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stabilizer group

Context

Group Theory

Representation theory

Contents

Idea

Given an action of a group on some space, and given a point or (or more generally some subspace), then the 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 (Klein 1872) in the context of what has come to be known the Erlangen program. For more on this aspect see at Klein geometry and Cartan geometry.

Definition

Traditional

Given an action G×XXG\times X\to X of a group GG on a set XX, for every element xXx \in X, the stabilizer subgroup of x (also called the isotropy group of xx) is the set of all elements in GG that leave xx fixed:

Stab G(x)={gGgx=x}. Stab_G(x) = \{g \in G \mid g\circ x = x\} \,.

If all stabilizer groups are trivial, then the action is called a free action.

Homotopy-theoretic formulation

We reformulate the tradtitional definition above from the nPOV, in terms of homotopy theory.

A group action ρ:G×XX\rho\colon G \times X \to X is equivalently encoded in its action groupoid fiber sequence in Grpd

XX/GBG, X \to X/G \to \mathbf{B}G \,,

where the X/GX/G is the action groupoid itself, BG\mathbf{B}G is the delooping groupoid of GG and XX is regarded as a 0-truncated groupoid.

This fiber sequence may be thought of as being the ρ\rho-associated bundle to the GG-universal principal bundle. (Here discussed for GG a discrete group but this discussion goes through verbatim for GG a cohesive group).

For

x:*X x\colon * \to X

any global element of XX, we have an induced element x:*XX/Gx\colon * \to X \to X/G of the action groupoid and may hence form the first homotopy group π 1(X/G,x)\pi_1(X/G, x). This is the stabilizer group. Equivalently this is the loop space object of X/GX/G at xx, given by the homotopy pullback

Stab G(x) * x * x X/G. \array{ Stab_G(x) &\to& * \\ \downarrow && \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X/G } \,.

This characterization immediately generalizes to stabilizer ∞-groups of ∞-group actions. This we discuss below

For \infty-group actions

Let H\mathbf{H} be an (∞,1)-topos and GGrp(G)G \in \infty Grp(G) be an ∞-group object in H\mathbf{H}. Write BGH\mathbf{B}G \in \mathbf{H} for its delooping object.

By the discussion at ∞-action we have the following.

Proposition

For XHX \in \mathbf{H} any object, an ∞-action of GG on XX is equivalently an object X/GX/G and a homotopy fiber sequence of the form

X X/G BG. \array{ X &\longrightarrow& X/G \\ && \downarrow \\ && \mathbf{B}G } \,.

Here X/GX/G is the homotopy quotient of the ∞-action

Remark

The action as a morphism X×GXX \times G \to X is recovered from prop. 1 by the (∞,1)-pullback

X×G X X X/G. \array{ X \times G &\to& X \\ \downarrow && \downarrow \\ X &\to& X/G } \,.
Definition

Given an ∞-action ρ\rho of GG on XX as in prop. 1, and given a global element of XX

x:*X x \colon \ast \to X

then the stabilizer \infty-group Stab ρ(x)Stab_\rho(x) of the GG-action at xx is the loop space object

Stab ρ(x)Ω x(X/G). Stab_\rho(x) \coloneqq \Omega_x (X/G) \,.
Remark

Equivalently, def. 1, gives the loop space object of the 1-image BStab ρ(x)\mathbf{B}Stab_\rho(x) of the morphism

*xXX/G. \ast \stackrel{x}{\to} X \to X/G \,.

As such the delooping of the stabilizer \infty-group sits in a 1-epimorphism/1-monomorphism factorization *BStab ρ(x)X/G\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G which combines with the homotopy fiber sequence of prop. 1 to a diagram of the form

* x X X/G epi mono BStab ρ(x) = BStab ρ(x) BG. \array{ \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 } \,.

In particular there is hence a canonical homomorphism of \infty-groups

Stab ρ(x)G. Stab_\rho(x) \longrightarrow G \,.

However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer Stab ρ(x)Stab_\rho(x) is not a sub-group of GG in general.

Examples

For a group acting on itself

For GG any ∞-group in an (∞,1)-topos H\mathbf{H}, its (right) action on itself is given by the looping/delooping fiber sequence

G*ρBG. G \to * \stackrel{\rho}{\to} \mathbf{B}G \,.

Clearly, for every point gGg \in G we have Stab ρ(g)*× ***Stab_{\rho}(g) \simeq * \times_* * \simeq * is trivial. Hence the action is free.

