# Symplectic frieze patterns

Sophie Morier-Genoud  Sophie Morier-Genoud, Sorbonne Universités, Université
Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche,
IMJ-PRG, F-75005, Paris, France<EMAIL_ADDRESS>
###### Abstract.

We introduce a new class of frieze patterns which is related to symplectic
geometry. On the algebraic side, 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 [CO]. The combinatorics and the main properties of the
symplectic friezes are similar to those of Coxeter frieze patterns and
$\mathrm{SL}$-frieze patterns.

###### Key words and phrases:

Frieze, Cluster algebra, Moduli space, Difference equation, Lagrangian
configuration

###### Contents

  1. 1 Main results
    1. 1.1 Combinatorial description and characterizations of the symplectic $2$-friezes
    2. 1.2 Symplectic $2$-friezes and difference equations
    3. 1.3 The algebraic variety of symplectic $2$-friezes
    4. 1.4 The cluster structure on the variety of symplectic $2$-friezes
    5. 1.5 Link with symplectic geometry
  2. 2 Examples
  3. 3 Symplectic 2-friezes, $\mathrm{SL}$-frieze patterns and difference equations
    1. 3.1 Notation for the symplectic 2-friezes
    2. 3.2 The tameness condition
    3. 3.3 $\mathrm{SL}$-friezes and difference equations
    4. 3.4 Determinantal formulas for the entries in a symplectic $2$-frieze
    5. 3.5 Proofs of Theorem 1, Theorem 2 and Theorem 3
  4. 4 Legendrian $n$-gons
    1. 4.1 Lift of the projective $n$-gons in $\mathbb{C}^{4}$
    2. 4.2 Symplectic $2$-friezes from Legendrian $n$-gons
    3. 4.3 Symplectic forms and $2$-friezes
    4. 4.4 Proof of Proposition 1.1
    5. 4.5 More correspondences
    6. 4.6 The case when $n$ is even
    7. 4.7 Legendrian $n$-gons and self-dual polygons
  5. 5 Cluster algebras
    1. 5.1 Preliminaries
      1. 5.1.1 Mutations from skew-symmetrizable matrices
      2. 5.1.2 Valued quivers
      3. 5.1.3 Mutation of valued quivers
      4. 5.1.4 Product of valued quivers
      5. 5.1.5 Bipartite belt
    2. 5.2 Cluster variables in the symplectic $2$-friezes
    3. 5.3 More clusters in the $2$-friezes
    4. 5.4 The cluster variety of tame symplectic 2-friezes
  6. 6 Symplectic 2-friezes of positive integers
  7. A Desnanot-Jacobi identity
  8. B Properties of $\mathrm{SL}_{k+1}$-friezes
    1. B.1 Definition of tame $\mathrm{SL}_{k+1}$-friezes
    2. B.2 Tame $\mathrm{SL}_{k+1}$-friezes and difference equations
    3. B.3 Coefficients of the difference equations from the entries of the frieze and vice versa
    4. B.4 Projective duality
    5. B.5 Gale duality

## Introduction

The notion of frieze patterns goes back to Coxeter in the early 70’s, [Cox71].
Friezes are arrays of numbers where neighboring values are related by a local
arithmetic rule modeled on the group $\mathrm{SL}_{2}$. Coxeter’s friezes are
surprisingly connected to many classical areas of mathematics such as
projective geomety, number theory, enumerative combinatorics, [Cox71], [CC73].
Many variants of friezes have been recently studied in connection with more
fields: cluster algebras, quiver representations, moduli spaces, integrable
systems, algebraic combinatorics, see e.g. [BM09], [ARS10], [ADSS12],
[MGOT12], [MGOST14], [BHJ17].

In [MGOST14], we establish a property called the “triality” that identifies
three different objects: $\mathrm{SL}_{k}$-friezes, configurations of points
in the projective spaces and periodic difference equations. In the rothcomimg
paper [CO], Lagrangian configurations of points in the symplectic spaces are
identified with symmetric periodic linear difference equations. The idea of
completing the triality in the case of Lagrangian configurations with a
combinatorial notion of “symplectic friezes” is due to Valentin Ovsienko.

In the present paper we focus on the case of 4-dimensional symplectic spaces.
This case is of particular interest for two reasons. Lagrangian configurations
in dimension 4 are interpreted as a discrete version of classical Legendrian
knots (“stick knots”). Symplectic frieze patterns provide a special
coordinates system on the space of Legendrian knots quotient by the
$\mathrm{Sp}_{4}$-action. Another interest of 4-dimensional symplectic friezes
is that the “local frieze rule” in this case can be simplified using $2\times
2$ determinants instead of $4\times 4$ ones. We define a variant of friezes,
called symplectic 2-friezes, for which the local rule mixes the rule of the
2-friezes of [MGOT12] and the rule of wall numbers [CG96].

The symplectic Lie group $\mathrm{Sp}_{4}$ appears in two ways. From an
algebraic point of view, the local rule for the symplectic $2$-friezes is
interpreted as cluster mutations involving the Dynkin type $\mathrm{C}_{2}$ of
the Lie group $\mathrm{Sp}_{4}$. From a geometric point of view, the
symplectic $2$-friezes parametrize particular configurations of points in the
4-dimensional symplectic space on which the Lie group $\mathrm{Sp}_{4}$ acts.

Symplectic 2-friezes enjoy nice periodicity properties which are very similar
to those of the classical Coxeter friezes. In particular, they are invariant
under the glide symmetry and therefore periodic. The periodicity of the
friezes is a case of Zamolodchikov periodicity; see [Kel13] and references
therein.

The paper is organized as follows.

In Section 1 we expose the main results of the paper.

Section 2 contains examples illustrating the main results.

In Section 3 we study the combinatorial structure of the symplectic 2-friezes
and characterize them in terms of $\mathrm{SL}$-friezes. We explain the
correspondence between symplectic $2$-friezes and symmetric difference
equations. We also give explicit determinantal formulas for the entries in the
friezes.

In Section 4 we explain the link between the $2$-friezes and symplectic
geometry. The results in this section are mainly based on [CO]. We identify
the $2$-friezes with Legendrian polygons in the projective space
${\mathbb{CP}}^{3}$. We discuss the relationship between symplectic forms and
symplectic $2$-friezes.

Section 5 makes the link between symplectic $2$-friezes and cluster algebras.
The corresponding cluster algebras are generated from a product of Dynkin
diagrams of type $\mathrm{C}_{2}$ and $\mathrm{A}_{w}$.

In Section 6, we discuss questions around symplectic $2$-friezes with positive
integer entries.

In Appendix A and B we collect important formulas and results on classical
$\mathrm{SL}$-friezes from [BR10] and [MGOST14] which are extensively used
througout the paper.

##  1\. Main results

###  1.1. Combinatorial description and characterizations of the symplectic
$2$-friezes

We call symplectic $2$-frieze an array of complex numbers (or polynomials,
rational functions, etc.) in the plane satisfying the following conditions:

  * •

The array has finitely many infinite rows;

  * •

The array starts and ends with a row of 1’s;

  * •

The entries are alternatingly colored in black and white;

  * •

The colored entries are subject to local rules

    * –

Every white entry is equal to the $2\times 2$-determinant of the matrix formed
by the four adjacent black entries;

    * –

Every square of a black entry is equal to the $2\times 2$-determinant of the
matrix formed by the four adjacent white entries.

Symplectic 2-friezes are represented as follows

$\begin{array}[]{ccccccccccccc}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots&\\\
&\vdots&&\vdots&&\vdots&&\vdots&&\vdots&&\\\
\cdots&\bullet&\circ&\bullet&\circ&\mathbf{B}&f&\bullet&\circ&\bullet&\circ&\cdots&\\\
&\circ&\bullet&\circ&\mathbf{A}&e&\mathbf{D}&h&\bullet&\circ&\bullet\\\
\cdots&\bullet&\circ&\bullet&\circ&\mathbf{C}&g&\bullet&\circ&\bullet&\circ&\cdots&\\\
&\vdots&&\vdots&&\vdots&&\vdots&&\vdots&&\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots&\\\
\end{array}$

in which the local rules read

$AD-BC=e,\quad eh-fg=D^{2},\quad\ldots$

The number of rows between the initial and final rows of 1’s is called the
width of the frieze.

We will consider “tame” friezes which are friezes satisfying an extra
condition of genericity (see Definition 3.1 for the details). For instance,
friezes with no zero entries are all tame.

The tame symplectic $2$-friezes share the same properties of symmetry as
Coxeter frieze patterns and classical $2$-friezes.

######  Theorem 1.

All tame symplectic $2$-friezes of width $w$ are $2(w+5)$-periodic. Moreover,
the arrays are all invariant under a glide reflection with respect to the
median line.

It turns out that the subarray consisting of the black entries of a symplectic
$2$-frieze form a classical $\mathrm{SL}_{4}$-frieze pattern. This allows the
following characterization in terms of $\mathrm{SL}_{k}$-friezes.

######  Theorem 2.

The set of tame symplectic $2$-friezes of width $w$ is in one-to-one
correspondence with each of the following set:

  1. (1)

The tame $\mathrm{SL}_{4}$-frieze patterns of width $w$ in which every
adjacent $3\times 3$-minors are equal to their central elements,

  2. (2)

The tame $\mathrm{SL}_{4}$-frieze patterns of width $w$ that are invariant
under a glide reflection with respect to the median line,

  3. (3)

The tame $\mathrm{SL}_{w+1}$-frieze patterns of width $3$ that are symmetric
with respect to the middle row.

Theorem 1 and Theorem 2 are proved in §3.5.

###  1.2. Symplectic $2$-friezes and difference equations

Consider the system of linear difference equations of the form

(1.1) $V_{i}=a_{i}\,V_{i-1}-b_{i}\,V_{i-2}+a_{i-1}V_{i-3}-V_{i-4},$

for $i\in\mathbb{Z}$, where $a_{i},b_{i}\in\mathbb{C}$ are given coefficients,
and $(V_{i})_{i}$ is a sequence of indeterminates or a “solution”.

Following [Kri14], [MGOST14], we call such difference equation $n$-super-
peridodic if

  * •

all the coefficients are $n$-periodic, i.e. $a_{i+n}=a_{i}$, $b_{i+n}=b_{i}$,
for all $i\in\mathbb{Z}$,

  * •

all solutions $(V_{i})_{i\in\mathbb{Z}}$ of the system are $n$-antiperiodic,
i.e. $V_{i+n}=-V_{i}$ for all $i\in\mathbb{Z}$.

The next result gives a a one-to-one correspondence between the set of tame
symplectic $2$-friezes of width $w=n-5$ and the set of $n$-super-periodic
difference equations of the form (1.1).

######  Theorem 3.

(i) In a tame symplectic $2$-friezes of width $w=n-5$, the first non-trivial
row $(\ldots,b_{i},\textbf{a}_{i},b_{i+1},\textbf{a}_{i+1},\ldots)$ are
coefficents of an $n$-super-periodic difference equations of the form (1.1);

(ii) Coefficients $(a_{i})_{i\in\mathbb{Z}},(b_{i})_{i\in\mathbb{Z}}$ of an
$n$-super-periodic difference equations of the form (1.1) define the first row
of a tame symplectic $2$-friezes of width $w=n-5$.

Theorem 3 is proved in §3.5.

###  1.3. The algebraic variety of symplectic $2$-friezes

One can give explicit algebraic conditions on the coefficients $a_{i}$ and
$b_{i}$ of the difference equation (1.1) in order to have the super-
periodicity property. This has been done in [MGOST14] for general linear
difference equations, and in [CO] for the particular case of equations with
symmetric coefficients. The conditions comes from the equation
$M=-\mathrm{Id}$ where $M$ is a monodromy matrix associated to the difference
equations.

