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\begin{document}

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\section*{representation theory}

\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{InHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{InHomotopyTypeTheory} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

\emph{Representation theory} is concerned with the study of [[algebra|algebraic]] [[structures]] via their [[representations]]. This concerns notably [[groups]], directly or in their incarnation as [[group algebras]], [[Hopf algebras]] or [[Lie algebras]], and usually concerns [[linear algebra|linear]] [[representations]], hence [[modules]] of these structures. But more generally representation theory also studies [[representations]]/[[modules]]/[[actions]] of generalizations of such structures, such as [[coalgebras]] via their [[comodules]] etc.

See also at \emph{[[geometric representation theory]]}.

\hypertarget{InHomotopyTypeTheory}{}\subsection*{{In homotopy type theory}}\label{InHomotopyTypeTheory}

The fundamental concepts of representation theory have a particular natural formulation in [[homotopy theory]] and in fact in [[homotopy type theory]], which also refines it from the study of [[representations]] of [[groups]] to that of [[∞-representations]] of [[∞-groups]]. This includes both [[discrete ∞-groups]] as well as [[geometric homotopy types]] such as [[smooth ∞-groups]], the higher analog of [[Lie groups]].

The key observation to this translation is that

\begin{enumerate}%
\item an [[∞-group]] $G$ is equivalently given by its [[delooping]] $\mathbf{B}G$ regarded with its canonical [[pointed object|point]] (see at [[looping and delooping]]), hence the universal $G$-[[principal ∞-bundle]]

\begin{displaymath}
\itexarray{
     G &\longrightarrow& \ast
     \\
     && \downarrow
     \\
     && \mathbf{B}G
  }
\end{displaymath}

\item an [[∞-action]] $\rho$ of $G$ on any [[geometric homotopy type]] $V$ is equivalently given by a [[homotopy fiber sequence]] of the form

\begin{displaymath}
\itexarray{
     V &\stackrel{}{\longrightarrow}& V//_\rho G
     \\
     && \downarrow
     \\
     && \mathbf{B}G
  }
  \,,
\end{displaymath}
hence by a $V$-[[fiber ∞-bundle]] over $\mathbf{B}G$ which is the $\rho$-[[associated ∞-bundle]] to the universal $G$-[[principal ∞-bundle]] (see at \emph{[[∞-action]]} for more on this).



\end{enumerate}
Under this identification, the representation theory of $G$ is equivalently

\begin{itemize}%
\item the [[homotopy theory]] in the [[slice (∞,1)-topos]] over $\mathbf{B}G$;


\item the [[homotopy type theory]] in the [[context]] of/[[dependent type theory|dependent on]] $\mathbf{B}G$.



\end{itemize}
More in detail, this yields the following identifications:

[[!include homotopy type representation theory -- table]]

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[representation]], [[action]], [[module]]

\begin{itemize}%
\item [[group]], [[group algebra]], [[groupoid]], [[algebraic group]], [[Lie algebra]]


\item [[vector space]], [[affine space]], [[symplectic vector space]]


\item [[character]]


\item [[action]], [[module]], [[equivariant object]]


\item [[bimodule]], [[Morita equivalence]], [[induced representation]]


\item [[Hilbert space]], [[Banach space]], [[Fourier transform]], [[functional analysis]]


\item [[weight (in representation theory)]]


\item [[orbit]], [[coadjoint orbit]], [[Killing form]]


\item [[geometric quantization]], [[coherent state]]


\item [[socle]], [[quiver]]


\item [[module algebra]], [[comodule algebra]], [[Hopf action]], [[measuring]]



\end{itemize}

\item [[representation ring]]


\item [[irreducible representation]]


\item [[Schur's lemma]]


\item [[Young diagram]]


\item [[Schur index]]


\item [[McKay correspondence]], [[ADE classification]]


\item [[geometric representation theory]]

\begin{itemize}%
\item [[Borel-Weil theorem]], [[Beilinson-Bernstein localization]]
\item [[D-module]], [[perverse sheaf]], [[BBDG decomposition theorem]]
\item [[Kazhdan-Lusztig theory]]
\item [[Dirac induction]]

\end{itemize}

\item [[Verma module]], [[BGG resolution]]


\item [[Grothendieck group]], [[lambda-ring]], [[symmetric function]], [[formal group]]


\item [[principal bundle]], [[torsor]], [[vector bundle]], [[Atiyah Lie algebroid]]


\item [[character sheaf]], [[Harish Chandra transform]]


\item [[geometric function theory]], [[groupoidification]]


\item [[Eilenberg-Moore category]], [[algebra over an operad]], [[actegory]], [[crossed module]]


\item [[reconstruction theorem]]s



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

Lecture notes include

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


\item [[Constantin Teleman]], \emph{Representation theory}, lecture notes 2005 (\href{https://math.berkeley.edu/~teleman/math/RepThry.pdf}{pdf})


\item Joel Robbin, \emph{Real, Complex and Quaternionic representations}, 2006 (\href{http://www.math.wisc.edu/~robbin/angelic/RCH-G.pdf}{pdf}, [[Robbin08RCHRep.pdf:file]])



\end{itemize}
Textbooks include

\begin{itemize}%
\item Charles Curtis, Irving Reiner, \emph{Representation theory of finite groups and associative algebras}, AMS 1962


\item [[William Fulton]], [[Joe Harris]], \emph{Representation Theory: a First Course}, Springer, Berlin, 1991 (\href{http://isites.harvard.edu/fs/docs/icb.topic1381051.files/fulton-harris-representation-theory.pdf}{pdf})


\item Klaus Lux, Herbert Pahlings, \emph{Representations of groups -- A computational approach}, Cambridge University Press 2010 (\href{http://www.math.rwth-aachen.de/~RepresentationsOfGroups/}{author page}, \href{http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521768078}{publisher page})



\end{itemize}
A list of texts on representation theory is maintained at

\begin{itemize}%
\item [[Mikhail Khovanov]], \emph{\href{http://www.math.columbia.edu/~khovanov/resources}{Resources}}.

\end{itemize}
The relation to [[number theory]] and the [[Langlands program]] is discussed in

\begin{itemize}%
\item [[Robert Langlands]], \emph{Representation theory: Its rise and its role in number theory} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.207.3303}{web})

\end{itemize}
[[!redirects linear representation theory]]



\end{document}