Equivariant Map of Group Representations
Definition1 2
Suppose we are given a group $G$ and two representations $\rho_{1} : G \to \operatorname{GL}(V)$, $\rho_{2} : G \to \operatorname{GL}(W)$. If a function $f : V \to W$ between two vector spaces satisfies the following, then $f$ is said to be equivariant with respect to $G$.
$$ f(\rho_{1}g (v)) = \rho_{2}g(f(v)), \qquad \forall g \in G, v \in V $$
Explanation
Names for such an $f$ include $G–$equivariant map, $G–$linear map,
If $f$ is an equivariant map, then applying a change on the domain and then substituting it into the function gives the same result as substituting into the function first and then applying a change on the range. Put simply, a function that yields the same result even when the order in which it and the given representation are applied is swapped is called an equivariant map.
Invariant Map
If $\rho_{2}g$ is the identity function for all $g$, then $f$ is said to be invariant. That is, the value of $f$ does not change even when the action of $G$ is applied to the domain.
$$ f(\rho_{1}g (v)) = f(v), \qquad \forall g \in G, v \in V $$
Example
- $G = \braket{R_{90^{\circ}}} = \left\{ R_{0^{\circ}}, R_{90^{\circ}}, R_{180^{\circ}}, R_{270^{\circ}} \right\}$ the cyclic rotation group
- $V = \mathbb{R}^{2}$ the coordinate plane
- $W = \mathbb{R}^{2}$ the coordinate plane
- $\rho_{1}: G \to \operatorname{GL}(V)$ is $$ \rho_{1}(R_{\theta^{\circ}}) = R_{\theta} \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$
- $\rho_{2}: G \to \operatorname{GL}(V)$ is $$ \rho_{2}(R_{\theta^{\circ}}) = R_{\theta} \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$
- $f : V \to W$ is the transformation that changes the magnitude of $(x,y)$ for some $\phi: \mathbb{R}^{2} \to \mathbb{R}$, given by $f(x,y) = \phi(\|(x, y)\|)(x,y)$
Then, since the following holds, $f$ is equivariant with respect to $G$.
$$ \begin{align*} f(R_{\theta}(x,y)) &= \phi(\|R_{\theta}(x,y)\|)R_{\theta}(x,y) \\ &= \phi(\|(x,y)\|)R_{\theta}(x,y) \\ &= R_{\theta}(\phi(\|(x,y)\|)(x,y)) \\ &= R_{\theta}f(x,y) \end{align*} $$
Properties
(a) If two functions $f$, $g : V \to W$ are equivariant with respect to $G$, then $f \circ g$ is also equivariant.
(b) If $\phi : V \to W$ is $G–$equivariant, then $\ker \phi$ and $\im \phi$ are subrepresentations. Here $\ker \phi$, $\im \phi$ are the kernel and image of $\phi$, respectively.
Statement (b) above is written concisely; spelled out in as much detail as possible, it reads as follows.
(b) For a group $G$, suppose we are given two representations $(\rho, V)$ and $(\sigma, W)$. For convenience, write $\rho_{g} = \rho(g)$. Suppose a linear transformation $\phi : V \to W$ is $G–$equivariant.
$$ \phi(\rho_{g} (v)) = \sigma_{g} (\phi(v)) \quad \forall v \in V, \quad g \in G $$
Then $\ker \phi$, $\im \phi$ are invariant subspaces of $(\rho, V)$ and $(\sigma, W)$, respectively.
$$ \rho_{g} (\ker \phi) \subset \ker \phi \\ \sigma_{g} (\im \phi) \subset \im \phi $$
Proof
(b)
$\ker \phi$ is a subrepresentation.
W.T.S.: $v \in \ker \phi \implies \rho_{g}(v) \in \ker \phi$
Let $v \in \ker \phi = \left\{ v : \phi(v) = 0_{W} \right\}$. Here $0_{W}$ is the zero vector of $W$. Then, since $\phi$ is $G–$invariant, the following holds.
$$ \begin{align*} \phi(\rho_{g}(v)) &= \sigma_{g}(\phi(v)) \\ &= \sigma_{g}(0_{W}) \\ &= 0_{W} \end{align*} $$
$$ \rho_{g}(v) \in \ker \phi $$
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$\im \phi$ is a subrepresentation.
W.T.S.: $w \in \im \phi \implies \sigma_{g}(w) \in \im \phi$
Let $w \in \im \phi$. That is, $w = \phi(v)$ for some $v$. Then, since $\phi$ is $G–$invariant, the following holds.
$$ \sigma_{g}(w) = \sigma_{g}(\phi(v)) \\ = \phi(\rho_{g}(v)) $$
$$ \sigma_{g}(w) \in \im \phi $$
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