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A $G$-invariant inner product is constructed by averaging an arbitrary initial inner product $\langle \cdot, \cdot \rangle$ over the group orbit. The new inner product is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.

Conditions: $G$ is a finite group.; $\rho: G \to GL(V)$ is a representation.; $\langle \cdot, \cdot \rangle$ is an arbitrary initial inner product on $V$.; $|G|$ is the order (cardinality) of the finite group $G$.

A $G$-invariant inner product is constructed by averaging an arbitrary initial inner product $\langle \cdot, \cdot \rangle$ over the group orbit. The new inner product is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.

Conditions: $G$ is a finite group.; $\rho: G \to GL(V)$ is a representation.; $\langle \cdot, \cdot \rangle$ is an arbitrary initial inner product on $V$.; $|G|$ is the order (cardinality) of the finite group $G$.

A group-invariant inner product on the representation space $V$ is defined by the condition that applying the same group element $g$ to both vectors does not change their inner product. Mathematically, this is expressed as $(\rho(g)v, \rho(g)w) = (v, w)$ for all $g \in G$ and all $v, w \in V$.

Conditions: $V$ is a vector space with an inner product $(\cdot, \cdot)$.; $\rho: G \to GL(V)$ is a representation.; The condition must hold for all $g \in G$ and all $v, w \in V$.

A group-invariant inner product on the representation space $V$ is defined by the condition that applying the same group element $g$ to both vectors does not change their inner product. Mathematically, this is expressed as $(\rho(g)v, \rho(g)w) = (v, w)$ for all $g \in G$ and all $v, w \in V$.

Conditions: $V$ is a vector space with an inner product $(\cdot, \cdot)$.; $\rho: G \to GL(V)$ is a representation.; The condition must hold for all $g \in G$ and all $v, w \in V$.