Symplectic Frieze Patterns

We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type $\mathrm{C}_{2}$ and $\mathrm{A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105–1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and $\mathrm{SL}$-friezes.