Asymptotically symmetric embeddings and symmetric quasicircles

A well-known characterization of quasicircles is the following: A Jordan curve in the complex plane is a quasicircle if and only if it is the image of the unit circle under a quasisymmetric embedding. In this paper we try to characterize a subclass of quasicircles, namely, symmetric quasicircles, by introducing the concept of asymptotically symmetric embeddings. We show that a Jordan curve in the complex plane is a symmetric quasicircle if and only if it is the image of the unit circle under an asymptotically symmetric embedding.

     

Tilted symmetric orders are symmetric orders

We prove in this note that a ring which is derived equivalent to a symmetric order is again a symmetric order. Group rings of finite groups over an integral domain of characteristic 0 are symmetric orders.     

The symmetric genus of alternating and symmetric groups

The symmetric genus of a finite group G has been defined by Thomas W. Tucker as the smallest genus of all surfaces on which G acts faithfully as a group of automorphisms (some of which may reverse the orientation of the surface). This note announces the symmetric genus of all finite alternating and symmetric groups.     

Covering a symmetric poset by symmetric chains

We prove a min-max result on special partially ordered sets, a conjecture of András Frank. As corollaries we deduce Dilworth's theorem and the well-known min-max formula for the minimum size edge cover of a graph.Research supported by the Netherlands Organization for Scientific Research (NWO)     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

Distributionally concave symmetric spaces and uniqueness of symmetric structure

We present new results concerning the uniqueness of symmetric structure of symmetric function spaces. Our methods are partly based on a detailed study of distributionally concave spaces and the tensor product operator.     

Locally 2-fold symmetric manifolds are locally symmetric

A manifold is locally k-fold symmetric if for any point and any k-dimensional vector subspace tangent to this point, there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that k-dimensional vector subspace is minus the identity. We show that for \(k \ge 2\), Riemannian, pseudoriemannian, and Finslerian locally k-fold symmetric manifolds are locally symmetric.     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Concentration of symmetric eigenfunctions

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.     

Embeddings of symmetric varieties

We generalize to the case of a symmetric variety the construction of the enveloping semigroup of a semisimple algebraic group due to E. B. Vinberg, and we establish a connection with the wonderful completion of the associated adjoint symmetric variety due to C. De Concini and C. Procesi.The author gratefully acknowledges the financial support of the Fonds FCAR and thanks the referees and V. Ginzburg for their comments.     

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification. Dedicated to the memory of C. Chevalley     

N-extendability of symmetric graphs

It is proved that a cyclically (k ? 1)(2n ? 1)-edge-connected edge transitive k-regular graph with even order is n-extendable, where k ≥ 3 and k ? 1 ≥ n ≥ ?(k + 1)/2?. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Properties of symmetric matrices

We find the extreme points and the smooth points of the unit ball of the Banach space n_℘ of all real symmetric (n×n)-matrices.     

Extensions of symmetric relations

Maximal accretive and dissipative extensions of symmetric relations defined by abstract boundary conditions are considered in Hubert space. Under the assumption that the resolvent of some extension has properties such as belonging to the Neumann-Shatten ideal or having given asymptotics of the s-numbers, either sufficient or necessary and sufficient conditions are found for a perturbation of the boundary conditions guaranteeing that the above properties are preserved for the resolvent of the perturbed extension.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 825–834, December, 1977.