In our situation, $M$ is a matrix of $\mathrm{Sp}_{4}(\mathbb{C})$ and its
entries can be expressed with the help of the following determinants

$\Delta_{i,j}=\left|\begin{array}[]{ccccccccccccc}a_{i}&b_{i+1}&a_{i+1}&1\\\\[3.0pt]
1&a_{i+1}&b_{i+2}&a_{i+2}&1\\\\[6.0pt]
&\ddots&\ddots&\ddots&\ddots&\ddots\\\\[6.0pt]
&&1&a_{j-2}&b_{j-1}&a_{j-1}\\\\[3.0pt] &&&1&a_{j-1}&b_{j}\\\\[3.0pt]
&&&&1&a_{j}\end{array}\right|.$

This allows us to obtain a complete system of polynomial equations for the set
of symplectic $2$-friezes of width $w$ over the complex numbers as an
algebraic subvariety of $\mathbb{C}^{2n}$, with $n=w+5$.

######  Proposition 1.1.

Let $(a_{i},b_{i})_{1\leq i\leq n}$ be coordinates on $\mathbb{C}^{2n}$. The
set of tame symplectic $2$-friezes of width $w=n-5$ is the algebraic
subvariety of $\mathbb{C}^{2n}$ of dimension $2w$ given by the system of 10
equations (with the convention $a_{0}=a_{n}$)

(1.2) $\left\\{\begin{array}[]{lcllcl}\Delta_{3,n-3}&=&a_{n},\\\\[4.0pt]
\Delta_{k+2,n-3+k}&=&1,&k=0,\,1,\\\\[4.0pt]
\Delta_{k+1,n-3+k}&=&0,&k=0,\,1,\,2,\\\\[4.0pt]
\Delta_{k,n-3+k}&=&0,&k=0,\,1,\,2,\,3.\\\\[4.0pt] \end{array}\right.$

The proof is given in §4.4.

###  1.4. The cluster structure on the variety of symplectic $2$-friezes

Two consecutive columns in a 2-frieze give a system of $2w$ coordinates on the
variety of symplectic $2$-friezes. The transition maps between the different
systems of coordinates are interpreted as sequences of cluster mutations
inside a cluster algebra. More precisely one has the following result.

######  Theorem 4.

The variety of tame symplectic $2$-friezes of width $w=n-5$ contains as an
open dense subset the cluster variety associated to the product of Dynkin
diagrams $\mathrm{C}_{2}\times\mathrm{A}_{w}$.

This theorem is reformulated more precisely as Proposition 5.6 and Proposition
5.11 which are proved in §5.2 and §5.4.

###  1.5. Link with symplectic geometry

In [CO] the variety of $n$-super-periodic linear difference equations of order
$2k$, with symmetric coefficients, has been identified with the moduli space
$\mathcal{L}_{2k,n}$ of Lagrangian configurations of $n$ points in
$\mathbb{C}^{2k}$. Applying the results and ideas of [CO] in the case $k=2$
gives a nice geometric interpretation of the symplectic $2$-friezes in a
symplectic space.

We present this geometric interpretation in a more naive terms.

######  Definition 1.2 ([CO], Lagrangian configurations of points).

Consider the projective space ${\mathbb{CP}}^{3}$ equipped with a contact
structure given by the data of hyperplanes $(H_{v})_{v\in{\mathbb{CP}}^{3}}$
attached to each point of the space. We call a Legendrian $n$-gon, every
sequence $(v_{i})_{i\in\mathbb{Z}}$ of points in ${\mathbb{CP}}^{3}$ such
that:

  * •

$v_{i+n}=v_{i}$, for all $i\in\mathbb{Z}$,

  * •

$v_{i-1}$ and $v_{i+1}$ belong to $H_{v_{i}}$ for all $i\in\mathbb{Z}$.

Legendrian $n$-gons are discrete analogues of Legendrian knots.

For a generic $n$-gon one has in addition that $v_{i+2}$ does not belong to
$H_{v_{i}}$.

We will consider the moduli space of generic Legendrian $n$-gons modulo
$\mathrm{P}\mathrm{Sp}_{4}$-equivalence.

######  Theorem 5.

When $n$ is odd, the space of tame symplectic $2$-friezes of width $w=n-5$
over the complex numbers is isomorphic to the moduli space of Legendrian
$n$-gons in ${\mathbb{CP}}^{3}$.

The correspondence is given in details in Section 4.

##  2\. Examples

In this section we work out examples to illustrate the results presented in
the previous section.

We start by giving examples of symplectic $2$-friezes as described in §1.1.

######  Example 2.1.

(a) Tame symplectic $2$-friezes of width 1 with integer entries

(2.3)
$\begin{array}[]{rrrrrrrrrrrrrr}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{0}&-1&\mathbf{1}&-2&\mathbf{-1}&-1&\mathbf{0}&-1&\mathbf{1}&-2&\mathbf{-1}&-1&\cdots\\\
\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

(2.4)
$\begin{array}[]{ccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{1}&2&\mathbf{3}&5&\mathbf{2}&1&\mathbf{1}&2&\mathbf{3}&5&\mathbf{2}&1&\cdots\\\
\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

(b) Tame symplectic $2$-friezes of width 1 with positive real entries and with
complex entries

$\begin{array}[]{ccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{2\sqrt{2}}&3&\mathbf{\sqrt{2}}&1&\mathbf{\sqrt{2}}&3&\mathbf{2\sqrt{2}}&3&\mathbf{\sqrt{2}}&1&\mathbf{\sqrt{2}}&3&\cdots\\\
\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

$\begin{array}[]{rrrrrrrrrrrrrrrr}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{i}&1&\mathbf{-2i}&-3&\mathbf{-i}&0&\mathbf{i}&1&\mathbf{-2i}&-3&\mathbf{-i}&0&\cdots\\\
\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

(c) Tame symplectic $2$-friezes of width 2 and 3 with positive integer entries

(2.5)
$\begin{array}[]{ccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\cdots&\mathbf{6}&14&\mathbf{3}&1&\mathbf{1}&2&\mathbf{3}&6&\mathbf{4}&5&\mathbf{2}&1&\mathbf{1}&3&\mathbf{6}&\cdots\\\
\cdots&6&\mathbf{4}&5&\mathbf{2}&1&\mathbf{1}&3&\mathbf{6}&14&\mathbf{3}&1&\mathbf{1}&2&\mathbf{3}&6&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$ (2.6)
$\begin{array}[]{ccccccccccccccccccccccc}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\cdots&1&\mathbf{2}&5&\mathbf{4}&6&\mathbf{4}&6&\mathbf{3}&2&\mathbf{1}&1&\mathbf{4}&30&\mathbf{10}&4&\mathbf{1}&1&\mathbf{2}&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{3}&14&\mathbf{10}&20&\mathbf{6}&3&\mathbf{1}&1&\mathbf{3}&14&\mathbf{10}&20&\mathbf{6}&3&\mathbf{1}&1&\cdots\\\
\cdots&2&\mathbf{1}&1&\mathbf{4}&30&\mathbf{10}&4&\mathbf{1}&1&\mathbf{2}&5&\mathbf{4}&6&\mathbf{4}&6&\mathbf{3}&2&\mathbf{1}&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\end{array}$

######  Example 2.2 (Glide reflection).

Recall that a glide reflection is the composition of a reflection about a line
and a translation along that line. The invariance mentioned in Theorem 1 can
be easily observed in the above examples. For instance, we can display a
fundamental domain in the array (2.6) that repeats under a glide reflection.
Note that the glide symmetry implies the periodicity of the arrays.

$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$5$$4$$6$$4$$6$$3$$2$$1$$1$$4$$30$$10$$4$$1$$1$$2$$1$$1$$3$$14$$10$$20$$6$$3$$1$$1$$3$$14$$10$$20$$6$$3$$1$$1$$2$$1$$1$$4$$30$$10$$4$$1$$1$$2$$5$$4$$6$$4$$6$$3$$2$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$5$$4$$6$$3$$14$$10$$1$$4$$30$
Figure 1. Glide symmetry in the frieze (2.6)

######  Example 2.3 (Corresponding $\mathrm{SL}$-friezes).

We illustrate Theorem 2. The $\mathrm{SL}_{4}$-frieze corresponding to a
symplectic 2-frieze is simply given by the subarray of black entries. Applying
the combinatorial Gale duality (see Appendix B.5) one obtains the
corresponding $\mathrm{SL}_{w+1}$-frieze of width 3.

0313612424313612612431361562114312431361200000001111111111111111000000000000000000000000000000000000000000111111111111111000000000000000000$\mathbf{V}=$$\mathbf{W}=$
Figure 2. The $\mathrm{SL}_{3}$-frieze of width 3 (left) and the
$\mathrm{SL}_{4}$-frieze of width 2 (right) corresponding to the symplectic
2-frieze (2.5).

The $\mathrm{SL}$-friezes corresponding to the symplectic 2-frieze (2.5) are
given in Figure 2. The friezes are related by Gale duality. One can check that
$WD\,(^{t}V)=0$ where $W$ and $V$ are respectively the $3\times 7$ and
$4\times 7$ matrices defined on Figure 2, and $D$ the $7\times 7$ diagonal
matrix with diagonal coefficients $(1,-1,1,-1,1,-1,1)$.

######  Example 2.4 (Corresponding difference equation).

We illustrate Theorem 3. The array (2.5) provides the following 7-periodic
sequences of coefficients:

$\displaystyle(\ldots,a_{0},a_{1},\ldots,a_{6},\ldots)=(\ldots,6,3,1,3,4,2,1,\ldots),$
$\displaystyle(\ldots,b_{0},b_{1},\ldots,b_{6},\ldots)=(\ldots,3,14,1,2,6,5,1,\ldots)$

Let us check that the associated recurrence (1.1) is indeed $7$-antiperiodic.
Choose the initial values $(V_{-3},V_{-2},V_{-1},V_{0})=(x,y,z,t)$ and compute

$\begin{array}[]{lclclcl}V_{1}&=&3V_{0}-14V_{-1}+6V_{-1}-V_{-2}&=&3t-14z+6y-x\\\
V_{2}&=&V_{1}-V_{0}+3V_{-1}-V_{-2}&=&2t-11z+5y-x\\\
V_{3}&=&3V_{2}-2V_{1}+V_{0}-V_{-1}&=&t-6z+3y-x\\\
V_{4}&=&4V_{3}-6V_{2}+3V_{1}-V_{0}&=&-x&=&-V_{-3}\\\
V_{5}&=&2V_{4}-5V_{3}+4V_{2}-V_{1}&=&-y&=&-V_{-2}\\\
V_{6}&=&V_{5}-V_{4}+2V_{3}-V_{2}&=&-z&=&-V_{-1}\\\
V_{7}&=&6V_{6}-3V_{5}+V_{4}-V_{3}&=&-t&=&-V_{0}\end{array}$

######  Example 2.5 (Equations of the variety).

For $w=1$ the system of equations (1.2) is

$a_{3}=a_{6},\quad\begin{array}[]{ccc}a_{3}a_{2}-b_{3}&=&1\\\
a_{4}a_{3}-b_{4}&=&1\\\
\end{array},\quad\begin{array}[]{ccc}a_{1}-a_{3}b_{2}+a_{2}&=&0\\\
a_{2}-a_{4}b_{3}+a_{3}&=&0\\\ a_{5}-a_{3}b_{5}+a_{4}&=&0\\\
\end{array},\quad\begin{array}[]{ccc}b_{1}-a_{1}a_{3}+1&=&0\\\
b_{2}-a_{2}a_{4}+1&=&0\\\ b_{5}-a_{4}a_{2}+1&=&0\\\ b_{6}-a_{5}a_{3}+1&=&0\\\
\end{array}.$

Note that using the first row expansion or last column expansion in
$\Delta_{i,j}$ the subsets of equations in the rows of the system (1.2) can be
immediately simplified using the previous subsets, in order to decrease the
degrees of the equations.

