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Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT, that there is an $n \in \mathbb{N}$ and a $G$-equivariant embedding $$\iota:X \hookrightarrow \mathbb{P}^n_k$$ where $G$ acts on $\mathbb{P}^n_k$ via a representation $\rho:G \rightarrow \mathrm{GL}_{n+1}$.

Can we always assume that $\rho$ is faithful?

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    $\begingroup$ I might be confused here, but can't you take the direct sum of $\rho$ with a faithful representation to get a faithful representation of $G$ into some larger $\text{GL}_{m+1}$? You can then embed $X$ into the associated larger projective space by embedding $\mathbb{P}^n_k$ as a linear subvariety in it. $\endgroup$
    – Andy Putman
    Commented Sep 28, 2022 at 20:56

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