Let $G$ be a group, let $T \le G$ be a subgroup, and let
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\begin{align*}
\rho:G\to GL(V)
\end{align*}
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be a finite-dimensional complex representation on a complex [vector space](/page/Vector%20Space) $V$. Regard $V$ as a representation of $T$ by restriction, and write $t\cdot v:=\rho(t)v$ for $t\in T$ and $v\in V$.
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A character of $T$ means a [group homomorphism](/page/Group%20Homomorphism) $\lambda:T\to\mathbb C^\times$. For a character $\lambda:T\to\mathbb C^\times$, define the $T$-weight space
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\begin{align*}
V_\lambda:=\{v\in V:t\cdot v=\lambda(t)v\text{ for every }t\in T\}.
\end{align*}