Stabilizers of shapes – Klein geometry

Let XX/GρBGX \to X/G \stackrel{\rho}{\to} \mathbf{B}G be an ∞-action of GG on XX.

Let YHY \in \mathbf{H} any other object, and regard it as equipped with the trivial GG-action YY×BGBGY \to Y \times \mathbf{B}G \to \mathbf{B}G. There is then an induced ∞-action ρ Y\rho_Y on the internal hom [Y,X][Y,X], the conjugation action, given by internal hom in the slice (∞,1)-topos over BG\mathbf{B}G:

[Y,X]BG[Y,X] /BGBG. [Y,X] \to \underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G \,.

Now given any f:YXf \colon Y \to X, then the stabilizer group Stab ρ Y(f)Stab_{\rho_Y}(f) is the stabilizer of YY “in” XX under this GG-action.

The morphism of \infty-groups

i f:Stab ρ Y(f)G i_f\colon Stab_{\rho_Y}(f) \to G

hence characterizes the (higher) Klein geometry induced by the GG-action and by the shape f:YXf\colon Y \to X. (See at Klein geometry – History.)

For completeness we notice that:

Proposition

Here (BG[Y,X] /BGBG)(\underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G ) is equivalently the (∞,1)-pullback ρ Y\rho_Y in

BG[Y,X] /BG [Y,X/G] ρ Y [Y,ρ] BG [Y,BG], \array{ \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] } \,,

where the bottom morphism is the internal hom adjunct of the projection Y×BGBGY \times \mathbf{B}G \to \mathbf{B}G.

Proof

We check the Hom adjunction property, that for any (A/GαBG)BG(A/G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{B}G we have

H /BG(A,[Y,X] BG)H /BG(A× BGY,X) \mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) \simeq \mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)

with [Y,X] /BG[Y,X]_{/\mathbf{B}G} replaced by the above pullback.

Notice that by the GG-action on YY being trivial, we have A× BGY(A/G×Yp 1A/GαBG)H /BGA \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 Hom-spaces in a slice to find H /BG(A,[Y,X] BG)\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G}) as the homotopy pullback on the left of

H /BG(A,[Y,X] BG) H(A/G,[Y,X]/G) H(A/G,[Y,X/G]) H(A/G,ρ Y) H(A/G,[Y,ρ]) * α H(A/G,BG) H(A/G,[Y,BG]), \array{ \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]) } \,,

Now using the Hom-adjunction in H\mathbf{H} itself, the fact that H(A/G,)\mathbf{H}(A/G,-) preserves homotopy pullbacks and the pasting law this is equivalent to

H /BG(A× BGY,X) H(A/G×Y,X/G) * α H(A/G,BG) H(A/G×Y,BG), \array{ \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) } \,,

Here the bottom map is indeed the name of αp 1\alpha \circ p_1 and so again by the pullback characterization of Hom-spaces in a slice this pasting diagram does exhibit H /BG(A× BGY,X)\mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X) as indicated.

For the canonical action on a coset space

Conversely, for any homomorphism HGH \to G of ∞-groups given, then the canonical GG-\infty-action for which HH is the stabilizer \infty-group of a point is the canonical action on (the “coset”) G/HG/H.

This follows from def. 1 by observing that the homotopy fiber sequence of prop. 1 for the GG-action on G/HG/H is just

G/H BH BG \array{ G/H &\stackrel{}{\longrightarrow}& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G }

so that for any point x:*G/Hx \colon \ast \to G/H we have

Stab(x)Ω x(BH)Ω *BHH. Stab(x) \simeq \Omega_{x}(\mathbf{B}H) \simeq \Omega_\ast \mathbf{B}H \simeq H \,.

Stabilizers of coshapes

Dually to “stabilizers of shapes”, as above one may consider stabilizers of “co-shapes”.

I.e. given a GG-action on XX, and given a map f:XAf \colon X \to A, then one may ask for the stabilizer of ff in the canonical GG-action on [X,A][X,A].

For instance if AA here is B nU(1) conn\mathbf{B}^{n}U(1)_{conn}, and f:XB nU(1) connf \colon X \to \mathbf{B}^n U(1)_{conn} is regarded as a prequantum n-bundle ,and [X,A][X,A] is replaced by its differential concretification, then these stabilizers are the quantomorphism n-groups.

to be expanded on

References

An early occurence of the concept of stabilizer subgroups is in p. 4 of

  • Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)

    translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)

(the “Erlangen program”).

Revised on December 18, 2014 17:09:52 by Urs Schreiber (127.0.0.1)