Landau–Ginzburg potentials via projective representations

We interpret the Landau–Ginzburg potentials associated to Gross–Hacking–Keel–Kontsevich’s partial compactifications of cluster varieties as F -polynomials of projective representations of Jacobian algebras. Along the way, we show that both the finite-dimensional projective and the finite-dimensional injective representations of Jacobian algebras are well behaved under Derksen–Weyman–Zelevinsky’s mutations of representations.