= Orbit dimension formula
{title2=$\dim\mathcal O_x=\dim G-\dim G_x$}
For an <algebraic group action>, a <group orbit> has <dimension>
$$
\dim\mathcal O_x=\dim G-\dim G_x,
$$
where $G_x$ is the <stabilizer subgroup>. The <orbit map> has <fibres> that are <cosets> of $G_x$, so the <fiber dimension theorem> gives the formula.
For the <base change action on quiver representations>, the <stabilizer subgroup> is the <automorphism group of a quiver representation>. It is a <Zariski-open set> in the <endomorphism ring> of that <quiver representation>, so their <dimensions> agree. The <Ringel form> then identifies orbit codimension with the <dimension> of the self-extension <group>. This open-unit-group argument applies to this representation-classification setting; a general vector <stabilizer subgroup> need not have that form.
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