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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 $v, w \in V$.