China
Africa
Explore chapters and articles related to this topic
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
An important concept we will see time and time again throughout this book is the class function. This is a function C:G→ℂ (recall that C denotes the complex numbers) which is constant on conjugacy classes. That is, its value is the same for all elements of each conjugacy class. This means that C(g)=C(h−1∘g∘h)orC(h∘g)=C(g∘h)
Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The concept of conjugacy classes is based on a classification of the elements of a group into a number of distinct subsets of the group. We start by defining that for two elements a and b of a group, b is conjugate to a if the group contains an element g such that () b=gag−1.We write this relation as b ~ a and it is easy to check (see Problem 4.6) that conjugacy is an equivalence relation, meaning that b ~ a implies a ~ b and that a ~ b together with b ~ c implies a ~ c. From the perspective of transformations, conjugacy has a very intuitive interpretation. If we can find a transformation g such that a is equivalent to first performing g, then b, and finally applying the inverse transformation of g, then a and b are conjugate to each other. The conjugacy class of a can now be defined as the set of all elements of the group that are conjugate to a.
Block conjugacy of irreducible toral automorphisms
Published in Dynamical Systems, 2019
Lennard F. Bakker, Pedro Martins Rodrigues
Under the conditions of Proposition 4.4 with k=2, if then and satisfy I=JY and and the conjugacy class of is associated with the arithmetic equivalence class of JX. So, every 2-block conjugacy is induced by an R-module isomorphism φ as in Proposition 4.2.
Vector and bidirector representations of magnetic point groups
Published in Phase Transitions, 2020
The real irreducible representations are labeled using the usual Mulliken notation. In addition, in case of gray groups, a prefix “m” is used to denote irreducible representations antisymmetric with respect to time reversal, as it is done for example on the Bilbao crystallographic server [12]. Characters are listed for each conjugacy class of the group.