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conjugation action

Contents

Idea

For GG a group and V 1V_1 and V 2V_2 two objects equipped with GG-action, the conjugation action on morphisms f:V 1V 2f : V_1 \to V_2 (not necessarily respecting the GG-action) is for gGg \in G

fgfg 1. f \mapsto g \circ f \circ g^{-1} \,.

The invariants of the conjugation action are the GG-action homomorphism.

Definition

Definition

Given a discrete group GG and two GG-actions ρ 1\rho_1 and ρ 2\rho_2 on sets S 1S_1 and S 2S_2, respectively, then the function set [S 1,S 2][S_1, S_2] is naturally equipped with the conjugation action

Ad:[S 1,S 2]×G[S 1,S 2] Ad \;\colon \; [S_1, S_2] \times G \longrightarrow [S_1,S_2]

which takes ((S 1fS 2),g)((S_1 \stackrel{f}{\to} S_2), g) to

ρ 2()(g)fρ 1()(g 1):S 1ρ 1()(g 1)S 1fS 2ρ 2()(g)S 2. \rho_2(-)(g)\circ f \circ \rho_1(-)(g^{-1}) \;\colon\; S_1 \stackrel{\rho_1(-)(g^{-1})}{\longrightarrow} S_1 \stackrel{f}{\longrightarrow} S_2\stackrel{\rho_2(-)(g)}{\longrightarrow} S_2 \,.
Proposition

The conjugation action construction of def. 1 is the internal hom in the category of actions.

Proof

We need to show that for any three permutation representations, functions

ϕ:S 3[S 1,S 2] \phi \;\colon\; S_3 \longrightarrow [S_1,S_2]

which intertwine the GG-action on S 3S_3 with the conjugation action on [S 1,S 2][S_1,S_2] are in natural bijection with functions

ϕ˜:S 3×S 1S 2 \tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2

which intertwine the diagonal action on the Cartesian product S 3×S 1S_3 \times S_1 with the action on S 2S_2.

The condition on ϕ\phi means that for all gGg\in G and s 3S 3s_3 \in S_3 it sends

ϕ:ρ 3(s 3)(g)(s 1ρ 2(ϕ(s 3)(ρ 1(s 1)(g 1)))(g)). \phi \;\colon\; \rho_3(s_3)(g) \mapsto \left( s_1 \mapsto \rho_2\left( \phi\left(s_3\right)\left( \rho_1\left(s_1\right)\left(g^{-1}\right) \right)\right)\left(g\right) \right) \,.

This is equivalently a function ϕ˜\tilde \phi of two variables which sends

ϕ˜:(ρ 3(s 3)(g),s 1)ρ 2(ϕ(s 3)(ρ 1(s 1)(g 1)))(g). \tilde \phi \;\colon\; (\rho_3(s_3)(g), s_1) \mapsto \rho_2 ( \phi(s_3)( \rho_1(s_1)(g^{-1}) ) )(g) \,.

Since this has to hold for all values of the variables, it has to hold when substituing s 1s_1 with ρ 1(s 1)(g)\rho_1(s_1)(g). After this substitution the above becomes

ϕ˜:(ρ 3(s 3)(g),ρ 1(s 1)(g))ρ 2(ϕ(s 3)(s 1))(g). \tilde \phi \;\colon\; (\rho_3(s_3)(g), \rho_1(s_1)(g)) \mapsto \rho_2(\phi(s_3)(s_1 ))(g) \,.

This is the intertwining condition on ϕ˜\tilde \phi. Conversely, given ϕ˜\tilde \phi satisfying this for all values of the variables, then running the argument backwards shows that its hom-adjunct ϕ\phi satisfies its required intertwining condition.

The following is immediate but conceptually important:

Proposition

The invariants of the conjugation action on [S 1,S 2][S_1,S_2] is the set of action homomorphisms/intertwiners.

Hence the inclusion of invariants into the conjugation action gives the inclusion of the external hom set of the category of GG-action into the set underlying the internal hom

GAct(ρ 1,ρ 2)[ρ 1,ρ 2]. G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2] \,.
Remark

Regarding the conjugation action as the internal hom of actions immediately gives the generalization of this concept to more general kinds of actions, notably to infinity-actions in general (infinity,1)-toposes. See at infinity-action – Conjugation action for more on this.

Revised on April 16, 2015 16:47:27 by Urs Schreiber (195.113.31.253)