On the Orthogonal Projections

For any rigid presentation $e$, we construct an orthogonal projection functor to $\mathrm{r}\mathrm{e}\mathrm{p}(e^{\perp })$ left adjoint to the natural embedding. We establish a bijection between presentations in $\mathrm{r}\mathrm{e}\mathrm{p}(e^{\perp })$ and presentations compatible with $e$. For quivers with potentials, we show that $\mathrm{r}\mathrm{e}\mathrm{p}(e^{\perp })$ 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 $\mathrm{r}\mathrm{e}\mathrm{p}(e^{\perp })$, 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.