Symplectic Frieze Patterns

TL;DR

This paper introduces symplectic friezes, a new class linked to symplectic geometry, cluster algebras, and moduli spaces, expanding the combinatorial and algebraic understanding of frieze patterns.

Contribution

It defines symplectic friezes and explores their connections to cluster algebras, symplectic geometry, and moduli spaces, providing a new perspective on frieze patterns.

Findings

Symplectic friezes relate to cluster algebras of types C2 and Am.

They are connected to moduli spaces of Lagrangian configurations in 4D symplectic space.

Symplectic friezes exhibit properties similar to Coxeter and SL-friezes.

Abstract

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 ${\rm C}_{2}$ and ${\rm A}_{m}$. 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 SL-friezes.