The dimension of the variety is 2. The above system can be solved generically
using two parameters $(a,b)\not=(0,0)$. E.g. choosing $(a,b)=(a_{3},b_{3})$ we
deduce $b_{4}$ from $b_{3}b_{4}$ using a combination of equations number 3, 5
and 2:

$b_{3}b_{4}=b_{3}(a_{4}a_{3}-1)=b_{3}a_{4}a_{3}-b_{3}=(a_{2}+a_{3})a_{3}-b_{3}=a_{3}^{2}+(a_{2}a_{3}-b_{3})=a_{3}^{2}+1.$

So that $b_{4}=\frac{1+a_{3}^{2}}{b_{3}}$. Then all the variables are deduced
one after each other using only one equation of the system, in the following
order

$\begin{array}[]{rclrclrcl}a_{2}&=&\frac{1+b_{3}}{a_{3}},&a_{4}&=&\frac{1+b_{4}}{a_{3}},\\\\[4.0pt]
b_{2}&=&a_{2}a_{4}-1,&a_{1}&=&a_{3}b_{2}-a_{2},&b_{1}&=&a_{1}a_{3}-{1},\\\\[4.0pt]
b_{5}&=&a_{2}a_{4}-1,&a_{5}&=&a_{3}b_{5}-a_{4},&b_{6}&=&a_{5}a_{3}-1,\\\\[4.0pt]
a_{6}&=&a_{3}.\\\ \end{array}$

Expressing everything in terms of $(a,b)=(a_{3},b_{3})$, one gets

$\begin{array}[]{rccclrccclrcl}b_{1}&=&b_{4}&=&\frac{1+a^{2}}{b}&&a_{1}&=&a_{4}&=&\frac{1+b+a^{2}}{ab}\\\\[6.0pt]
b_{2}&=&b_{5}&=&\frac{(1+b)^{2}+a^{2}}{a^{2}b},&&a_{2}&=&a_{5}&=&\frac{1+b}{a}\\\\[6.0pt]
b_{3}&=&b_{6}&=&b&&a_{3}&=&a_{6}&=&a\\\ \end{array}.$

######  Example 2.6 (Corresponding Legendrian $n$-gon).

We illustrate Theorem 5. We can associate a Legendrian heptagon to the
$2$-frieze of width 2 given in (2.5). We choose a $4\times 7$-subarray of
black entries in the frieze, e.g. the block displayed in Figure 2, and denote
by $V_{i}$ its columns, so that:

$\blockarray{cccccccc}V_{1}&V_{2}V_{3}V_{4}V_{5}V_{6}V_{7}\\\
\block{(ccccccc)c}1431000\\\ 0121100\\\ 0011310\\\ 0001641\\\ $

We extend the sequence by periodicity $V_{i+7}=V_{i}$.

Consider the symplectic form on $\mathbb{C}^{4}$ given by

$\omega=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&-4&1\\\ -1&4&0&0\\\
0&-1&0&0\end{array}\right)$

Attach to each $V_{i}$ the hyperplane $H_{i}:=\\{V_{i}\\}^{\perp_{\omega}}$
formed by the orthogonal vectors. One can easily check that $V_{i-1},V_{i+1}$
belong to $H_{i}$ for all $i$. The 7-periodic sequence
$v=(\mathbb{C}V_{i})_{i}$ forms a symplectic heptagon in ${\mathbb{CP}}^{3}$.

From the Legendrian $n$-gon, one can compute the entries in the corresponding
frieze using the values of

$\omega(V_{i},V_{j}).$

E.g. in this example the black entries of the first row (which is the same as
the second row by glide symmetry) of (2.5) is
$\omega(V_{2},V_{5}),\omega(V_{3},V_{6}),\omega(V_{4},V_{7}),\ldots$.

If we move the initial $4\times 7$-subarray along the North-East diagonal, the
sequence changes by a shift of indices. If we move the initial $4\times
7$-subarray along the North-West diagonal, the sequence changes under the
action of $\mathrm{Sp}_{4}$. Therefore any choice of $4\times 7$-subarrays
leads to the same Legendrian $n$-gon, modulo a shift of indices of the
vertices and modulo $\mathrm{Sp}_{4}$-action.

######  Example 2.7 (Cluster variables in the frieze).

Every symplectic $2$-frieze is generically determined by the entries into 2
consecutive columns. All the other entries can be expressed as rational
fractions of the initial entries using the frieze rule. These expressions can
be recognized as cluster variables of type $\mathrm{C}_{2}\times A_{w}$, where
$w$ is the width of the frieze. For instance, a frieze of width $w=1$ will
produce the 6 cluster variables of type $\mathrm{C}_{2}$.

(2.7)
$\begin{array}[]{ccccccccccccccccccc}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\\[4.0pt]
\cdots&x_{1}&\mathbf{x_{2}}&\frac{1+x_{2}^{2}}{x_{1}}&\mathbf{\frac{1+x_{1}+x_{2}^{2}}{x_{1}x_{2}}}&\frac{(1+x_{1})^{2}+x_{2}^{2}}{x_{1}x_{2}^{2}}&\mathbf{\frac{1+x_{1}}{x_{2}}}&x_{1}&\mathbf{x_{2}}&\cdots\\\\[4.0pt]
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\end{array}$

##  3\. Symplectic 2-friezes, $\mathrm{SL}$-frieze patterns and difference
equations

###  3.1. Notation for the symplectic 2-friezes

We complete the description of the 2-friezes given in §1.1 by introducing a
labelling of the entries and more convention.

The entries in a symplectic $2$-frieze of width $w$ are denoted by
$(d_{i,j})$, ${i,j\in\mathbb{Z}}$ for the “black entries” and
$(d_{i+\frac{1}{2},j+\frac{1}{2}})$, ${i,j\in\mathbb{Z}}$ for the “white
entries”. The local rules around a white entry is

$d_{i+\frac{1}{2},j+\frac{1}{2}}=d_{i,j}d_{i+1,j+1}-d_{i+1,j}d_{i,j+1},$

and the local rule around a black entry is

$d_{i,j}^{2}=d_{i-\frac{1}{2},j-\frac{1}{2}}d_{i+\frac{1}{2},j+\frac{1}{2}}-d_{i+\frac{1}{2},j-\frac{1}{2}}d_{i-\frac{1}{2},j+\frac{1}{2}}.$

The boundary conditions are

  * •

$d_{i,i-1}=d_{i+\frac{1}{2},i-1+\frac{1}{2}}=1$

  * •

$d_{i,i+w}=d_{i+\frac{1}{2},i+w+\frac{1}{2}}=1$.

Note that the index set for the entries is

$\\{(i,j)\;|\,i\in\mathbb{Z},\;i-1\leq j\leq i+w\\}.$

However, we will often consider the $2$-friezes as infinite arrays by adding
the extra conventional conditions:

  * •

$d_{i,i-\ell}=d_{i+\frac{1}{2},i-\ell+\frac{1}{2}}=0$, for $\ell=2,3,4$,

  * •

$d_{i,j+w+5}=-d_{i,j}$ for all $i,j$,

which means that we add three rows of zeros at the top and the bottom of the
frieze and extend it to an antiperiodic infinite array.

Figure 3 shows how the entries are organized in the plane. Note that a row in
the frieze consists in the entries $d_{i,j}$ where $j-i$ is a fixed constant
whereas $i$ fixed or $j$ fixed form the diagonals of the frieze.

$\textstyle{d_{i-\frac{3}{2},j+\frac{3}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-\frac{3}{2},j+\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-1,j+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-\frac{1}{2},j+\frac{3}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-\frac{3}{2},j-\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-1,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-\frac{1}{2},j+\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i,j+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{d_{i-1,j-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i-\frac{1}{2},j-\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i+\frac{1}{2},j+\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i+1,j+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i,j-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i+\frac{1}{2},j-\frac{1}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i+1,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d_{i+1,j-1}}$
Figure 3. Coordinate labelling in a symplectic $2$-frieze. Black entries are
connected with plain lines and the white entries with dashed lines.

###  3.2. The tameness condition

We define the notion of “tame” 2-friezes. This notion is an analog of the
tameness condition for the classical $\mathrm{SL}$-friezes, see Appendix B.1.
It is understood as a genericity condition, see Proposition 3.2 and Remark
3.3.

######  Definition 3.1.

A symplectic $2$-frieze is called tame if the following three conditions are
satisfied

  1. (i)

All $3\times 3$-minors of adjacent black entries are equal to their central
elements, i.e. for all $i,j\in\mathbb{Z}$

(3.8) $\left|\begin{array}[]{lll}d_{i-1,j-1}&d_{i-1,j}&d_{i-1,j+1}\\\\[4.0pt]
d_{i,j-1}&d_{i,j}&d_{i,j+1}\\\\[4.0pt]
d_{i+1,j-1}&d_{i+1,j}&d_{i+1,j+1}\end{array}\right|=d_{i,j}.$

  2. (ii)

All $4\times 4$-minors of adjacent black entries are equal to 1, i.e. for all
$i,j\in\mathbb{Z}$

(3.9)
$\left|\begin{array}[]{llll}d_{i-1,j-1}&d_{i-1,j}&d_{i-1,j+1}&d_{i-1,j+2}\\\\[6.0pt]
d_{i,j-1}&d_{i,j}&d_{i,j+1}&d_{i,j+2}\\\\[6.0pt]
d_{i+1,j-1}&d_{i+1,j}&d_{i+1,j+1}&d_{i+1,j+2}\\\\[6.0pt]
d_{i+2,j-1}&d_{i+2,j}&d_{i+2,j+1}&d_{i+2,j+2}\\\ \end{array}\right|=1.$

  3. (iii)

All $5\times 5$-minors of adjacent black entries are equal to 0, i.e. for all
$i,j\in\mathbb{Z}$

(3.10)
$\left|\begin{array}[]{cccc}d_{i-1,j-1}&d_{i-1,j}&\ldots&d_{i-1,j+3}\\\\[6.0pt]
d_{i,j-1}&d_{i,j}&\ldots&\vdots\\\\[6.0pt]
\vdots&\vdots&\ddots&\vdots\\\\[6.0pt]
d_{i+3,j-1}&\ldots&\ldots&d_{i+3,j+3}\\\ \end{array}\right|=0.$

The first examples of tame symplectic $2$-friezes are given by the friezes
containing no zero entries. However the conditions of tame friezes allow to
have zero entries, see Examples 3.4 and 3.5 below.

######  Proposition 3.2.

If there are no zero entries in a symplectic $2$-frieze then it is tame.

###### Proof.

The key formula is the Desnanot-Jacobi determinantal identity, see Appendix A.
Consider the following piece of symplectic $2$-frieze, where the upper case
(resp. lower case) entries are considered black entries (resp. white entries):

$\begin{array}[]{ccccccccccccccc}&&&&D\\\\[3.0pt] &&&C&c&H\\\\[3.0pt]
&&B&b&G&f&L\\\\[3.0pt] &A&a&F&e&K&i&P\\\\[3.0pt] &&E&d&J&h&O\\\\[3.0pt]
&&&I&g&N\\\\[3.0pt] &&&&M\end{array}$

By Desnanot-Jacobi identity one has the following relation between the minors

(3.11) $\left|\begin{array}[]{ccc}A&B&C\\\ E&F&G\\\
I&J&K\end{array}\right|F=\left|\begin{array}[]{ccc}A&B\\\ E&F\\\
\end{array}\right|\left|\begin{array}[]{ccc}F&G\\\
J&K\end{array}\right|-\left|\begin{array}[]{ccc}E&F\\\
I&J\end{array}\right|\left|\begin{array}[]{ccc}B&C\\\ F&G\\\
\end{array}\right|$

Using the frieze rule in the above equation one obtains the relation

$\left|\begin{array}[]{ccc}A&B&C\\\ E&F&G\\\ I&J&K\end{array}\right|F=ae-
bd=F^{2},$

from which we deduce (3.8) by cancelling out $F\not=0$:

$\left|\begin{array}[]{ccc}A&B&C\\\ E&F&G\\\ I&J&K\end{array}\right|=F.$

Using again Desnanot-Jacobi identity, and then (3.8) one obtains:

$\displaystyle\left|\begin{array}[]{cccc}A&B&C&D\\\ E&F&G&H\\\ I&J&K&L\\\
M&N&O&P\end{array}\right|\cdot\left|\begin{array}[]{ccc}F&G\\\
J&K\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}A&B&C\\\ E&F&G\\\
I&J&K\end{array}\right|\cdot\left|\begin{array}[]{ccc}F&G&H\\\ J&K&L\\\
N&O&P\end{array}\right|-\left|\begin{array}[]{ccc}E&F&G\\\ I&J&K\\\
M&N&O\end{array}\right|\cdot\left|\begin{array}[]{ccc}B&C&D\\\ F&G&H\\\
J&K&L\end{array}\right|$ $\displaystyle=$ $\displaystyle FK-JG,$

from which we deduce (3.9) by cancelling out $FK-JG=e\not=0$:

$\left|\begin{array}[]{cccc}A&B&C&D\\\ E&F&G&H\\\ I&J&K&L\\\
M&N&O&P\end{array}\right|=1.$

Applying again Desnanot-Jacobi identity to compute the $5\times 5$-minors and
using (3.8) and (3.9) already established one gets that all $5\times 5$-minors
vanish. Hence (3.10). ∎

######  Remark 3.3.

Note that generically the three conditions for the tameness property of a
symplectic 2-frieze are automatically satisfied according to the arguments
given in the proof of Proposition 3.2. More precisely,

– (3.8) is satisfied for a given $(i,j)$ whenever $d_{i,j}\not=0$;

– when (3.8) is satisfied, then (3.9) is automatically satisfied for a given
$(i,j)$ whenever $d_{i+\frac{1}{2},j+\frac{1}{2}}\not=0$;

– when (3.8) and (3.9) are satisfied, then (3.10) is automatically satisfied
for a given $(i,j)$ whenever $d_{i+1,j+1}\not=0$.

