Magnetic point groups and space groups

We review the notion of magnetic symmetry of finite structures as well as infinite periodic and quasiperiodic crystals. We describe one of the most direct consequences of having magnetic symmetry in crystals which is the extinction of magnetic Bragg peaks in neutron diffraction patterns. We finish by mentioning the generalization of magnetic groups to spin groups and color groups.Comment: Written for the Encyclopedia of Condensed Matter Physics. Contains 2 color figures - gray scale version available from the author's website: http://www.tau.ac.il/~ronlif

Fixed Point Polynomials of Permutation Groups

In this paper we study, given a group $G$ of permutations of a finite set, the so-called fixed point polynomial $\sum_{i=0}^{n}f_{i}x^{i}$, where $f_{i}$ is the number of permutations in $G$ which have exactly $i$ fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014. </jats:p

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions

The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{\alpha,A}((z,t)) = \left(|z|^\alpha + A |t|^{\alpha/2}\right)^{1/\alpha}$, for $\alpha \ge 2$ and $A>0$. This natural family includes the canonical Cygan-Kor\'anyi norm, corresponding to $\alpha =4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $\alpha=2$ is the best possible, in all dimensions