Orbit dimension formula ($\dim {\mathcal{O}}_x=\dim G-\dim G_x$)

For an algebraic group action, a group orbit has dimensionwhere $G_x$ 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.