nLab conjugation action

1. Idea

For $G$ a group and $V_1, V_2$ pair of objects equipped with $G$-action, the conjugation action of $G$ on a morphism $f \colon V_1 \to V_2$ (not necessarily respecting the $G$-action) is for $g \in G$ given by

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

The invariants (fixed points) of the conjugation action are the maps which do respect the $G$-action, hence the homomorphisms for the $G$-action, hence the $G$-equivariant maps.

In the case that the $G$-action on $V_2$ is trivial, the conjugation action becomes the precomposition action and in the case that the action on $V_1$ is trivial it becomes the postcomposition action.

In matrix calculus conjugation actions are also known as similarity transformations.

2. Definition

Definition 2.1. Given a discrete group $G$ and two $G$-actions $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then the function set $[S_1, S_2]$ is naturally equipped with the conjugation action

$$Ad \;\colon \; [S_1, S_2] \times G \longrightarrow [S_1,S_2]$$

which takes $((S_1 \stackrel{f}{\to} S_2), g)$ to

$$\rho_2(-)(g)\circ f \circ \rho_1(-)(g^{-1}) \;\colon\; S_1 \stackrel{\rho_1(-)(g^{-1})}{\longrightarrow} S_1 \stackrel{f}{\longrightarrow} S_2\stackrel{\rho_2(-)(g)}{\longrightarrow} S_2 \,.$$

Proposition 2.2. The conjugation action construction of def. 2.1 is the internal hom in the category of actions.

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

$$\phi \;\colon\; S_3 \longrightarrow [S_1,S_2]$$

which intertwine the $G$-action on $S_3$ with the conjugation action on $[S_1,S_2]$ are in natural bijection with functions

$$\tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2$$

which intertwine the diagonal action on the Cartesian product $S_3 \times S_1$ with the action on $S_2$.

The condition on $\phi$ means that for all $g\in G$ and $s_3 \in S_3$ it sends

$$\phi \;\colon\; \rho_3(s_3)(g) \mapsto \left( s_1 \mapsto \rho_2\left( \phi\left(s_3\right)\left( \rho_1\left(s_1\right)\left(g^{-1}\right) \right)\right)\left(g\right) \right) \,.$$

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

$$\tilde \phi \;\colon\; (\rho_3(s_3)(g), s_1) \mapsto \rho_2 ( \phi(s_3)( \rho_1(s_1)(g^{-1}) ) )(g) \,.$$

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

$$\tilde \phi \;\colon\; (\rho_3(s_3)(g), \rho_1(s_1)(g)) \mapsto \rho_2(\phi(s_3)(s_1 ))(g) \,.$$

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

The following is immediate but conceptually important:

Proposition 2.3. The invariants of the conjugation action on $[S_1,S_2]$ 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 $G$-action into the set underlying the internal hom

$$G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2] \,.$$

Remark 2.4. 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.

3. Related concepts

• inner automorphism

• adjoint action, free loop space of classifying space

• conjugate subgroup