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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{stabilizer group} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{TraditionalDefinition}{Traditional}\dotfill \pageref*{TraditionalDefinition} \linebreak \noindent\hyperlink{GeneralAbstract}{Homotopy-theoretic formulation}\dotfill \pageref*{GeneralAbstract} \linebreak \noindent\hyperlink{ForInfinityGroupActions}{For $\infty$-group actions}\dotfill \pageref*{ForInfinityGroupActions} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{for_a_group_acting_on_itself}{For a group acting on itself}\dotfill \pageref*{for_a_group_acting_on_itself} \linebreak \noindent\hyperlink{KleinGeometry}{Stabilizers of shapes -- Klein geometry}\dotfill \pageref*{KleinGeometry} \linebreak \noindent\hyperlink{ForCanonicalActionOnCosetSpace}{For the canonical action on a coset space}\dotfill \pageref*{ForCanonicalActionOnCosetSpace} \linebreak \noindent\hyperlink{StabilizersOfCoShapes}{Stabilizers of coshapes}\dotfill \pageref*{StabilizersOfCoShapes} \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} Given an [[action]] of a [[group]] on some [[space]], and given a point or (or more generally some subspace), then the \emph{stabilizer group} of that point (that subspace) is the [[subgroup]] whose action leaves the point (the subspace) fixed, [[invariant]]. The importance of stabilizer subgroups for the general development of [[geometry]] was famously highlighted in (\hyperlink{Klein1872}{Klein 1872}) in the context of what has come to be known the \emph{[[Erlangen program]]}. For more on this aspect see at \emph{[[Klein geometry]]} and \emph{[[Cartan geometry]]}. Sometimes (such as in the context of [[Wigner classification]]) stabilizer groups are called \emph{little groups}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{TraditionalDefinition}{}\subsubsection*{{Traditional}}\label{TraditionalDefinition} Given an [[action]] $G\times X\to X$ of a [[group]] $G$ on a set $X$, for every element $x \in X$, the \textbf{stabilizer subgroup} of $x$ (also called the \textbf{isotropy group} of $x$) is the set of all elements in $G$ that leave $x$ fixed: \begin{displaymath} Stab_G(x) = \{g \in G \mid g\circ x = x\} \,. \end{displaymath} If all stabilizer groups are trivial, then the action is called a \emph{free action}. \hypertarget{GeneralAbstract}{}\subsubsection*{{Homotopy-theoretic formulation}}\label{GeneralAbstract} We reformulate the traditional definition \hyperlink{TraditionalDefinition}{above} from the [[nPOV]], in terms of [[homotopy theory]]. A [[group]] [[action]] $\rho\colon G \times X \to X$ is equivalently encoded in its [[action groupoid]] [[fiber sequence]] in [[Grpd]] \begin{displaymath} X \to X//G \to \mathbf{B}G \,, \end{displaymath} where the $X//G$ is the [[action groupoid]] itself, $\mathbf{B}G$ is the [[delooping]] [[groupoid]] of $G$ and $X$ is regarded as a [[0-truncated]] groupoid. This fiber sequence may be thought of as being the $\rho$-[[associated bundle]] to the $G$-[[universal principal bundle]]. (Here discussed for $G$ a [[discrete group]] but this discussion goes through verbatim for $G$ a \href{/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#InfinGroups}{cohesive group}.) For \begin{displaymath} x\colon * \to X \end{displaymath} any [[global element]] of $X$, we have an induced element $x\colon * \to X \to X//G$ of the action groupoid and may hence form the first [[homotopy group]] $\pi_1(X//G, x)$. This is the stabilizer group. Equivalently this is the [[loop space object]] of $X//G$ at $x$, given by the [[homotopy pullback]] \begin{displaymath} \itexarray{ Stab_G(x) &\to& * \\ \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X//G } \,. \end{displaymath} This characterization immediately generalizes to stabilizer [[∞-groups]] of \href{/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#GroupRepresentations}{∞-group actions}. This we discuss \hyperlink{ForInfinityGroupActions}{below} \hypertarget{ForInfinityGroupActions}{}\subsubsection*{{For $\infty$-group actions}}\label{ForInfinityGroupActions} Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $G \in \infty Grp(G)$ be an [[∞-group]] object in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its [[delooping]] object. By the discussion at \emph{[[∞-action]]} we have the following. \begin{prop} \label{InfinityAction}\hypertarget{InfinityAction}{} For $X \in \mathbf{H}$ any object, an [[∞-action]] of $G$ on $X$ is equivalently an object $X/G$ and a [[homotopy fiber sequence]] of the form \begin{displaymath} \itexarray{ X &\longrightarrow& X//G \\ && \downarrow \\ && \mathbf{B}G } \,. \end{displaymath} \end{prop} Here $X/G$ is the [[homotopy quotient]] of the [[∞-action]] \begin{remark} \label{}\hypertarget{}{} The action as a morphism $X \times G \to X$ is recovered from prop. \ref{InfinityAction} by the [[(∞,1)-pullback]] \begin{displaymath} \itexarray{ X \times G &\to& X \\ \downarrow && \downarrow \\ X &\to& X//G } \,. \end{displaymath} \end{remark} \begin{defn} \label{StabilizerInInfinityTopos}\hypertarget{StabilizerInInfinityTopos}{} Given an [[∞-action]] $\rho$ of $G$ on $X$ as in prop. \ref{InfinityAction}, and given a [[global element]] of $X$ \begin{displaymath} x \colon \ast \to X \end{displaymath} then the \textbf{stabilizer $\infty$-group} $Stab_\rho(x)$ of the $G$-action at $x$ is the [[loop space object]] \begin{displaymath} Stab_\rho(x) \coloneqq \Omega_x (X//G) \,. \end{displaymath} \end{defn} \begin{defn} \label{}\hypertarget{}{} Equivalently, def. \ref{StabilizerInInfinityTopos}, gives the [[loop space object]] of the [[1-image]] $\mathbf{B}Stab_\rho(x)$ of the morphism \begin{displaymath} \ast \stackrel{x}{\to} X \to X//G \,. \end{displaymath} As such the [[delooping]] of the stabilizer $\infty$-group sits in a [[1-epimorphism]]/[[1-monomorphism]] factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X//G$ which combines with the homotopy fiber sequence of prop. \ref{InfinityAction} to a diagram of the form \begin{displaymath} \itexarray{ \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X//G \\ \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow \\ \mathbf{B} Stab_\rho(x) &=& \mathbf{B} Stab_\rho(x) &\longrightarrow& \mathbf{B}G } \,. \end{displaymath} In particular there is hence a canonical homomorphism of $\infty$-groups \begin{displaymath} Stab_\rho(x) \longrightarrow G \,. \end{displaymath} However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a \emph{sub}-group of $G$ in general. \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{for_a_group_acting_on_itself}{}\subsubsection*{{For a group acting on itself}}\label{for_a_group_acting_on_itself} For $G$ any [[∞-group]] in an [[(∞,1)-topos]] $\mathbf{H}$, its (right) [[action]] on itself is given by the [[looping and delooping|looping/delooping]] [[fiber sequence]] \begin{displaymath} G \to * \stackrel{\rho}{\to} \mathbf{B}G \,. \end{displaymath} Clearly, for every point $g \in G$ we have $Stab_{\rho}(g) \simeq * \times_* * \simeq *$ is trivial. Hence the action is free. \hypertarget{KleinGeometry}{}\subsubsection*{{Stabilizers of shapes -- Klein geometry}}\label{KleinGeometry} Let $X \to X//G \stackrel{\rho}{\to} \mathbf{B}G$ be an [[∞-action]] of $G$ on $X$. Let $Y \in \mathbf{H}$ any other object, and regard it as equipped with the trivial $G$-action $Y \to Y \times \mathbf{B}G \to \mathbf{B}G$. There is then an induced [[∞-action]] $\rho_Y$ on the [[internal hom]] $[Y,X]$, the [[conjugation action]], given by internal hom in the [[slice (∞,1)-topos]] over $\mathbf{B}G$: \begin{displaymath} [Y,X] \to \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G \,. \end{displaymath} Now given any $f \colon Y \to X$, then the stabilizer group $Stab_{\rho_Y}(f)$ is the stabilizer of $Y$ ``in'' $X$ under this $G$-action. The morphism of $\infty$-groups \begin{displaymath} i_f\colon Stab_{\rho_Y}(f) \to G \end{displaymath} hence characterizes the ([[higher Klein geometry|higher]]) [[Klein geometry]] induced by the $G$-action and by the shape $f\colon Y \to X$. (See at \href{Klein+geometry#History}{Klein geometry -- History}.) For completeness we notice that: \begin{prop} \label{ReformulationOfRightSidedConjugationAction}\hypertarget{ReformulationOfRightSidedConjugationAction}{} Here $(\underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G )$ is equivalently the [[(∞,1)-pullback]] $\rho_Y$ in \begin{displaymath} \itexarray{ \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} &\to& [Y, X//G] \\ \downarrow^{\mathrlap{\rho_Y}} && \downarrow^{\mathrlap{[Y, \rho]}} \\ \mathbf{B}G &\to& [Y, \mathbf{B}G] } \,, \end{displaymath} where the bottom morphism is the [[internal hom]] [[adjunct]] of the [[projection]] $Y \times \mathbf{B}G \to \mathbf{B}G$. \end{prop} \begin{proof} We check the Hom adjunction property, that for any $(A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{B}G$ we have \begin{displaymath} \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) \simeq \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X) \end{displaymath} with $[Y,X]_{/\mathbf{B}G}$ replaced by the above pullback. Notice that by the $G$-action on $Y$ being trivial, we have $A \times_{\mathbf{B}G} Y \simeq (A//G \times Y \stackrel{p_1}{\to} A//G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$. Then use the characterization of \href{over-%28infinity%2C1%29-category#HamSpacesInASlice}{Hom-spaces in a slice} to find $\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})$ as the [[homotopy pullback]] on the left of \begin{displaymath} \itexarray{ \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) &\longrightarrow& \mathbf{H}(A//G, [Y,X]//G) &\to& \mathbf{H}(A//G,[Y, X//G]) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{H}(A//G,\rho_Y)}} && \downarrow^{\mathrlap{\mathbf{H}(A//G,[Y, \rho])}} \\ \ast &\stackrel{\vdash \alpha}{\longrightarrow}& \mathbf{H}(A//G,\mathbf{B}G) &\to& \mathbf{H}(A//G,[Y, \mathbf{B}G]) } \,, \end{displaymath} Now using the Hom-adjunction in $\mathbf{H}$ itself, the fact that $\mathbf{H}(A//G,-)$ preserves [[homotopy pullbacks]] and the [[pasting law]] this is equivalent to \begin{displaymath} \itexarray{ \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X) &\longrightarrow& &\longrightarrow& \mathbf{H}(A//G \times Y, X //G) \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{\vdash \alpha}{\longrightarrow}& \mathbf{H}(A//G,\mathbf{B}G) &\to& \mathbf{H}(A//G \times Y, \mathbf{B}G) } \,, \end{displaymath} Here the bottom map is indeed the name of $\alpha \circ p_1$ and so again by the pullback characterization of \href{over-%28infinity%2C1%29-category#HamSpacesInASlice}{Hom-spaces in a slice} this pasting diagram does exhibit $\mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)$ as indicated. \end{proof} \hypertarget{ForCanonicalActionOnCosetSpace}{}\subsubsection*{{For the canonical action on a coset space}}\label{ForCanonicalActionOnCosetSpace} Conversely, for any [[homomorphism]] $H \to G$ of [[∞-groups]] given, then the canonical $G$-$\infty$-action for which $H$ is the stabilizer $\infty$-group of a point is the canonical action on (the ``[[coset]]'') $G/H$. This follows from def. \ref{StabilizerInInfinityTopos} by observing that the homotopy fiber sequence of prop. \ref{InfinityAction} for the $G$-action on $G/H$ is just \begin{displaymath} \itexarray{ G/H &\stackrel{}{\longrightarrow}& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G } \end{displaymath} so that for any point $x \colon \ast \to G/H$ we have \begin{displaymath} Stab(x) \simeq \Omega_{x}(\mathbf{B}H) \simeq \Omega_\ast \mathbf{B}H \simeq H \,. \end{displaymath} \hypertarget{StabilizersOfCoShapes}{}\subsubsection*{{Stabilizers of coshapes}}\label{StabilizersOfCoShapes} Dually to ``stabilizers of shapes'', as \hyperlink{KleinGeometry}{above} one may consider stabilizers of ``co-shapes''. I.e. given a $G$-action on $X$, and given a map $f \colon X \to A$, then one may ask for the stabilizer of $f$ in the canonical $G$-action on $[X,A]$. For instance if $A$ here is $\mathbf{B}^{n}U(1)_{conn}$, and $f \colon X \to \mathbf{B}^n U(1)_{conn}$ is regarded as a [[prequantum n-bundle]] ,and $[X,A]$ is replaced by its [[differential concretification]], then these stabilizers are the [[quantomorphism n-groups]]. \begin{quote}% to be expanded on \end{quote} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[normalizer subgroup]] \item [[centralizer subgroup]] \item [[stable point]] \item [[Klein geometry]], [[Cartan geometry]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} An early occurence of the concept of stabilizer subgroups is in p. 4 of \begin{itemize}% \item [[Felix Klein]], \emph{Vergleichende Betrachtungen \"u{}ber neuere geometrische Forschungen} (1872) translation by M. W. Haskell, \emph{A comparative review of recent researches in geometry} , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (\href{http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf}{retyped pdf}, [[KleinRetyped.pdf:file]], \href{http://math.ucr.edu/home/baez/erlangen/erlangen.pdf}{scan of original}) \end{itemize} (the ``[[Erlangen program]]''). [[!redirects stabilizer group]] [[!redirects stabilizer groups]] [[!redirects stabiliser group]] [[!redirects stabiliser groups]] [[!redirects stabilizer]] [[!redirects stabiliser]] [[!redirects stabilizers]] [[!redirects stabilisers]] [[!redirects stabilizer ∞-group]] [[!redirects stabilizer infinity-group]] [[!redirects stabiliser ∞-group]] [[!redirects stabiliser infinity-group]] [[!redirects stabilizer ∞-groups]] [[!redirects stabilizer infinity-groups]] [[!redirects stabiliser ∞-groups]] [[!redirects stabiliser infinity-groups]] [[!redirects homotopy stabilizer group]] [[!redirects homotopy stabilizer groups]] [[!redirects stabilizer subgroup]] [[!redirects stabiliser subgroup]] [[!redirects stabilizer subgroups]] [[!redirects stabiliser subgroups]] [[!redirects isotropy group]] [[!redirects isotropy groups]] [[!redirects isotropy subgroup]] [[!redirects isotropy subgroups]] [[!redirects stabilizer 2-group]] [[!redirects stabilizer 2-groups]] [[!redirects little group]] [[!redirects little groups]] \end{document}