The symmetry of the thermoelastic properties of quasicrystals

The thermoelastic properties of quasicrystals, possessing 5-th, 8-th, 10-th and 12-th order rotational axes, not characteristic of crystalline bodies, are considered. Using group analysis, canonical representations of the second and fourth rank tensors, characterizing the properties of quasicrystals, are constructed. The fact that orthogonal, decagonal and dodecagonal quasicrystals belong to the class of transversely isotropic materials and the isotropy of icosahedric quasicrystals and fullerenes are proved.     

On invariants of symmetry groups of regular complex polytopes

We consider algebras of invariants of symmetry groups for regular complexn-dimensional polyhedraM _n (n>2). We present a positive solution to the “vertex problem’ for the polyhedraM _n . Translated fromDinamicheskic Sistemy. Vol. 12. pp. 74–79, 1993.     

On the symmetry of images of word maps in groups

Word maps on a group are defined by substitution of formal words. Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inversion. We also study this behavior in nilpotent groups.     

Spencer cohomologies and symmetry groups

We propose the construction of a spectral sequence converging to Spencer cohomologies. By using symmetry groups of differential equations systems, we manage to unify computations by reduction to the invariant systems over a homogeneous space. The conditions of coincidence of Spencer cohomologies with the cohomologies of an invariant Spencer complex we obtain from the arithmetic of a -characteristic manifold with respect to fundamental weights of the homogeneous space.     

Two-parameter families of quantum symmetry groups

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O⁺(p,q), B⁺(p,q), S⁺(p,q) and H⁺(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H⁺(p,q) the situation is different and we show that , where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus.     

Critical groups of graphs with reflective symmetry

The critical group of a graph is a finite Abelian group whose order is the number of spanning forests of the graph. For a graph G with a certain reflective symmetry, we generalize a result of Ciucu–Yan–Zhang factorizing the spanning tree number of G by interpreting this as a result about the critical group of G. Our result takes the form of an exact sequence, and explicit connections to bicycle spaces are made.     

The symmetry groups of periodic isonemal fabrics

An isonemal fabric is a weaving in which the symmetry group is transitive on the strands. Its symmetry groups is a layer group. In this paper we determine the 21 possible layer groups which occur as symmetry groups of (2-way, 2-fold) periodic isonemal fabrics. In addition, utilizing the induced strand subgroup, we classify the fabrics into 42 symmetry types.     

Hyperbolic surfaces with prescribed infinite symmetry groups

For any countable group whatsoever, there is a complete hyperbolic surface whose isometry group is .

     

Classification of affine symmetry groups of orbit polytopes

Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups \({\text {GL}}(Gv)\) of orbits \(Gv\subseteq V\), where the linear symmetry group \({\text {GL}}(S)\) of a subset \(S\subseteq V\) is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups \({\text {GL}}(Gv)\) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group \({\text {GL}}(Gw)\) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in \({\text {GL}}(V)\).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331–337, 1977.  https://doi.org/10.1007/BF02429904).     

Fundamental groups of positively curved manifolds with symmetry

We obtain a classification for the fundamental groups of closed $n$ -manifolds of positive sectional curvature which admit an isometric $T^k$ -action with $k \ge \frac{n}{6}+1 (n \ne 11, 15)$ .     

Multipliers for the symmetry groups of p-adic spacetime

The consequences for particle classification of the Volovich hypothesis that spacetime geometry is non-archimedean at the Planck scale are explored. The multiplier groups and universal topological central extensions of the p-adic Poincaré and Galilean groups are determined. The text was submitted by the author in English.