nLab conjugacy class

Definition

For $G$ a group and $g \in G$ an element, the conjugacy class of $g$ is the orbit of $g$ under the adjoint action of $G$ on itself, hence the subset $\{ h g h^{-1} | h \in G \} \subset G$ of all elements in $G$ obtained from $g$ by conjugation with another group element.

Similarly there is the conjugacy classes of larger subsets of a group, and in particular the conjugacy classes of subgroups.

Examples

• The conjugacy class of the neutral element consists of precisely the neutral element itself.
• In an abelian group, conjugacy classes are singletons, one for each element of the group.

Properties

For discrete groups

For any finite group the number of its conjugacy classes is equal to the number of its irreducible representations. For finite groups of Lie type this result can be strengthened to show that, in heuristic terms, there is a canonical way to match conjugacy classes of a group $G$ to the irreducible representations of its dual group $G^{\ast}$. The dual group is defined in terms of the root datum? of $G$ in Deligne-Lusztig theory.

Taking symmetric groups as general linear groups over the field with one element, we see they are self-dual, and so there is a correspondence between their conjugacy classes and their irreducible representations. This correspondence is encoded by Young diagrams.

For Lie groups

For $G$ a connected compact Lie group, then its conjugacy classes are in bijection with the quotient $T/W(G,T)$ of its maximal torus by the action of the Weyl group. See at maximal torus – Properties.

The conjugacy classes of a Lie group with binary invariant pairing are the leaves of a Dirac structure on the Lie group, the Cartan-Dirac structure.

Regarding the Lie group as the target space of the WZW model, the conjugacy classes correspond to the D-branes of the model.

Related concepts

• coset space

• character

• conjugacy class of subgroups