For any ${ \rm E}$-rigid presentation $e$, we construct an orthogonal projection functor to ${ \rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${ \rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${ \rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $\delta$-vectors of positive and negative complements and the dimension vectors of simple modules in ${ \rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.
Categories: math.RT math.RA