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The group $\mathrm{GL}(\mathbf{n})={\prod }_i{\mathrm{GL}}_{n_i}(k)$ acts by $(g\cdot f{)}_{\rho }=g_{t(\rho )}{f}_{\rho }{g}_{s(\rho )}^{-1}$. Inverse entries are regular on a general linear group, so this is an algebraic group action. Its stabilizer is the automorphism group of a quiver representation.

An indecomposable of dimension vector $\mathbf{n}$ is a brick module and has $q_Q(\mathbf{n})=1$ by the previous parts. Its automorphism group of a quiver representation has dimension one. ThereforeIt follows that its orbit is open and dense in the irreducible quiver representation space. Two different indecomposables of the same dimension vector would give two disjoint nonempty open orbits. Nonempty open subsets of an irreducible space must intersect, so this is impossible. Hence an indecomposable is uniquely determined by its dimension vector, up to isomorphism. This is the positive definite Tits form indecomposable classification.

The stabilizer is the automorphism group of a quiver representation, an open set in its endomorphism algebra. Thus $\dim {\mathcal{O}}_X={\sum }_i{n}_i^2-\dim {\mathrm{End}}_Q(X)$. The Ringel form gives orbit codimension $\dim {\mathrm{Ext}}_Q^1(X,X)$. Since orbits are locally closed and the representation space is irreducible, rigidity is equivalent to an open dense orbit.