Rigid quiver representation (${\mathrm{Ext}}_Q^1(X,X)=0$)

A finite-dimensional quiver representation is rigid when ${\mathrm{Ext}}_Q^1(X,X)=0$. Its extension complex of quiver representations then has surjective differential. Equivalently, it has an open orbit under base change. Rigidity constrains arrow ranks and path identities.

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.

A base-change invariant polynomial identity holding on a rigid representation holds on its dense orbit and hence throughout the representation space. A path through nonzero vertex spaces cannot vanish identically there: assign every involved arrow a rank-one map carrying a chosen source vector to the chosen target vector. This works for repeated arrows. Consequently a nonzero path image in a rigid representation cannot be killed by an outgoing arrow into a nonzero vertex space.