Conway Symbols of Frieze Patterns of Dynkin Types

Abstract

In this paper, we classify the Conway symbols of the Euclidean symmetry groups of frieze patterns of Dynkin types $A_n$, $B_n$, $C_n$, $D_n$, $F_4$ and $G_2$. First, we show that the Conway symbol of a Conway--Coxeter frieze is determined by the bilateral and the central symmetry of the underlying triangulation. Secondly, we show that a frieze pattern of non-simply-laced type has the symbol ${*}\infty\infty$ or $\infty\infty$ according as its first interior row is palindromic or not, with the single exception of type $C_3$, which realises four symbols. Thirdly, we show that the same dichotomy holds for type $D_n$ with $n \ge 5$, the symbol being read off the tagged triangulation of the once-punctured polygon attached to the frieze together with the value on its puncture edges, and that type $D_4$ realises five symbols.