######  Example 3.4.

(a) In the following array, containing a black zero entry, the frieze rule is
satisfied for any values of $x$.

$\begin{array}[]{rrrrrrrrrcrrrr}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&-1&\mathbf{1}&-2&\mathbf{-1}&-1&\mathbf{0}&-1&\mathbf{x}&-1-x^{2}&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

The next two entries on the right can be computed if $x(1+x^{2})\not=0$. If we
want the array to be tame, the condition (3.8) imposes

$\left|\begin{array}[]{ccc}-1&1&0\\\ 1&0&1\\\ 0&1&x\end{array}\right|=0$

so that it determines $x=1$. This will lead to the (6-periodic) tame frieze
(2.3).

(b) In the following array, containing a white zero entry, the frieze rule is
satisfied for any values of $x$ (that will then uniquely determine $y$ and
$z$).

(3.13)
$\begin{array}[]{ccccccccccccccccccccccccc}\cdots&1&&\mathbf{1}&&1&\mathbf{1}&1&&\mathbf{1}&&1&\cdots\\\\[4.0pt]
\cdots&\mathbf{\sqrt{2}}&&1&&\mathbf{\sqrt{2}}&0&\mathbf{-\sqrt{2}+1}&&x&&\mathbf{y}&\cdots\\\\[4.0pt]
\cdots&-1&&\mathbf{1}&&-2&\mathbf{-2+\sqrt{2}}&-3+2\sqrt{2}&&\mathbf{1}&&z&\cdots\\\\[4.0pt]
\cdots&\mathbf{1}&&1&&\mathbf{1}&1&\mathbf{1}&&1&&\mathbf{1}&\cdots\\\
\end{array}$

The tameness condition (3.8) is also satisfied for any values of $x$.
Condition (3.9) imposes

$\left|\begin{array}[]{cccc}\sqrt{2}&1&0&0\\\\[2.0pt]
1&\sqrt{2}&1&0\\\\[2.0pt] 1&-2+\sqrt{2}&-\sqrt{2}+1&1\\\\[2.0pt]
0&1&1&y\end{array}\right|=1$

so that one solves $y=\sqrt{2}-1$, and then $x=-4+2\sqrt{2}$, $z=1$.
Therefore, there is a unique way to extend the array to get a tame symplectic
2-frieze.

######  Example 3.5.

(a) The following symplectic $2$-frieze

(3.14)
$\begin{array}[]{ccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\cdots&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&\cdots\\\
\cdots&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

is not tame because Condition (3.8) fails

$\left|\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right|\not=0$

(b) The following symplectic $2$-frieze

$\begin{array}[]{ccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\cdots&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&\cdots\\\
\cdots&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\cdots\\\
\cdots&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&0&\mathbf{0}&\cdots\\\
\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\end{array}$

is not tame because Condition (3.9) fails

$\left|\begin{array}[]{cccc}0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\end{array}\right|\not=1.$

(c) There is only one tame symplectic $2$-frieze with constant first row equal
to 0. It is the following frieze of width 7:

(3.15)
$\begin{array}[]{rrrrrrrrrrrrrrrrrrrrrr}\cdots&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&\cdots\\\
\cdots&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&\cdots\\\
\cdots&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&\cdots\\\
\cdots&1&\mathbf{-1}&&1&\mathbf{-1}&&1&\mathbf{-1}&&1&\mathbf{-1}&&1&\mathbf{-1}&\cdots\\\
\cdots&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&\cdots\\\
\cdots&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&&0&\mathbf{0}&\cdots\\\
\cdots&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&&\mathbf{0}&0&\cdots\\\
\cdots&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&&1&\mathbf{1}&\cdots\\\
\end{array}$

######  Remark 3.6.

The conventional choice to extend the array of a symplectic $2$-frieze with
three rows of zeros at the top and bottom and then by antiperiodicty
$d_{i+w+5,j}=d_{i,j+w+5}=-d_{i,j}$ (c.f. §3.1) is justified as the only choice
that extends a $2$-frieze with no zero entry to an infinite tame array.

###  3.3. $\mathrm{SL}$-friezes and difference equations

By definition, the tameness of the symplectic frieze forces the subarray of
the black entries to form a tame $\mathrm{SL}_{4}$-frieze. Hence, many
properties of the symplectic friezes are immediate consequences of known
results on $\mathrm{SL}$-friezes, see Appendix B.

######  Proposition 3.7.

Let $w$ be the width of the friezes.

(i) The subarray of the black entries in a tame symplectic $2$-frieze forms a
tame $\mathrm{SL}_{4}$-frieze in which the ajacent $3\times 3$ minors are
equal to the central element. Conversely, every such $\mathrm{SL}_{4}$-frieze
determines a unique tame symplectic $2$-frieze.

(ii) The subarray of the black entries in a tame symplectic $2$-frieze forms a
tame $\mathrm{SL}_{4}$-frieze invariant under the glide reflection along the
median line:

$d_{j-w-2,i-3}=d_{i,j}=d_{j+3,i+w+2},\quad i\leq j\leq i+w-1.$

Conversely, every such $SL_{4}$-frieze determines a unique tame symplectic
$2$-frieze.

(iii) In a tame symplectic $2$-frieze the entries $a_{i}:=d_{i,i}$,
$b_{i}:=d_{i-\frac{1}{2},i-\frac{1}{2}}$ define the coefficients of a
$(w+5)$-superperiodic equation of the form (1.1). Moreover, the sequences on
the diagonals $(d_{i_{0},i})_{i\in\mathbb{Z}}$, for every fixed
$i_{0}\in\mathbb{Z}$, are solutions of the equation (1.1), i.e. for all
$i_{0}\in\mathbb{Z}$

$d_{i_{0},i}=a_{i}\,d_{i_{0},i-1}-b_{i}\,d_{i_{0},i-2}+a_{i-1}d_{i_{0},i-3}-d_{i_{0},i-4},\quad\text{for
all }i\in\mathbb{Z}.$

Conversely, every $n$-superperiodic equation of the form (1.1) determines a
tame symplectic $2$-frieze.

###### Proof.

We already know that the $\mathrm{SL}_{4}$-frieze corresponds to a
superperiodic difference equation of order 4, see Proposition B.1. We first
show the following lemma stating that the two additional symmetries on the
$\mathrm{SL}_{4}$-frieze and the symmetry on the coefficients of the
corresponding difference equation are three equivalent conditions.

######  Lemma 3.8.

Given a tame $\mathrm{SL}_{4}$-frieze, the following properties are
equivalent:

  1. (1)

All $3\times 3$-minors of adjacent black entries are equal to their central
elements,

  2. (2)

The array is invariant under a glide reflection along the median line,

  3. (3)

The corresponding superperiodic equation is of the form (1.1).

###### Proof.

We use the formulas given in Appendix B (in our case $k=3$). Formula (B.48)
gives

$d_{i,j}=\left|\begin{array}[]{ccc}d_{j-w-3,i-4}&d_{j-w-3,i-3}&d_{j-w-3,i-2}\\\\[6.0pt]
d_{j-w-2,i-4}&d_{j-w-2,i-3}&d_{j-w-2,i-2}\\\\[6.0pt]
d_{j-w-1,i-4}&d_{j-w-1,i-3}&d_{j-w-1,i-2}\\\\[6.0pt] \end{array}\right|.$

From which we deduce that the property on the $3\times 3$ minors of (1) is
equivalent to the glide symmetry of (2). The equation is of the form
$V_{i}=a_{i}\,V_{i-1}-b_{i}\,V_{i-2}+c_{i}V_{i-3}-V_{i-4}$ and Formula (B.32)
gives $c_{i}=a_{i}^{3}$ as a $1\times 1$-minor and $a_{i}=a_{i}^{1}$ as a
$3\times 3$ minor,

$c_{i}=d_{i+2,i+w+1},\quad\text{ and }\quad
a_{i-1}=\left|\begin{array}[]{cccc}d_{i+1,i+w}&1&0\\\\[4.0pt]
d_{i+2,i+w}&d_{i+2,i+w+1}&1\\\\[4.0pt]
d_{i+3,i+w}&d_{i+3,i+w+1}&d_{i+j+1,i+w+2}\end{array}\right|.$

Hence, the property on the $3\times 3$ minors of (1) implies the symmetry in
property (3). Finally, if (3) is satisfied i.e. $c_{i}=a_{i-1}$, Formulas
(B.39) and (B.46) respectively give

$d_{i,j}=\left|\begin{array}[]{cccclcc}a_{i}&1&\\\\[8.0pt]
b_{i+1}&a_{i+1}&1&\\\\[4.0pt] a_{i+1}&\ddots&\ddots&\ddots\\\\[6.0pt]
1&&&\ddots&\ddots\\\\[4.0pt] &\ddots&\ddots&&a_{j}&1\\\\[8.0pt]
&&1&a_{j-1}&b_{j}&a_{j}\end{array}\right|,\;d_{j+3,i+w+2}=\left|\begin{array}[]{cccccc}a_{i}&b_{i+1}&a_{i+1}&\;\;1&&\\\\[8.0pt]
1&a_{i+1}&b_{i+2}&a_{i+2}&\\!\\!1&\\\\[8.0pt]
&1&\ddots&\ddots&&\\!\\!1\\\\[8.0pt] &&\ddots&&\ddots&a_{j-1}\\\\[12.0pt]
&&&\;1&a_{j-1}&b_{j}\\\\[8.0pt] &&&&\\!\\!\\!\\!1&a_{j}\end{array}\right|.$

From which we deduce the glide symmetry (2). ∎

By definition, the black array of tame symplectic $2$-frieze is a tame
$\mathrm{SL}_{4}$-frieze in which the $3\times 3$-minors are equal to their
middle elements. Therefore, the direct statements in (i), (ii), (iii) of
Proposition 3.7 are immediate consequences of the above lemma.

Conversely, starting from a tame $\mathrm{SL}_{4}$-frieze pattern
$(d_{i,j})_{i,j\in\mathbb{Z}}$ one can complete the array by computing all the
$2\times 2$-minors
$d_{i-\frac{1}{2},j-\frac{1}{2}}:=d_{i,j}d_{i-1,j-1}-d_{i-1,j}d_{i,j-1}$. If
the initial $\mathrm{SL}_{4}$-frieze has all its $3\times 3$-minors equal to
their central elements then the completed array will satisfy the $2$-frieze
rule thank to the relation (3.11). This establishes all the converse
statements in (i), (ii), (iii) of Proposition 3.7.

∎

###  3.4. Determinantal formulas for the entries in a symplectic $2$-frieze

The entries of a tame symplectic $2$-frieze are given by multi-diagonals
determinants.

