Symplectic frieze patterns (Q2280428)

This paper introduces a class of friezes called \emph{sympletic \(2\)-friezes} and studies their algebraic, combinatorial and geometric properties. An example of a symplectic 2-frieze is shown below (Example~3.1(c) from the paper). \setcounter{MaxMatrixCols}{20} \[ \arraycolsep=3.8pt \begin{matrix} \dotsb & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & \dotsb \\ \dotsb & 6 & \textcolor{red}{14} & 3 & \textcolor{red}{1} & 1 & \textcolor{red}{2} & 3 & \textcolor{red}{6} & 4 & \textcolor{red}{5} & 2 & \textcolor{red}{1} & 1 & \textcolor{red}{3} & 6 & \dotsb \\ \dotsb & \textcolor{red}{6} & 4 & \textcolor{red}{5} & 2 & \textcolor{red}{1} & 1 & \textcolor{red}{3} & 6 & \textcolor{red}{14} & 3 & \textcolor{red}{1} & 1 & \textcolor{red}{2} & 3 & \textcolor{red}{6} & \dotsb \\ \dotsb & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \textcolor{red}{1} & 1 & \dotsb \end{matrix} \] Formally, a symplectic \(2\)-frieze is an array made up of finitely many rows of complex numbers, each infinite in both directions, with rows of 1's at the top and bottom, subject to the following rules. The entries of the array are coloured either black or red (black or white in the paper) in an alternating fashion, such that the four neighbours above, below, to the left and to the right of each entry are coloured differently to the entry itself. Furthermore, any red entry is equal to the two-by-two determinant of the four adjacent black entries, as follows: \[ \arraycolsep=3.6pt \begin{matrix} &b &\\ a & \textcolor{red}{x}& d\\ & c & \end{matrix} \qquad\qquad \textcolor{red}{x}= ad-bc. \] Also, the \emph{square} of any black entry is equal to the two-by-two determinant of the four adjacent red entries, as follows: \[ \arraycolsep=3.6pt \begin{matrix} &\textcolor{red}{b} &\\ \textcolor{red}{a} & x& \textcolor{red}{d}\\ & \textcolor{red}{c} & \end{matrix} \qquad\qquad x^2= \textcolor{red}{ad}-\textcolor{red}{bc}. \] The \emph{width} of the frieze is the number of rows minus 2 (discount the top and bottom rows of 1's). Sympletic 2-friezes over rings other than the complex numbers are also considered. This work is motivated by the triality programme described in [\textit{S. Morier-Genoud} et al., Forum Math. Sigma 2, Paper No. e22, 45 p. (2014; Zbl 1297.39004)], which identifies a correspondence between certain tame \(\text{SL}_k\)-friezes, configurations of points in projective space, and periodic linear difference equations. The second two parts of this programme were considered in [\textit{C. H. Conley} and \textit{V. Ovsienko}, Math. Ann. 375, No. 3--4, 1105--1145 (2019; Zbl 1429.53097)] for configurations of points in symplectic spaces and certain symmetric periodic linear difference equations. The present paper completes the triality programme for these particular classes of objects (in the case \(k=4\)) by introducing symplectic 2-friezes. The leading theorems of the paper describe the relationship between symplectic 2-friezes and other friezes and algebraic and geometric structures; we summarise some (but not all) of the results. First, it is shown (Theorem~2.1) that, like the classical friezes of positive integers studied by Coxeter [\textit{H. S. M. Coxeter}, Acta Arith. 18, 297--310 (1971; Zbl 0217.18101)], symplectic 2-friezes are periodic and invariant under a glide reflection. It is also proven (Theorem~2.2) that, by restricting to the black entries of a symplectic 2-frieze, we can see that there is a one-to-one correspondence between tame sympectic 2-friezes of width \(w\) and tame \(\text{SL}_4\)-friezes of width \(w\) that are invariant under a glide reflection. This correspondence is extended to other classes of friezes (using, for example, Gale duality from [\textit{S. Morier-Genoud} et al., Forum Math. Sigma 2, Paper No. e22, 45 p. (2014; Zbl 1297.39004)]). Next, Theorem~2.3 states that there is a correspondence between the set of tame sympectic 2-friezes of a given width and a certain class of linear difference equations that have strong periodicity properties, called \emph{\(n\)-superperiodic difference equations}. Theorem~2.5 relates symplectic 2-friezes to cluster algebras. It says that the variety of tame symplectic 2-friezes of width \(w\) contains as an open dense subset the cluster variety associated to an orientation of the product of Dynkin diagrams \(C_2 \times A_w\). The last of the main results, Theorem~2.7, finds a correspondence between sympletic 2-friezes and certain Lagrangian configurations of lines considered in [\textit{C. H. Conley} and \textit{V. Ovsienko}, Math. Ann. 375, No. 3--4, 1105--1145 (2019; Zbl 1429.53097)]. Briefly, in the language of the paper under review, consider projective space \(\mathbb{CP}^3\) equipped with a contact structure of hyperplanes \(H_v\), for each vector \(v\in\mathbb{CP}^3\). The author defines a \emph{Legendrian \(n\)-gon} to be a bi-infinite sequence of points \((v_i)\) in \(\mathbb{CP}^3\) with period \(n\) such that \(v_{i-1},v_{i+1}\in H_{v_i}\), for each \(i\in\mathbb{Z}\). (The further restriction \(v_{i+2}\notin H_{v_i}\) is also assumed.) Theorem~2.7 states that, for odd integers \(n\geqslant 5\), there is a one-to-one correspondence between tame symplectic 2-friezes of width \(n-5\) and the moduli space of Legendrian \(n\)-gons under the action of the projective symplectic group \(\text{PSp}_2(\mathbb{C})\). The paper is furnished with plenty of illustrating examples, two appendices contain key techniques from the subject, and there is a final section of open problems for further study.