Affine Automorphism Group Research Articles

Affine Automorphism Group of Polar Codes

Affine Automorphism Group of Polar Codes

Computing periodic points on Veech surfaces

A non-square-tiled Veech surface has finitely many periodic points, i.e. points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. Our algorithm serves as a new proof of the finiteness of periodic points for non-square-tiled Veech surfaces. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

New families of flag-transitive linear spaces

In this paper, we construct new families of flag-transitive linear spaces with q2n points and q2 points on each line that admit a one-dimensional affine automorphism group. We achieve this by building a natural connection with permutation polynomials of Fq2 of a particular form and following the scheme of Pauley and Bamberg in (2008) [14].

Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

The space of convex domains in complex Euclidean space

In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kahler–Einstein metric are all continuous on the space of convex domains. The group of affine automorphisms acts on this space and we also describe the orbit closures for some special classes of domains.

Symmetry and Special Relativity

We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.

A criterion of smoothness at infinity for an arithmetic quotient of the future tube

Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.

The affine automorphism group of A3 is not a maximal subgroup of the tame automorphism group

The affine automorphism group of A3 is not a maximal subgroup of the tame automorphism group

The solvable Lie group [formula omitted]: An example of an almost [formula omitted]-C*-algebra

Motivated by the description of the C*-algebra of the affine automorphism group N6,28 of the Siegel upper half-plane of degree 2 as an algebra of operator fields defined over the unitary dual N6,28ˆ of the group, we introduce a family of C*-algebras, which we call almost C0(K), and we show that the C*-algebra of the group N6,28 belongs to this class.

Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

AbstractLet F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;2 is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Open image in new window be the quaternion Heisenberg group, and let P be the affine automorphism group of Open image in new window. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on Open image in new window. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on Open image in new window. A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on Open image in new window both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Open image in new window.

Radial wavelet and radon transform on the Heisenberg group

Let be the Heisenberg group, and let denote the affine automorphism group of . The theory of continuous wavelet transforms on the Heisenberg group associated with has been studied in the viewpoint of square integral group representations [J.X. He and H.P. Liu, Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane, J. Math. Anal. Appl. 208 (1997), pp. 58–70]. In this paper, we construct a type of radial wavelets on , the Calderón reproducing formula is also valid. In addition, we devise a subspace of Schwartz functions on which the Radon transform is a bijection. Furthermore, we introduce two subspaces of such that the Radon transform and inverse Radon transform hold by using the wavelet transforms. In our new formulae, the inverse Radon transforms are associated with the sub-Laplacian on , and the smoothness on f can be neglected if wavelet functions are differential.

Representations in L2-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L2-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”

Automorphismes polynomiaux des variétés affines

Let (M, ▽) be an affine manifold which is compact and complete, or compact and unimodular. We show that the connected component of the identity of the group of polynomial automorphisms of (M, ▽) is a nilpotent Lie group whose action on M is locally free. We also show that the orbits of the connected component of the identity of the group of affine automorphisms of (M, ▽) are complete affine immersed submanifolds.

Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane

LetP=NAMbe the minimal parabolic subgroup ofSU(n+1,1), which can be regarded as the affine automorphism group of the Siegel upper half-planeUn+1,Palso acts on the Heisenberg groupHn, the boundary ofUn+1. ThereforePhas a natural representationUonL2(Hn). We decomposeL2(Hn) into the direct sum of the irreducible invariant closed subspaces underU. The restrictions ofUon these subspaces are square-integrable. We give the characterization of the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform gives isometric operators from the irreducible invariant closed subspaces ofL2(Hn) toL2(P).

On block-transitive designs with affine automorphism group

Detailed necessary and sufficient conditions for a k-subset of AG( d, 3) to generate the block set of a block-transitive t-design with automorphism group AGL( d, 3) are derived for t = 3, 4, 5. Similar necessary conditions are found for the existence of a block-transitive design with automorphism group AGL( d, p) when p is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with k or v − k not exceeding 1000 were for t = 4 and ( d, p) = (7, 3), (8, 3), and (3, 7). Examples of block-transitive 4-designs admitting AGL(7, 3) are found for each of the values k = 115, 116, 230, 437, and 552.

Block-transitive 3-designs with affine automorphism group

A t-(v, k, λ) design is a pair D = (X,B) where X is a set of v points and B is a set of k-subsets of X called blocks such that any t points are contained in exactly λ > 0 blocks. If B contains all k-subsets of X then D is said to be trivial. Let G = AutD = {g ∈ Sym(X) | B = B}. That is, G is the set of all permutations of X that fix B setwise. The group G is said to be block-transitive if B = B for B ∈ B, where B = {B | g ∈ G}. In this case we say that D is block-transitive. Similarly, G is point-transitive if X = x for some x ∈ X, where x = {x | g ∈ G}.

Affine distance-transitive groups with alternating or symmetric point stabiliser

We determine all finite graphs which admit a distance-transitive, primitive, affine automorphism group G such that a point stabilizer in G is an alternating or symmetric group (modulo scalars). This work forms part of a programme to classify all finite distance-transitive graphs.

Affine manifolds with nilpotent holonomy

An affine manifold is a differentiable manifold together with an atlas of coordinate charts whose coordinate changes extend to affine automorphisms of Euclidean space. These charts are called atline coordinates. A map between affine manifolds is called affine it its expression in affine coordinates is the restriction of an aftine map between vector spaces. Thus we form the category of affine manifolds and affine m_nps. Let M be a connected affine manifold of dimension n-> 1, locally isomorphic to the vector space E. Its universal covering/~/ inheri ts a unique affine structure for which the covering projection/~/--~ M is an aifine immersion. The group ~r of deck transformations acts on /V/by afline automorphisms. It is well known that there is an affine immersion D :/~5/--~ E, called the developing map. This follows, for example, from Chevalley's Monodromy Theorem; a proof is outlined in Section 2. Such an immersion is unique up to composition with an atiine automorphism of E. Thus for every g ~ n there is a unique affine automorphism a(g) of E such that D o g =ct(g)oD. The resulting homomorphism a : ~r --~ Aft (E) from 7r into the group of affine automorphisms of E is called the affine holonomy representation. It is unique up to inner automorphisms of Aft (E). The composition A : 7r ~ G L (E) is called the linear holonomy representation. The affine structure on M is completely determined by the pair (D, a) . M is called complete when D is a homeomorphism. This is equivalent to geodesic completeness of the connection on M (in which parallel transport is locally defined by affine charts as ordinary parallel transport in E). It is notorious that compactness does not imply completeness. The main results of this paper are about aftine manifolds whose affine holonomy groups a(Tr) are nilpotent. An important class of such manifolds are the affine nilmanifolds 7r\G. Here 7r is a discrete subgroup of a simply connected nilpoint Lie group G. It is assumed that G has a left-invariant afline structure; the space of right cosets of r then inherits an affine structure.