######  Proposition 3.9.

Let $(a_{i},b_{i})_{i}$ be the coefficients of the difference equation
corresponding to a tame symplectic $2$-frieze. One has

$\begin{array}[]{lllll}d_{i,j}=&d_{i-\frac{1}{2},j-\frac{1}{2}}=\\\\[8.0pt]
\left|\begin{array}[]{ccccccccccccc}a_{i}&b_{i+1}&a_{i+1}&1\\\\[3.0pt]
1&a_{i+1}&b_{i+2}&a_{i+2}&1\\\\[6.0pt]
&\ddots&\ddots&\ddots&\ddots&\ddots\\\\[6.0pt]
&&1&a_{j-2}&b_{j-1}&a_{j-1}\\\\[3.0pt] &&&1&a_{j-1}&b_{j}\\\\[3.0pt]
&&&&1&a_{j}\end{array}\right|,&\left|\begin{array}[]{ccccccccccccc}b_{i}&a_{i}&1\\\\[3.0pt]
a_{i}&b_{i+1}&a_{i+1}&1\\\\[3.0pt] 1&a_{i+1}&b_{i+2}&a_{i+2}&1\\\\[6.0pt]
&\ddots&\ddots&\ddots&\ddots&\ddots\\\\[6.0pt]
&&1&a_{ij-2}&b_{j-1}&a_{j-1}\\\\[3.0pt] &&&1&a_{j-1}&b_{j}\\\
\end{array}\right|,\end{array}$

for all $i,j\in\mathbb{Z}$. In particular, $a_{i}=d_{i,i}$,
$b_{i}=d_{i-\frac{1}{2},i-\frac{1}{2}}$ are the entries in the non trivial
bottom row of the symplectic $2$-frieze.

###### Proof.

The formula for $d_{i,j}$ is given by (B.39). Using the frieze rule
$d_{i-\frac{1}{2},j-\frac{1}{2}}=d_{i,j}d_{i-1,j-1}-d_{i-1,j}d_{i,j-1}$ and
Desnanot-Jacobi identity in the determinant of $d_{i-1,j}$ one deduces the
above formula for $d_{i-\frac{1}{2},j-\frac{1}{2}}$.

∎

###  3.5. Proofs of Theorem 1, Theorem 2 and Theorem 3

The theorems are mainly consequences and reformulations of Proposition 3.7.

Theorem 1 is an immediate consequence of Proposition 3.7 (ii).

Theorem 2 (1) and (2) are also immediate consequences of Proposition 3.7 (i)
and (ii). Theorem 2 (3) is obtained by applying the Gale duality, see Appendix
B.5. More precisely, if $a_{i},b_{i}$ are the coefficients of the associated
difference equation then the corresponding $\mathrm{SL}_{w+1}$-frieze of width
3 is given as the following array

$\begin{array}[]{cccccccccccc}&1&&1&&1&&1&&1&&\\\
\cdots&&a_{0}&&a_{1}&&a_{2}&&a_{3}&&\cdots\\\
&b_{0}&&b_{1}&&b_{2}&&b_{3}&&\cdots\\\
\cdots&&a_{0}&&a_{1}&&a_{2}&&a_{3}&&\cdots\\\ &1&&1&&1&&1&&1&&\\\
\end{array}.$

Theorem 3 is deduced from Proposition 3.7 (iii) and the fact that the
coefficients of the equation are also entries in the $2$-frieze, see
Proposition 3.9 (note that the first row and the last row of a $2$-frieze
contain the same sequence by glide symmetry).

##  4\. Legendrian $n$-gons

###  4.1. Lift of the projective $n$-gons in $\mathbb{C}^{4}$

Consider the symplectic space $(\mathbb{C}^{4},\omega_{0})$ where $\omega_{0}$
is the symplectic form given by the matrix

$\Omega_{0}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\end{array}\right).$

This naturally equipps the projective space ${\mathbb{CP}}^{3}$ with a
standard contact structure $(H_{v})_{v\in{\mathbb{CP}}^{3}}$, where $H_{v}$ is
the projectivization of the hyperplane $v^{\bot}$ orthogonal to the line $v$
in $(\mathbb{C}^{4},\omega_{0})$.

A Legendrian $n$-gons in ${\mathbb{CP}}^{3}$, see Definition 1.2, can be
lifted to $(\mathbb{C}^{4},\omega_{0})$ in a unique way (up to a sign) by
imposing some normalization conditions. The normalized lift provides us with a
system of $2n$ parameters for the Legendrian $n$-gons. The general statement
in higher dimension can be found in [CO]. We specify the construction in the
particular case of dimension 4 and sketch the proof in this case.

######  Proposition 4.1 ([CO]).

Let $n$ be an odd integer. Let $v=(v_{i})_{i\in\mathbb{Z}}$ be a generic
Legendrian $n$-gon in $({\mathbb{CP}}^{3},\omega_{0})$.

  1. (1)

There exists a lift $V=(V_{i})_{i\in\mathbb{Z}}$ in $\mathbb{C}^{4}$ of $v$
satisfying

    1. (a)

$V_{i+n}=-V_{i}$, for all $i$,

    2. (b)

$\omega_{0}(V_{i},V_{i+1})=0$ for all $i$,

    3. (c)

$\omega_{0}(V_{i},V_{i+2})=1$ for all $i$.

  2. (2)

The sequences $V$ and $-V$ are the unique lifts of $v$ satisfying the above
conditions.

  3. (3)

The sequence $V$ satisfies

(4.16) $V_{i}=a_{i}\,V_{i-1}-b_{i}\,V_{i-2}+a_{i-1}V_{i-3}-V_{i-4},$

for all $i\in\mathbb{Z}$ for some unique $n$-periodic sequences of complex
numbers $a(v)=(a_{i})_{i\in\mathbb{Z}}$ and $b(v)=(b_{i})_{i\in\mathbb{Z}}$.

  4. (4)

A Legendrian $n$-gon $v^{\prime}$ satisfies $a(v^{\prime})=a(v)$ and
$b(v^{\prime})=b(v)$ if and only if $v^{\prime}$ and $v$ are
$\mathrm{P}\mathrm{Sp}_{4}$-equivalent.

###### Proof.

We start with an arbitrary lift $(V_{1},\ldots,V_{n})$ of
$(v_{1},\ldots,v_{n})$. The sequence automatically satisfies
$\omega_{0}(V_{i},V_{i+1})=0$ since by definition the Legendrian $n$-gons
satisfy $v_{i+1}\in H_{v_{i}}$.

Generically $v_{i+2}\not\in H_{v_{i}}$ so we get $n$ non zero constants
$\gamma_{i}:=\omega_{0}(V_{i},V_{i+2})$, $1\leq i\leq n-2$,
$\gamma_{n-1}:=\omega_{0}(V_{n-1},V_{1})$,
$\gamma_{n}:=\omega_{0}(V_{n},V_{2})$. When $n$ is odd, the following system
of $n$ equations

(4.17)
$\left\\{\begin{array}[]{lclcrccccc}\omega_{0}(\lambda_{1}V_{1},\lambda_{3}V_{3})&=&\lambda_{1}\lambda_{3}\gamma_{1}&=&1\\\
\vdots&\vdots&&&\vdots\\\
\omega_{0}(\lambda_{n-2}V_{n-2},\lambda_{n}V_{n})&=&\lambda_{n-2}\lambda_{n}\gamma_{n-2}&=&1\\\
\omega_{0}(\lambda_{n-1}V_{n-1},\lambda_{1}V_{1})&=&\lambda_{n-1}\lambda_{1}\gamma_{n-1}&=&-1\\\
\omega_{0}(\lambda_{n}V_{n},\lambda_{2}V_{2})&=&\lambda_{n}\lambda_{2}\gamma_{n}&=&-1\\\
\end{array}\right.$

has exactly two opposite solutions $\pm(\lambda_{1},\ldots,\lambda_{n})$ in
terms of $\gamma_{i}$, $1\leq i\leq n$. We rescale the $V_{i}$’s to
$\lambda_{i}V_{i}$ by using one of the solutions of the system. The rescaled
sequence $(V_{i})$ satisfies the condition (1c). We extend the sequence by
antiperiodicity, i.e. we set $V_{i+kn}=(-1)^{k}V_{i}$, for all $1\leq i\leq
n$, $k\in\mathbb{Z}$. The sequence $V$ and its opposite $-V$ are the only ones
satisfying the conditions of (1) by unicity, up to a sign, of the solution of
the system (4.17).

Items (1) and (2) are proved.

Every four consecutive points of the normalized sequence $V$ form a basis of
$\mathbb{C}^{4}$, so that there is a unique sequence of coefficients
$(a_{i},b_{i},c_{i},d_{i})_{i}$, which is $n$-periodic and satisfies

(4.18) $V_{i}=a_{i}\,V_{i-1}-b_{i}\,V_{i-2}+c_{i}V_{i-3}-d_{i}V_{i-4},$

for all $i\in\mathbb{Z}$.

Applying $\omega_{0}(V_{i-2},-)$ in (4.18) and using the normalization
conditions of (1), we get $d_{i}=1$ for all $i\in\mathbb{Z}$.

Applying $\omega_{0}(V_{i-3},-)$ in (4.18), we obtain
$a_{i}=\omega_{0}(V_{i-3},V_{i})$ for all $i\in\mathbb{Z}$, that gives by a
shift $a_{i-1}=\omega_{0}(V_{i-4},V_{i-1})$. And finally applying
$\omega_{0}(-,V_{i-1})$ in (4.18), we obtain
$c_{i}=\omega_{0}(V_{i-4},V_{i-1})=a_{i-1}$. Item (3) is proved.

Let $v$ and $v^{\prime}$ be two Legendrian $n$-gons such that their normalized
lifts $V$ and $V^{\prime}$ satisfy the same recurrence relation (4.16). Define
$T$ in $\mathrm{GL}_{4}$ such that $TV_{i}=V^{\prime}_{i}$ for $i=1,2,3,4$.
The recurrence relation implies $TV_{i}=V^{\prime}_{i}$ for all $i$. From the
normalization conditions and the fact that
$a_{4}=\omega_{0}(V_{1},V_{4})=\omega_{0}(V^{\prime}_{1},V^{\prime}_{4})$, one
can see that the matrix $T$ preserves $\omega_{0}$. Hence, $T$ is an element
of $\mathrm{Sp}_{4}$ that transforms $v$ to $v^{\prime}$.

Conversely if $v^{\prime}$ and $v$ are $\mathrm{P}\mathrm{Sp}_{4}$-equivalent
then there exists $T\in\mathrm{Sp}_{4}$ that transforms the normalized lifts
$V$ to $V^{\prime}$. Therefore the sequences $V$ and $V^{\prime}$ satisfy the
same recurrence relation. Item (4) is proved. ∎

######  Remark 4.2 (Proof of Theorem 5).

From Proposition 4.1 one can identify the Legendrian $n$-gons, modulo
$\mathrm{P}\mathrm{Sp}_{4}$-equivalence, with $n$-super-periodic equations of
the form (4.16), provided $n$ is odd. Therefore Theorem 5 is an immediate
consequence of Theorem 3.

###  4.2. Symplectic $2$-friezes from Legendrian $n$-gons

One can obtain a tame symplectic $2$-frieze from the lift of a Legendrian
$n$-gon using the canonical symplectic form $\omega_{0}$ on $\mathbb{C}^{4}$.

######  Proposition 4.3.

Let $v=(v_{i})$ be a Legendrian $n$-gon in ${\mathbb{CP}}^{3}$ and $V=(V_{i})$
be its normalized lift in $(\mathbb{C}^{4},\omega_{0})$ given by Proposition
4.1. The following assignment

$d_{i,j}:=\omega_{0}(V_{i-3},V_{j}),\qquad\text{for all }i,j\in\mathbb{Z}$

defines the black array of a tame symplectic $2$-frieze of width $w=n-5$.

###### Proof.

Using the normalization properties of $V=(V_{i})$ and the linearity of
$\omega_{0}$ one deduces that the sequences

$d_{i,\bullet}=(0,\ldots,0,1,d_{i,i},d_{i,i+1},\ldots,d_{i,i+w-1},1)$

are solutions of the same $w+5$-superperiodic equation of the form (4.16) with
different initial conditions

$(V_{i-4},\ldots,V_{i-1})=(0,\ldots,0,1).$

By Proposition B.1 this implies that $(d_{i,j})$ form a tame
$\mathrm{SL}_{4}$-frieze. And since the coefficients of the equation are
symmetric it implies by Proposition 3.7 that this $\mathrm{SL}_{4}$-frieze is
indeed the black array of a tame symplectic $2$-frieze. ∎

######  Remark 4.4.

Alternatively, one can compute the symplectic $2$-frieze using $4\times
4$-determinants involving four vertices of the $n$-gon:

$d_{i,j}=\det(V_{i-4},V_{i-3},V_{i-2},V_{j}),\qquad
d_{i-\frac{1}{2},j-\frac{1}{2}}=\det(V_{i-4},V_{i-3},V_{j-1},V_{j}).$

This is obtained as a particular case of the general correspondence between
$n$-gons and $\mathrm{SL}_{k}$-friezes without considering the extra
symplectic structure, see e.g. Formula (B.49).

