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\begin{document}

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\section*{conjugacy class}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_discrete_groups}{For discrete groups}\dotfill \pageref*{for_discrete_groups} \linebreak
\noindent\hyperlink{for_lie_groups}{For Lie groups}\dotfill \pageref*{for_lie_groups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For $G$ a [[group]] and $g \in G$ an element, the \emph{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 \emph{[[conjugacy classes of subgroups]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The conjugacy class of the neutral element consists of precisely the neutral element itself.
\item In an abelian group, conjugacy classes are singletons, one for each element of the group.

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{for_discrete_groups}{}\subsubsection*{{For discrete groups}}\label{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]].

\hypertarget{for_lie_groups}{}\subsubsection*{{For Lie groups}}\label{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 \emph{\href{maximal+torus#Properties}{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 \emph{[[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.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[coset space]]


\item [[character]]


\item [[conjugacy class of subgroups]]



\end{itemize}
[[!redirects conjugacy classes]]



\end{document}