Cyclic symmetries of AnAn-quiver representations

This note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's S-construction). Our specific motivation (in the spirit of expectations of Kontsevich) is a combinatorial construction of categorical quantizations of Lagrangian skeleta of symplectic manifolds. The main result of this paper gives an immediate solution in the one-dimensional case of ribbon graphs.