###  4.3. Symplectic forms and $2$-friezes

Consider the following family of matrices of symplectic forms

$\Omega_{a}=\left(\begin{array}[]{cccc}0&0&1&a\\\ 0&0&0&1\\\ -1&0&0&0\\\
-a&-1&0&0\end{array}\right),\qquad\check{\Omega}_{a}=\left(\begin{array}[]{cccc}0&0&1&0\\\
0&0&-a&1\\\ -1&a&0&0\\\
0&-1&0&0\end{array}\right)=\,^{t}\Omega_{a}^{-1}=-\Omega_{a}^{-1},$

where $a$ is a complex parameter.

######  Proposition 4.5.

In a symplectic $2$-friezes, every $4\times 4$ submatrices of black entries

(4.19) $D_{i,j}:=\left(\begin{array}[]{llll}d_{i,j-3}&\ldots&d_{i,j}\\\
\vdots&&\vdots\\\ d_{i+3,j-3}&\ldots&d_{i+3,j+3}\\\ \end{array}\right)$

satisfies

${}^{t}\\!D_{i,j}\check{\Omega}_{a_{i}}D_{i,j}={\Omega}_{a_{j}},$

where $a_{i}=d_{i,i}$ are the black entries in the first (and last) row of the
$2$-frieze.

###### Proof.

It is easy to check that the property holds for the matrices $D_{i,i}$ since
$D_{i,i}=\Omega_{a_{i}}$. By the recurrence relation in the frieze one has
$D_{i,j+1}=D_{i,j}E_{j+1}$, where $E_{j+1}$ is the matrix

(4.20) $E_{j+1}:=\left(\begin{array}[]{cccr}0&0&0&-1\\\ 1&0&0&a_{j}\\\
0&1&0&-b_{j}\\\ 0&0&1&a_{j}\end{array}\right).$

Hence, the property for all $D_{i,j}$ follows by easy induction on $j$. ∎

###  4.4. Proof of Proposition 1.1

Let $M$ be a monodromy matrix associated to the equation (1.1). It can be
defined as the product $M=E_{1}E_{2}\cdots E_{n}$, where $E_{i}$ are the
matrices (4.20). The condition of superperiodicity of (1.1) translates as
$M=-\mathrm{Id}$. We want to prove that the algebraic conditions on the
coefficients $(a_{i})$, $(b_{i})$ given by $M=-\mathrm{Id}$ are equivalent to
the one given in system (1.2).

We apply the matrix $M$ on the “initial” fragment of $2$-frieze given as the
matrix

$D_{0,0}=\left(\begin{array}[]{cccc}0&0&1&a_{0}\\\ 0&0&0&1\\\ -1&0&0&0\\\
-a_{0}&-1&0&0\end{array}\right).$

The recurrence relations in the frieze imply that $D_{0,0}M=D_{0,n}$. Hence,
$M=-\mathrm{Id}$ is equivalent to $D_{0,n}=-D_{0,0}$. This gives 16 equations
when identifying the entries of the matrices. However, since $D_{0,n}$
satisfies the relation
${}^{t}\\!D_{0,n}\check{\Omega}_{a_{0}}D_{0,n}={\Omega}_{a_{0}}$ according to
Proposition 4.5 it is enough to identify the triangular lower parts of the
matrices, i.e.

$\left(\begin{array}[]{llll}d_{0,n-3}&\cdot&\cdot&\cdot\\\
d_{1,n-3}&d_{1,n-2}&\cdot&\cdot\\\ d_{2,n-3}&d_{2,n-2}&d_{2,n-1}&\cdot\\\
d_{3,n-3}&d_{3,n-2}&d_{3,n-2}&d_{3,n}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}0&\cdot&\cdot&\cdot\\\
0&0&\cdot&\cdot\\\ 1&0&0&\cdot\\\ a_{0}&1&0&0\end{array}\right),$

which will give the system (1.2) using the expressions for the entries given
in Proposition 3.9.

###  4.5. More correspondences

We explain different ways to obtain symplectic $2$-friezes, Legendrian
$n$-gons, and difference equations one from another.

If one knows the $(a_{i},b_{i})$-parameters of the Legendrian $n$-gon as
defined in Proposition 4.1, or equivalently the $(a_{i},b_{i})$-coefficients
of the associated difference equation (1.1), then the corresponding $2$-frieze
can be computed in two alternative easy ways.

1:

One can write down the coefficients in the first non-trivial row
$(\ldots,b_{i},\textbf{a}_{i},b_{i+1},\textbf{a}_{i+1},\ldots)$ of the array,
and compute the rest of the entries by applying the frieze rule.

2:

One can recursively compute the diagonals of the black entries of the
$2$-frieze using the equations (1.1) and initial conditions
$(0,\ldots,0,1)=(V_{i-4},\ldots,V_{i-1})=(d_{i,i-4},\ldots,d_{i,i-1})$.

The first procedure is limited to the case of generic elements, while the
second one can be applied in all cases.

Now, we describe the correspondence between symplectic $2$-friezes and
Legendrian $n$-gons without using explicitly the coefficients of the
recurrence relations. For the rest of the section, we assume $n$ is odd.

Starting from a Legendrian $n$-gon $v=(v_{i})_{i\in\mathbb{Z}}$, we first
consider the normalized lift $V=(V_{i})$ given by Proposition 4.1. We
transform $(V_{i})$ to $(W_{i})$ where $W_{i}=TV_{i}$ for the matrix $T$ of
$\mathrm{Sp}_{4}$ defined by

$T(V_{-3})\,,\,T(V_{-2})\,,\,T(V_{-1})\,,\,T(V_{0})=\left(\begin{array}[]{cccc}0\\\
0\\\ -1\\\ -a_{0}\end{array}\right)\,,\,\left(\begin{array}[]{cccc}0\\\ 0\\\
0\\\ -1\end{array}\right)\,,\,\left(\begin{array}[]{cccc}1\\\ 0\\\ 0\\\
0\end{array}\right)\,,\,\left(\begin{array}[]{cccc}a_{0}\\\ 1\\\ 0\\\
0\end{array}\right).$

with $a_{0}=\omega_{0}(V_{-3},V_{0})$.

The sequence $W$ will have the following form

(4.21)
$\blockarray{ccccccccccccccc}W_{-3}&W_{-2}W_{-1}W_{0}W_{1}W_{2}W_{3}\cdots\cdots
W_{n-3}W_{n-2}W_{n-1}W_{n}\\\
\block{(cccccccccccccc)c}001a_{0}\bullet\cdots\bullet 1000-1-a_{0}\\\
0001a_{1}\bullet\cdots\bullet 1000-1\\\ -10001a_{2}\bullet\cdots\bullet
1000\\\ -a_{0}-10001a_{3}\bullet\cdots\bullet 100\\\ $

and can uniquely be extended in a tame $\mathrm{SL}_{4}$-frieze. Adding the
$2\times 2$ minors inside the $\mathrm{SL}_{4}$-frieze will lead to a
symplectic $2$-frieze.

Note that the sequence $W$ does not satisfy the normalization given in
Proposition 4.1 (1) for $\omega_{0}$ but for the form given by
$\Omega_{a_{0}}$ of §4.3

Conversely, if one cuts a $4\times n$ block of the form (4.21) in the black
subarray of a symplectic $2$-frieze it gives $n$ points in $\mathbb{C}^{4}$
satisfying the condition of Proposition 4.1 1 for some symplectic form
$\Omega_{a}$. After renormalizing the points they will project to a Legendrian
$n$-gons in $({\mathbb{CP}}^{3},\omega_{0})$.

###  4.6. The case when $n$ is even

In the case when $n$ is even the system of equations (4.17) splits into two
subsystems: one for the $\lambda_{i}$ with $i$ odd and one for the
$\lambda_{i}$ with $i$ even. If $n$ is a multiple of 4 the set of solutions
for the subsystems is in general empty, otherwise each subsystem has two
opposite sets of solutions $\pm(\lambda_{1},\lambda_{3},\ldots,\lambda_{n-1})$
and $\pm(\lambda_{2},\lambda_{4},\ldots,\lambda_{n})$. So that we get four
sequences of lifted points $\pm(V_{1},\pm V_{2},\ldots,V_{2i-1},\pm
V_{2i},\ldots)$ satisfying the normalisation conditions of Proposition 4.1,
which define the same parameters $(a_{i},b_{i})$ up to a sign for the
coefficients $a_{i}$. This will lead to two distinct $2$-friezes that differ
under the action by conjugation of the matrix

$\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{array}\right),$

on the black subarray.

For instance, the $2$-frieze of width 3 given in (2.6) and the following one

$\begin{array}[]{ccccccccccccccccccccccc}\cdots&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&\mathbf{-1}&1&\mathbf{-2}&5&\mathbf{-4}&6&\mathbf{-4}&6&\mathbf{-3}&2&\mathbf{-1}&1&\mathbf{-4}&30&\mathbf{-10}&4&\mathbf{-1}&1&\cdots\\\
\cdots&3&\mathbf{1}&1&\mathbf{3}&14&\mathbf{10}&20&\mathbf{6}&3&\mathbf{1}&1&\mathbf{3}&14&\mathbf{10}&20&\mathbf{6}&3&\mathbf{1}&\cdots\\\
\cdots&\mathbf{-3}&2&\mathbf{-1}&1&\mathbf{-4}&30&\mathbf{-10}&4&\mathbf{-1}&1&\mathbf{-2}&5&\mathbf{-4}&6&\mathbf{-4}&6&\mathbf{-3}&2&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$

correspond to the same class of symplectic $8$-gons.

###  4.7. Legendrian $n$-gons and self-dual polygons

Let $v=(v_{i})$ be a Legendrian $n$-gon in ${\mathbb{CP}}^{3}$, for some odd
integer $n$. The symmetry of the coefficients in the corresponding difference
equation (4.16) (or equivalently the symmetry in the corresponding
$\mathrm{SL}_{4}$-friezes) implies that the polygon $v$ can be identified with
its dual polygon $v^{*}$ in $({\mathbb{CP}}^{3})^{*}$. More precisely there is
a projective map that sends $v_{i}$ to $v^{*}_{i+n-2}$. The shift can be
computed in terms of $\mathrm{SL}_{4}$-frieze using the Formula (B.48) and the
glide symmetry.

Legendrian $n$-gons are $(n-2)$-self-dual polygons, in the terminology of
[FT09]. The space of Legendrian $n$-gons is of dimension $2(n-5)$ and provides
a $3$-dimensional analog of the space $\mathcal{M}_{m,n}$ of [FT09].

##  5\. Cluster algebras

###  5.1. Preliminaries

Cluster algebras are due to Fomin and Zelevinsky [FZ02], [FZ03]. They are
commutative associative algebras defined by generators and relations inside a
field of fractions. The generators and relations of a cluster algebra are not
given from the begining but they are produced recursively.

A seed in a cluster algebra is a couple $\Sigma=((u_{1},\ldots,u_{m}),B)$,
where $(u_{1},\ldots,u_{m})$ is a transcendence basis of
$\mathbb{C}(x_{1},\ldots,x_{m})$ over $\mathbb{C}$ and $B$ is a skew-
symmetrizable matrix of integer entries (alternatively the matrices can be
replaced by quivers). The initial seed
$\Sigma_{0}=((x_{1},\ldots,x_{m}),B_{0})$ is the initial data needed to
generate the entire cluster algebra. All seeds are created by a process of
mutation from $\Sigma_{0}$. The variables created by mutations are called
cluster variables, they come grouped in a $m$-tuple called cluster. The
complex cluster algebra $\mathcal{A}_{\Sigma_{0}}$ is the subalgebra of
$\mathbb{C}(x_{1},\ldots,x_{m})$ generated by all the cluster variables.

