PhyslibAlpha.ProbabilisticTheory.Representation.Schur

Let $E$ be a real vector space and $G$ be a group acting on $E$ via a map $(g, x) \mapsto g \cdot x$. A submodule $W \subseteq E$ is said to be a **Schur block** if every linear endomorphism $f: E \to E$ that is $G$-equivariant on $W$ (meaning $f(g \cdot x) = g \cdot f(x)$ for all $g \in G, x \in W$) and leaves $W$ invariant ($f(W) \subseteq W$) acts as a scalar multiplication on $W$. That is, there exists a constant $c \in \mathbb{R}$ such that $f(x) = c \cdot x$ for all $x \in W$.

Let $E$ be a real vector space and $G$ be a group acting on $E$ via a map $(g, x) \mapsto g \cdot x$. Suppose $E$ is the internal direct sum of a family of $G$-invariant submodules $\{W_i\}_{i \in \iota}$ (i.e., $E = \bigoplus_{i \in \iota} W_i$ and $g \cdot W_i \subseteq W_i$ for all $g \in G$). If each $W_i$ is a **Schur block**, then for any $G$-equivariant linear map $f: E \to E$ that preserves each block (i.e., $f(W_i) \subseteq W_i$), there exists a collection of scalars $\{c_i\}_{i \in \iota}$ in $\mathbb{R}$ such that for every $i \in \iota$, the restriction of $f$ to $W_i$ is multiplication by $c_i$: $$\forall i \in \iota, \forall x \in W_i, f(x) = c_i x$$

Let $E$ be a real vector space and $G$ a group acting on $E$ through a symmetry representation $\rho: G \to \text{Sym}(E)$. Let $\phi: E \to E$ be a channel that is covariant with respect to $\rho$, meaning $\phi(\rho(g)x) = \rho(g)\phi(x)$ for all $g \in G$ and $x \in E$. Suppose $E$ admits a direct sum decomposition $E = \bigoplus_{i \in \iota} W_i$ such that: 1. Each submodule $W_i$ is invariant under the action of $G$ (i.e., $\rho(g)x \in W_i$ for all $g \in G$ and $x \in W_i$). 2. Each $W_i$ is a **Schur block**, meaning any $G$-equivariant linear map that preserves $W_i$ acts as a scalar multiplication on $W_i$. 3. The channel $\phi$ preserves each block $W_i$ (i.e., $\phi(x) \in W_i$ for all $x \in W_i$). Then there exist real scalars $c_i \in \mathbb{R}$ for each $i \in \iota$ such that for all $x \in W_i$, the channel acts as $\phi(x) = c_i x$.