Dual Representation Preserves Natural Pairings Between Vectors and Their Duals
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Definition
Where $\rho$ is a representation of $G$ on a finite-dimensional vector space $V$ -- that is, $\rho$ is a homomorphism from $G$ to $GL(V)$ -- and the dual space $V^*$ is the set of homomorphisms of $V$, let the dual representation $\rho:V^* \rightarrow V^*$, a permutation of $V^*$, be defined by the formula $\rho^*(g)=\rho(g^{-1})^T$.
Recognizing the natural pairing between elements of $V$ and $V^*$, we will refer to a pairing function, $n(v^*)=v$, between these elements, which we will also refer to by the notation $\langle v^*, v\rangle$.
Theorem
We want to show that for all $g \in G$, $v \in V$, and $v^* \in V^*$ such that $\langle v^*, v\rangle$, $ \langle \rho^*(g)(v^*),\rho(g)(v) \rangle$; in other words, that $n(\rho^*(g)(v^*)) = \rho(g)(v)$. To that end, we will use properties and definitions to rewrite $n(\rho^*(g)(v^*))$.
| \(\ds n(\rho^*(g)(v^*))\) | \(=\) | \(\ds n(\rho(g^{-1})^T(v*))\) | Definition of $\rho^*(g)$ from above | |||||||||||
| \(\ds \) | \(=\) | \(\ds n(\rho(g)^{-1 T}(v^*))\) | Homomorphisms preserve inverses | |||||||||||
| \(\ds \) | \(=\) | \(\ds n(\rho(g)(v^*))\) | The transpose of a permutation is its inverse | |||||||||||
| \(\ds \) | \(=\) | \(\ds \rho(g)n(v^*)\) | Homomorphisms preserve natural pairings | |||||||||||
| \(\ds \) | \(=\) | \(\ds \rho(g)v\) | from above: $n(v^*)=v$ |
Applying transitivity, we have $n(\rho^*(g)(v^*)) = \rho(g)v$; that is, $\langle \rho^*(g)(v^*) = \rho(g)v \rangle$ as intended.