nodal domains. The main reswts are estimates (most l.V under convexity ronditions) of the fust eigenvalue in whose eigenspace a fued irreducible representation of 0(2) occurs, and estimates of the dimensions of a rbitrary eigenspaces. The irreducible representations of 0(2), much in the same \I'av as !host' of 50(2), are of dimension one or two, S~ 2.1. Each irreducible representation of dimension two is characterized by the winding number k E IN of the corresponding representation of 50(2); we denote it by pl<. By po we denote the trivial representation of dimension one. The eigenfunctions in an irreducible component or EA will be proved to have the following particwarly simplt' form:

A projection operator technique for determining selection rules

The charge conjugation C and the parity P are assumed to generate a finite group without assuming any apriori commutation or anti-commutation relation. The group then generated is the di-hedral group D/sub n/ (n being even) which has two dimensional irreducible representations. This implies that we could have in principle, elementary particles with parity and charge-conjugation doublet structure.

New bounds on the number of unit spheres that can touch a unit sphere in n dimensions

The coupled magnon—Phonon modes in magnetic crystals with shubnikov symmetry

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

The proofs presented in this chapter can be grouped as follows. We prove lower and upper estimates for the contact numbers of packings of n unit balls in Euclidean 3-space. One can regard this problem as a combinatorial relative of the Kepler problem on the densest unit sphere packings. Next, we give lower estimates for the surface volume of Voronoi cells in packings of unit balls in Euclidean d-space for all d ≥ 2 and then we improve those estimates in dimensions d ≥ 8. All these results imply upper bounds for the usual density of unit ball packings. Returning to the 3-dimensional Euclidean space we give lower bounds for the average surface area (resp., average edge curvature) of the cells in an arbitrary normal tiling with each cell holding a unit ball. On the one hand, it leads to a new version of the Kepler problem on unit sphere packings on the other hand, it generates a new relative of Kelvin’s foam problem. Finally, we find sufficient conditions for sphere packings being uniformly stable, a property that holds for all densest lattice sphere packings up to dimension 8.KeywordsUnit BallVoronoi CellKepler ProblemRigidity MatrixSurface VolumeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove that a hypersurface in $$E^n$$ is of constant slope if and only if its normalized position vector provides a solution of the eikonal equation on the unit sphere. Explicit parametric equation of the constant slope hypersurface is given. We show that the constant slope hypersurfaces are based on the polar pair of hypersurfaces in the unit sphere.

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz equations from an $n$-dimensional closed Riemannian manifold $\mathbb{T}$ or a bounded domain in $\mathbb{R}^n$ into a unit sphere $S_\mathfrak{g}(1)$ in $\mathfrak{g}$. In particular, we consider the Hamiltonian system associated with the nonlocal energy--{\it micromagnetic energy} defined on a bounded domain of $\mathbb{R}^3$ and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

The use of permutation representations in structural computations in large finite matrix groups