To fix notation and convention we recall briefly the main definitions related
to cluster algebras but refer to the original papers [FZ02], [FZ03] or the
introductory text [FWZ16] for details. We also use the approach of [Dup08] and
[Kel13].

####  5.1.1. Mutations from skew-symmetrizable matrices

The mutation $\mu_{k}$, $1\leq k\leq m$, of a seed
$\Sigma=((u_{1},\ldots,u_{m}),B)$ where $B=(b_{i,j})_{1\leq i,j\leq m}$, is
skew-symmetrizable matrix with integer entries, is a new seed
$\Sigma^{\prime}=\mu_{k}(\Sigma)=((u^{\prime}_{1},\ldots,u^{\prime}_{m}),B^{\prime})$
where the new variables are given by

$\begin{array}[]{ccl}u^{\prime}_{k}&=&\displaystyle\frac{1}{x_{k}}(\prod_{i:b_{ik}>0}u_{i}^{b_{ik}}+\prod_{i:b_{ik}<0}u_{i}^{-b_{ik}})\\\\[20.0pt]
u^{\prime}_{i}&=&u_{i},\quad i\not=k,\\\ &&\end{array}$

and the new matrix $B^{\prime}=(b^{\prime}_{i,j})_{1\leq i,j\leq m}$ given by

(5.22) $b^{\prime}_{ij}=\begin{cases}-b_{ij}&\text{if $i=k$ or
$j=k$;}\\\\[3.61371pt] b_{ij}+b_{ik}b_{kj}&\text{if $b_{ik}>0$ and
$b_{kj}>0$;}\\\\[3.61371pt] b_{ij}-b_{ik}b_{kj}&\text{if $b_{ik}<0$ and
$b_{kj}<0$;}\\\\[3.61371pt] b_{ij}&\text{otherwise.}\end{cases}$

####  5.1.2. Valued quivers

The valued quiver $Q_{v}(B)$ associated to a skew-symmetrizable matrix
$B=(b_{i,j})_{1\leq i,j\leq m}$ with integer entries, is defined as follows:

  * •

$\\{1,\ldots,m\\}$ is the set of vertices,

  * •

each entry $b_{i,j}>0$ gives a weighted arrow $i\to j$ of weight
$(|b_{i,j}|,|b_{j,i}|)$.

When there is a weighted arrow between the vertices $i$ and $j$ the weight
$|b_{i,j}|$ is “attached” to the vertex $i$ and the weight $|b_{j,i}|$
“attached” to the vertex $j$. On the graph of a valued quiver we write the
weights on the top or bottom of the arrows. Note that
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{j}$is
the same weighted arrow as
$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(b,a)}$$\textstyle{i}$.

In the case when $(|b_{i,j}|,|b_{j,i}|)=(b,b)$, we replace the weighted arrow
by $b$ arrows. It may be convenient to adopt the following convention:
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,0)}$$\textstyle{j}$means
no arrow between $i$ and $j$, and
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-a,-b)}$$\textstyle{j}$means
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{j}$.

######  Example 5.1.

For example the matrix

$\left(\begin{array}[]{rrrr}0&1&0&-1\\\ -2&0&1&4\\\ 0&-1&0&-2\\\
1&-2&1&0\end{array}\right)$

gives the following valued quiver
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,2)}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,2)}$(2,4)$\textstyle{3}$

####  5.1.3. Mutation of valued quivers

The mutation at vertex $k$ of a valued quiver $Q_{v}$ gives a valued quiver
$\mu_{k}(Q_{v})$ defined by the following transformations:

  1. (1)

For all paths $i\to k\to j$ in $Q$, change the weighted arrows

$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(c,d)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(e,f)}$$(a,b)$$\textstyle{j}$
to
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(c,d)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(e+ac,\,f+bd)}$$(a,b)$$\textstyle{j}$

and

$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(c,d)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g,h)}$$(a,b)$$\textstyle{j}$
to
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(c,d)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g-ac,\,h-bd)}$$(a,b)$$\textstyle{j}$

where $a,b,c,d>0$ and $e,f,g,h\geq 0$.

  2. (2)

Reverse the weighted arrows incident with $k$, i.e. change all

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{k}$
to
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{j}$

and

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.76042pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
7.52203pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{(c,d)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.76042pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.76042pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{j}$}}}}}}}\ignorespaces}}}}\ignorespaces\qquad\text{to}\qquad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.76042pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.76042pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
7.52203pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{(c,d)}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.76042pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{j}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

This rule gives the correspondence $\mu_{k}(Q_{v}(B)=Q_{v}(\mu_{k}(B))$.
Examples of mutations can be found in Example 5.2 and 5.3.

####  5.1.4. Product of valued quivers

We follow the notation of [Kel13]. Consider the valued quivers associated to
the skew-symmetrizable, resp. skew-symmetric, companion matrices corresponding
to the Dynkin type $\mathrm{C}_{2}$, resp. $\mathrm{A}_{w}$, i.e.

$\textstyle{\vec{\mathrm{C}}_{2}:\;1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;(1,2)}$$\textstyle{2}$$\textstyle{\vec{\mathrm{A}}_{w}:\;1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w}$

We define the square product ${\mathrm{C}_{2}}\square\mathrm{A}_{w}$ as the
following valued quiver

(5.23)
$\textstyle{\overset{w+1}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{w+2}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{w+3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{2w-1}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{2w}{\circ}}$$\textstyle{\mathrm{C}_{2}\square\mathrm{A}_{w}=}$$\textstyle{\underset{1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{2}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{3}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{w-1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{w}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$

It is the valued quiver associated to the following $2w\times 2w$ skew-
symmetrizable matrix

$B_{\mathrm{C}_{2}\square\mathrm{A}_{w}}=\left(\begin{array}[]{rrrrr|rrrrrr}0&-1&0&\;\;0&\cdots&1&0&\;\;0&0&\cdots\\\\[4.0pt]
1&0&1&0&\cdots&0&-1&0&0&\cdots\\\\[4.0pt]
0&-1&0&-1&\cdots&0&0&1&0&\cdots\\\\[4.0pt]
\vdots&&&\vdots&&\vdots&&&\vdots&\\\\[4.0pt]
\hline\cr-2&0&0&0&\cdots&0&1&0&0&\cdots\\\\[4.0pt]
0&2&0&0&\cdots&-1&0&-1&0&\cdots\\\\[4.0pt]
0&0&-2&0&\cdots&0&1&0&1&\cdots\\\\[4.0pt]
\vdots&&&\vdots&&\vdots&&&\vdots&\\\\[4.0pt] \end{array}\right)$

######  Example 5.2.

(a) For $w=1$, $\mathrm{C}_{2}\square\mathrm{A}_{1}$ is just the valued quiver
$\textstyle{\vec{\mathrm{C}}_{2}:\;1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;(1,2)}$$\textstyle{2}$associated
to the matrix $\left(\begin{array}[]{cc}0&1\\\ -2&0\end{array}\right)$.

(b) For $w=2$, one can show that $\mathrm{C}_{2}\square\mathrm{A}_{2}$ is
mutation equivalent to a valued quiver of Dynkin type $\mathrm{F}_{4}$.
Indeed,

$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{4}{\circ}}$$\textstyle{\overset{\mu_{1}}{\longrightarrow}}$$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(2,1)$$\textstyle{\overset{4}{\circ}}$$\textstyle{\overset{\mu_{3}}{\longrightarrow}}$$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(2,1)$$\textstyle{\overset{4}{\circ}}$$\textstyle{\overset{\mu_{1}}{\longrightarrow}}$$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{4}{\circ}}$$\textstyle{\underset{1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{2}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{2}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{2}{\bullet}}$$\textstyle{\underset{1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{\underset{2}{\bullet}}$

######  Example 5.3.

Let $\Sigma=((x_{1},x_{2},\ldots,x_{2w}),\mathrm{C}_{2}\square\mathrm{A}_{w})$
be a seed. A mutation at vertex $i$ for $1\leq i\leq w$ in (5.23) locally
changes the graph around the vertices $i$ and $w+i$ into

$\textstyle{w+i-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{w+i+1}$$\textstyle{i-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i+1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$

if $i$ is even (and a similar picture if $i$ is odd but with reversed arrows),
and changes the variable $x_{i}$ to

(5.24) $x_{i}^{\prime}=\dfrac{x_{i-1}x_{i+1}+x_{w+i}^{2}}{x_{i}}.$

A mutation at vertex $w+i$ for $1\leq i\leq w$ in (5.23) locally changes the
graph around the vertices $i$ and $w+i$ into

$\textstyle{w+i-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+i+1}$$\textstyle{i-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$(1,2)$$\textstyle{i+1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$

if $i$ is even (and a similar picture if $i$ is odd but with reversed arrows),
and changes the variable $x_{w+i}$ to

(5.25) $x_{w+i}^{\prime}=\dfrac{x_{w+i-1}x_{w+i+1}+x_{i}}{x_{w+i}}.$

####  5.1.5. Bipartite belt

A bipartite quiver is such that the vertices can be colored in two colors so
that vertices of the same color are not connected by any arrow. For instance,
the quive (5.23) is bipartite.

Let $\mathcal{Q}$ be a bipartite quiver. Following Fomin-Zelevinsky [FZ07],
consider the iterated mutations

$\mu_{+}=\prod_{\text{white }k}\;\mu_{k},\qquad\mu_{-}=\prod_{\text{black
}k}\;\mu_{k}.$

Note that the mutation $\mu_{k}$’s with $k$ of a fixed color commute with each
other since the vertices are not connected.

The mutations $\mu_{+}$ and $\mu_{-}$ on $Q$ give the same quiver with
reversed orientation:

$\mu_{+}(\mathcal{Q})=\mathcal{Q}^{\hbox{op}},\qquad\mu_{-}(\mathcal{Q}^{\hbox{op}})=\mathcal{Q}.$

Let $\Sigma_{0}=(\underline{x},\mathcal{Q})$ be an initial seed. The bipartite
belt, see [FZ07], of the cluster algebra $\mathcal{A}_{\Sigma_{0}}$ is the
collection of seeds obtained from $\Sigma_{0}$ by applying successively
$\mu_{+}$ or $\mu_{-}$:

$\ldots,\;\mu_{-}\mu_{+}\mu_{-}(\Sigma_{0}),\quad\mu_{+}\mu_{-}(\Sigma_{0}),\quad\mu_{-}(\Sigma_{0}),\quad\Sigma_{0},\quad\mu_{+}(\Sigma_{0}),\quad\mu_{-}\mu_{+}(\Sigma_{0}),\quad\mu_{+}\mu_{-}\mu_{+}(\Sigma_{0}),\;\ldots$

######  Remark 5.4.

In general, the bipartite belt is infinite. It has been proved in [Kel10],
that for a bipartite (valued) quiver constructed as a product of two Dynkin
quivers $\Delta\square\Delta^{\prime}$ the mutation $\mu_{+}\mu_{-}$ has
finite order, more precisely

$(\mu_{+}\mu_{-})^{2(h+h^{\prime})}=(\mu_{-}\mu_{+})^{2(h+h^{\prime})}=\mathrm{Id},$

where $h$ and $h^{\prime}$ are the Coxeter numbers of the Dynkin quivers
$\Delta$, $\Delta^{\prime}$ respectively. This property is known as
Zamolodchikov periodicity.

###  5.2. Cluster variables in the symplectic $2$-friezes

Let $F_{w}$ be the tame symplectic $2$-frieze of width $w$ of rational
fractions computed with the frieze rule from two consecutive columns
containing the free variables
$(x_{1},\ldots,x_{w},\mathbf{x_{w+1}},\ldots,\mathbf{x_{2w}})$ organized in a
zig-zag as follows.

