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\begin{document}

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\section*{regular representation}

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

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{mathematics}{}\paragraph*{{Mathematics}}\label{mathematics}

[[!include mathematicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Many algebraic objects have a [[representation theory]] in which they act on other things; for example, [[groups]] often act on sets and [[algebras]] act on [[modules]]. In these situations, the acting object can often be viewed as an example of the type of thing acted upon: a group has an underlying set and an algebra has an underlying module. The multiplication in the acting object then defines an action of the object on this unstructured copy of itself. This is called the \textbf{regular representation} and is an extremely useful representation to study as it only involves the object itself, whence is in a sense \emph{canonical}, but contains a lot of information, as opposed to, say, the [[trivial representation]] (which is also canonical).

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

The definition is the same in each case, but we shall give the actual definitions for the usual suspects.

\begin{defn}
\label{grpreg}\hypertarget{grpreg}{}
Let $G$ be a [[group]] with multiplication $\mu$. Let us write ${|G|}$ for the underlying set of $G$. The \textbf{left regular representation} of $G$ is

\begin{enumerate}%
\item as a [[permutation representation]]: the action $G \times {|G|} \to {|G|}$ defined by $g \cdot h = \mu(g,h)$;


\item as a [[linear representation]]: the corresponding representation on the [[linear span]] of $G$.



\end{enumerate}
The \textbf{right regular representation} is defined analogously.

\end{defn}
\begin{defn}
\label{algreg}\hypertarget{algreg}{}
Let $A$ be an [[associative unital algebra]] with multiplication $\mu$. Let us write ${|A|}$ for the underlying module of $A$. The \textbf{left regular representation} of $A$ is the action $A \otimes {|A|} \to {|A|}$ defined by $a \cdot m = \mu(a,m)$.

The \textbf{right regular representation} is defined analogously.

\end{defn}
These can be seen as examples of a more general concept.

\begin{defn}
\label{monrep}\hypertarget{monrep}{}
Let $(C,\otimes,I)$ be a [[monoidal category]]. Let $M = ({|M|},\mu,\eta)$ be a [[monoid]] in $C$, where ${|M|}$ is the underlying object of $M$ in $C$. The \textbf{regular representation} of $M$ is the action of $M$ on ${|M|}$ induced by the product $\mu$.

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

\begin{itemize}%
\item The regular representation of $G$ as a linear representation is the [[induced representation]] $Ind_{1}^G 1$ of the trivial representation along the inclusion of the trivial [[subgroup]].

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

\begin{itemize}%
\item [[trivial representation]]


\item [[alternating representation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes include

\begin{itemize}%
\item [[Tammo tom Dieck]], section 1.10 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})

\end{itemize}
[[!redirects regular representation]] [[!redirects regular representations]]



\end{document}