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\begin{document}

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\section*{projective representation}

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

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

[[!include representation theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{GroupExtensionAndCocycle}{The group extension and its cocycle}\dotfill \pageref*{GroupExtensionAndCocycle} \linebreak
\noindent\hyperlink{RelationToGenuineRepresentations}{Relation to genuine representations}\dotfill \pageref*{RelationToGenuineRepresentations} \linebreak
\noindent\hyperlink{as_twisted_linear_representations}{As twisted linear representations}\dotfill \pageref*{as_twisted_linear_representations} \linebreak
\noindent\hyperlink{as_genuine_representations_after_extensions}{As genuine representations after extensions}\dotfill \pageref*{as_genuine_representations_after_extensions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \emph{projective representation} of a [[group]] $G$ is a [[representation]] up to a central term: a [[group]] [[homomorphism]] $G\longrightarrow PGL(V)$, to the [[projective general linear group]] of some $\mathbb{K}$-[[vector space]] $V$.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{GroupExtensionAndCocycle}{}\subsubsection*{{The group extension and its cocycle}}\label{GroupExtensionAndCocycle}

By construction, there is a [[short exact sequence]]

\begin{displaymath}
1 \to \mathbb{K}^\times \longrightarrow GL(V) \longrightarrow PGL(V)\to 1
\end{displaymath}
which exhibits $GL(V)$ as a [[group extension]] of $PGl(V)$ by the [[group of units]] $\mathbb{K}^\times$ of the [[ground field]]. This is classified by a 2-[[cocycle]] $c$ in [[group cohomology]] $H^2_{Grp}(PGL(V),\mathbb{K}^\times)$.

It is useful to re-express this equivalently in terms of [[homotopy theory]] via the discussion at [[looping and delooping]], by which [[group]] [[homomorphisms]] $\phi \colon G\longrightarrow H$ are equivalently maps $\mathbf{B}\phi \colon \mathbf{B}G\longrightarrow \mathbf{B}H$ between their [[delooping]] [[groupoids]].

In terms of this the above [[group extension]] and its classifying cocycle is exhibited by a [[homotopy fiber sequence]] of [[deloopings]] of the form

\begin{displaymath}
\itexarray{
    \mathbf{B}\mathbb{K}^*& \longrightarrow & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \ast &\longrightarrow&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
  \,.
\end{displaymath}
\hypertarget{RelationToGenuineRepresentations}{}\subsubsection*{{Relation to genuine representations}}\label{RelationToGenuineRepresentations}

Via the projection $GL(V)\to PGL(V)=GL(V)/\mathbb{K}^\times$, every linear representation of $G$ induces a projective representation.

By the [[universal property]] of the [[homotopy pullback]], the discussion \hyperlink{GroupExtensionAndCocycle}{above} means that the [[obstruction]] to lift a given projective representation $\mathbf{B}\rho \colon \mathbf{B}G\longrightarrow \mathbf{B} PGL(V)$ to a linear representation $\hat \rho$

\begin{displaymath}
\itexarray{
    &  & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    & {}^{\mathllap{\mathbf{B}\hat \rho}}\nearrow & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
\end{displaymath}
is the class of the 2-[[cocycle]] $c(\rho)\coloneqq c \circ \mathbf{B}\rho$ in the [[group cohomology]] class $H^2_{Grp}(G,\mathbb{K}^\times)$.

\hypertarget{as_twisted_linear_representations}{}\subsubsection*{{As twisted linear representations}}\label{as_twisted_linear_representations}

By the \hyperlink{GroupExtensionAndCocycle}{above} and the discussion at \emph{\href{group+extension#FormulationInHomotopyTheory}{group extension -- Central group extensions -- Formulation in homotopy theory}} the cocycle map $\mathbf{B}c \colon \mathbf{B}PGL(V)\to \mathbf{B}^2 \mathbb{K}^\times$ of [[homotopy types]] may be represented by a [[zigzag]] (``[[infinity-anafunctor|2-anafunctor]]'') of [[crossed complexes]] as

\begin{displaymath}
\itexarray{
    \mathbf{B}(\mathbb{K}^\times \to GL(V)) &\longrightarrow& \mathbf{B}(\mathbb{K}^\times \to 1) 
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}PGL(V)
  }
  \,.
\end{displaymath}
Here the [[2-groupoid]] $\mathbf{B}(\mathbb{K}^\times \to GL(V))$ looks schematically like

\begin{displaymath}
\left\{   
     \itexarray{
       && \ast
       \\
       & {}^{\mathllap{p_1}}\nearrow &\Downarrow^{\mathrlap{k}}& \searrow^{\mathrlap{p_2}}
       \\
       \ast && \underset{k p_1 p_2}{\longrightarrow}&& \ast
     }
  \right\}
\end{displaymath}
This shows that a map $\mathbf{B}G\to \mathbf{B}PGL(V)$ may equivalently be represented by two [[functions]] (not group homomorphisms in general!)

\begin{enumerate}%
\item $\rho \colon G\to GL(V)$


\item $\lambda \colon G\times G\to \mathbb{K}^\times$



\end{enumerate}
such that for all $g,h \in G$

\begin{enumerate}%
\item $\rho(g)\rho(h)=\lambda(g,h)\rho(g h)$


\item $\lambda$ is a [[group cohomology|group 2-cocycle]] on $G$ with values in $\mathbb{K}^*$ representing the above cohomology class of $c_\rho$.



\end{enumerate}
This is the form in which projective representations are often discussed in the literature.

\hypertarget{as_genuine_representations_after_extensions}{}\subsubsection*{{As genuine representations after extensions}}\label{as_genuine_representations_after_extensions}

Alternatively, one may consider the \hyperlink{RelationToGenuineRepresentations}{above} diagram

\begin{displaymath}
\itexarray{
    &  & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    & & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
\end{displaymath}
and form the further [[pullback]] along $\mathbf{B}\rho$. By the [[pasting law]] this is (the [[delooping]] of) the [[group extension]] $\hat G$ of $G$ which is classified by $c(\rho)$:

\begin{displaymath}
\itexarray{
    \mathbf{B}\hat G  & \stackrel{\mathbf{B}\tilde \rho}{\longrightarrow} & \mathbf{B}GL(V) &\longrightarrow& \ast
    \\
    \downarrow & & \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times
}
\end{displaymath}
This way the projective representation $\rho$ of $G$ induces a genuine linear representation $\tilde \rho$ of $\hat G$. One finds (this is a special case of the general discussion at [[twisted infinity-bundle]]) that this constitutes an equivalence between projective representations of $G$ and genuine representations of $\hat G$.

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

\begin{itemize}%
\item [[projectively flat connection]]

\end{itemize}
[[!redirects projective representations]]



\end{document}