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

For an algebraic group action, the orbit dimension is the dimension of the group minus that of its stabilizer. For a finite-dimensional representation, the stabilizer is an automorphism group, open in its endomorphism ring, so their dimensions agree. In a quiver representation space, the Ringel form turns the orbit codimension into the dimension of the self-extension group.