[[Group representation]]
# Symplectic representation

A **symplectic representation** $\mathfrak{X}$ of $G$ is a [[group homomorphism]] $\mathfrak{X} : G \to \opn{Sp}(V)$ into the [[symplectic group]] #m/def/rep2 
where $V$ is a [[symplectic vector space]].
Thus $\mathfrak{X}$ is a [[group representation]] of $G$ carried by $V$ such that
$$
\begin{align*}
\omega(\mathfrak{X}(g) v, \mathfrak{X}(g) w) = 0
\end{align*}
$$
for all $g \in G$ and $v, w \in V$ where $\omega$ is the symplectic form.

## Properties

- [[A group representation is self-dual iff it preserves a nondegenerate bilinear form]]

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#state/tidy | #lang/en | #SemBr