(5.26) $F_{w}:\begin{array}[]{cccccccc}\cdots&1&1&1&1&1&1&\cdots\\\\[4.0pt]
&&x_{1}&\mathbf{x_{w+1}}&x^{\prime}_{1}&\mathbf{x^{\prime}_{w+1}}&&\\\\[4.0pt]
&&\mathbf{x_{w+2}}&x_{2}&\mathbf{x^{\prime}_{w+2}}&x^{\prime}_{2}&\\\\[4.0pt]
&&x_{3}&\mathbf{x_{w+3}}&x^{\prime}_{3}&\mathbf{x^{\prime}_{w+3}}&\\\\[4.0pt]
&&\vdots&\vdots&\vdots&\vdots&\\\ \cdots&1&1&1&1&1&1&\cdots\end{array}$

######  Remark 5.5.

The frieze $F_{w}$ is well defined. The operation used to compute the next
entry is subtraction free so that the expressions can not vanish. This allows
to compute recursively all the entries of the frieze as elements of
$\mathbb{C}(x_{1},\ldots,x_{2w})$. Moreover, since the entries are non zero,
the frieze $F_{w}$ is tame by Proposition 3.2.

######  Proposition 5.6.

Let $\mathcal{A}=\mathcal{A}_{\Sigma_{0}}$ be the cluster algebra generated
from the initial seed $\Sigma_{0}=(x_{1},x_{2},\ldots,x_{2w},\;\mathcal{Q})$,
where $\mathcal{Q}=\mathrm{C}_{2}\square\mathrm{A}_{w}$ is the valued quiver
(5.23). Let $F_{w}$ be the formal frieze (5.26) containing the same initial
variables $(x_{1},x_{2},\ldots,x_{2w})$.

  1. (i)

The entries of $F_{w}$ are all cluster variables of $\mathcal{A}$;

  2. (ii)

The pairs of consecutive columns of $F_{w}$ give all the clusters of the
bipartite belt of $\mathcal{A}$;

###### Proof.

(i) is a trivial consequence of (ii). Statement (ii) is established by direct
computations. Indeed, let
$(x^{\prime}_{1},x^{\prime}_{w+2},x^{\prime}_{3},\ldots)$ and
$(x^{\prime}_{w+1},x^{\prime}_{2},x^{\prime}_{w+3},\ldots)$ be the third and
fourth column in $F_{w}$. The entries are deduced using the frieze rule so
that we have for $i$ odd, $1\leq i\leq w$,

$x_{i}^{\prime}=\dfrac{x_{i-1}x_{i+1}+x_{w+i}^{2}}{x_{i}},$

and $i$ even, $1\leq i\leq w$,

$x_{w+i}^{\prime}=\dfrac{x_{w+i-1}x_{w+i+1}+x_{i}}{x_{w+i}}.$

Comparing with the result of the mutation $\mu_{+}$, see (5.24), (5.25), one
immediately gets

$\displaystyle(x^{\prime}_{1},x_{2},x^{\prime}_{3},\dots,x^{\prime}_{2w-1},x_{2w},\quad\mathcal{Q}^{op})$
$\displaystyle=$ $\displaystyle\mu_{+}\Sigma_{0}$
$\displaystyle(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},\dots,x^{\prime}_{2w-1},x^{\prime}_{2w},\quad\mathcal{Q})$
$\displaystyle=$ $\displaystyle\mu_{-}\mu_{+}\Sigma_{0}$

and so on. ∎

######  Remark 5.7.

From the cluster algebra theory Theorem 1 becomes an easy consequence of
Proposition 5.6 and the Zamolodchikov periodicity, see Remark 5.4 (recall that
the Coxeter number for $\mathrm{C}_{2}$ is 4 and the one for $\mathrm{A}_{w}$
is $w+1$). However, the periodicity of the 2-friezes or the periodicity of the
$\mathrm{SL}$-friezes can be established with elementary arguments using the
identification with superperiodic difference equations, see [MGOST14].

###  5.3. More clusters in the $2$-friezes

From $F_{w}$ one can extract more clusters of $\mathcal{A}$ than the ones of
the bipartite belt. These clusters are given as “double zig-zags” in the
array. We follow [MGOT12].

We call double zig-zag in the 2-frieze (5.26) a sequence
$\zeta=(\widetilde{x}_{1},\ldots,\widetilde{x}_{w},\widetilde{y}_{1},\ldots,\widetilde{y}_{w})$
where $\widetilde{x}_{i}$’s are white entries and $\widetilde{y}_{i}$’s black
entries that are locally in the following 6 configurations

(5.27)
---
$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i+1}}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{x}_{i+1}}$$\textstyle{\widetilde{x}_{i+1}}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{x}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{x}_{i+1}}$$\textstyle{\widetilde{x}_{i+1}}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{y}_{i+1}}$$\textstyle{\widetilde{x}_{i+1}}$

######  Example 5.8.

The following sequence forms a double zig-zag in a frieze of width 5.

$\begin{matrix}\cdots&1&1&1&1&1&1&\cdots\\\\[4.0pt]
&&&\widetilde{x}_{1}&\widetilde{y}_{1}&&&\\\\[4.0pt]
&&&&\widetilde{x}_{2}&\widetilde{y}_{2}&&\\\\[4.0pt]
&&&\widetilde{x}_{3}&\widetilde{y}_{3}&&&\\\\[4.0pt]
&&&\widetilde{y}_{4}&\widetilde{x}_{4}&&&\\\\[4.0pt]
&&\widetilde{y}_{5}&\widetilde{x}_{5}&&&&\\\\[4.0pt]
\cdots&1&1&1&1&1&1&\cdots\end{matrix}$

To every double zig-zag $\zeta$ one associates a valued quiver
$\mathcal{Q}_{\zeta}$ defined over the set of vertices $\\{1,2,\ldots,2w\\}$
as follows. Each of the 6 configurations of (5.27) gives respectively the
following set of weighted arrows:

$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+{i+1}}$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+{i+1}}$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{w+{i+1}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{w+{i+1}}$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+{i+1}}$$\textstyle{w+i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w+{i+1}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$(1,2)$$\textstyle{{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$

when $i$ is odd, and same pictures with reversed arrows when $i$ is even.

######  Example 5.9.

The quiver associated to the double zig-zag of Example 5.8 is

$\textstyle{6\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{7\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{8\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{9\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\\!(2,1)$$\textstyle{10}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$(1,2)$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(1,2)$

One can extend Proposition 5.6 with the following statement.

######  Proposition 5.10.

For every double zig-zag $\zeta$ in $F_{w}$, the couple
$(\zeta,\mathcal{Q}_{\zeta})$ is a seed of $\mathcal{A}_{\Sigma_{0}}$.

###### Proof.

The statement follows from the mutation rule and the frieze rule. Every double
zig-zag can be redressed in two consecutive columns by elementary moves that
correspond to mutations. ∎

###  5.4. The cluster variety of tame symplectic 2-friezes

Let $\mathcal{A}=\mathcal{A}_{\Sigma_{0}}$ be the cluster algebra generated
from the initial seed
$\Sigma_{0}=(x_{1},x_{2},\ldots,x_{2w},\;\mathrm{C}_{2}\times\mathrm{A}_{w})$,
and let $\chi=(u_{1},\ldots,u_{2w})$ be an arbitrary cluster of $\mathcal{A}$.
We know from the cluster theory that all the cluster variables of
$\mathcal{A}$ can be expressed as Laurent monomials with positive integer
coefficients in the variables $u_{i}$. Denote by $F_{w}(u_{1},\ldots,u_{2w})$
the frieze of Laurent monomials obtained by expressing all the entries of
$F_{w}$ in terms of $u_{i}$. The following statement is immediate.

######  Proposition 5.11.

For every cluster $\chi=(u_{1},\ldots,u_{2w})$ of $\mathcal{A}$, the map

$\phi_{\chi}:(z_{1},\ldots,z_{2w})\mapsto
F_{w}(u_{1},\ldots,u_{2w})_{|u_{i}=z_{i}}$

defines a biregular homomorphism from $(\mathbb{C}^{*})^{2w}$ to the variety
of tame symplectic $2$-friezes.

We address the following natural question.

######  Open problem 1.

Is the variety of tame symplectic $2$-friezes of width $w$ entirely covered by
the images of the maps $\phi_{\chi}$?

The same question in the case of tame Coxeter’s friezes has been answered by
Cuntz and Holm [CH17].

######  Definition 5.12.

A symplectic $2$-frieze is cluster-singular if it is not in any of the images
of the maps $\phi_{\chi}$.

We know that every double zig-zag forms a cluster in $\mathcal{A}$ so that a
cluster-singular frieze necessarily has no double zig-zag of non-zero entries.

For $w=1,2$, one can give the following answers.

######  Proposition 5.13.

(i) There are only two cluster-singular friezes of width 1 given by

(5.28)
$\begin{array}[]{rrrrrrrrrcrrrr}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\cdots&-1&\mathbf{0}&-1&\mathbf{0}&-1&\mathbf{0}&-1&\mathbf{0}&-1&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&\cdots\\\
\end{array}$ (5.29)
$\begin{array}[]{rrrrrrrrrcrrrrrr}\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\cdots&0&\mathbf{i}&0&\mathbf{-i}&0&\mathbf{i}&0&\mathbf{-i}&0&\mathbf{i}&\cdots\\\
\cdots&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\mathbf{1}&1&\cdots\\\
\end{array}$

The frieze (5.28) is tame whereas (5.29) is not.

(ii) There is no cluster-singular tame frieze of width 2.

###### Proof.

For $w=1$, the cluster algebra is of type $\mathrm{C}_{2}$ and has exactly 6
clusters. This clusters are given as two consecutive entries in the unique
row, see (2.7). Using the frieze rule it is then easy to check that there are
only two ways to not get two consecutive non zero entries, leading to (5.28)
and (5.29). One checks the tameness property using Remark 3.3. The only
relevant calculations are

$\left|\begin{array}[]{cccccccccccc}0&1&0\\\ 1&0&1\\\ 0&1&0\\\
\end{array}\right|=0,\qquad\left|\begin{array}[]{cccccccccccc}0&1&0&0&0\\\
1&0&1&0&0\\\ 0&1&0&1&0\\\ 0&0&1&0&1\\\ 0&0&0&1&0\end{array}\right|=0,$

which allow to conclude that (5.28) is tame. Whereas

$\left|\begin{array}[]{cccccccccccc}i&1&0&0\\\ 1&-i&1&0\\\ 0&1&i&1\\\
0&0&1&-i\\\ \end{array}\right|=-1,$

shows that (5.29) is not tame. Hence (i).

For $w=2$, we will show that there always exists a double zig-zag of non-zero
entries. We know by Proposition 5.10 that they form a cluster.

Case (1): there are two consecutive zeroes on the first row of the frieze. The
frieze rule will imply that the two rows consist of zeroes. This will lead to
the array (3.14), which is known to be not tame. So this case does not lead to
a singular frieze.

Case (2): the first (or similarly the last) row has a sequence of the form
$(x,0,y,0,z,0,t)$ where $xyzt\not=0$ are either black or white entries. The
frieze rule will imply that the array is of the form

$\begin{array}[]{cccccccccccc}1&1&1&1&1&1&1&1&&\\\ x&0&y&0&z&0&t&r\\\
r&xy&s&yz&u&zt&v\\\ 1&1&1&1&1&1&1&1&\end{array}$

with $rsuv\not=0$ so that the specified entries in the second row are non
zero. By glide symmetry, one would have the entry $r$ next to $t$ in the first
row so that $(t,zt,r,v)$ forms a non-zero double zig-zag.

Case (3): the first row has two consecutive non zero entries $(x,y)$. If
$(x,y)$ is on the top of $(0,0)$ then it is absurd according to Case (1). If
$(x,y)$ is on the top of $(s,t)$ with $st\not=0$ then one has a double zig-zag
of non zero entries. We can assume that the array is locally of the form

$\begin{array}[]{cccccccccccc}1&1&1&1&\\\ x&y&z&\\\ 0&t&u&v&\\\
1&1&1&1\end{array}$

with $t\not=0$. If $u\not=0$ then $(x,t,y,u)$ forms a non-zero double zig-zag.
If $u=0$ then $v\not=0$ (otherwise it goes back to Case (1)) and then
$z=tv\not=0$. By induction we show that the next entries on the first row are
non zero and the one of the second row alternate zero and non-zero, so that it
goes back to Case (2).

∎

######  Remark 5.14.

The frieze (5.28) is the point of coordinates
$(a_{1},\ldots,a_{6})=(0,\ldots,0)$, $(b_{1},\ldots,b_{6})=(-1,\ldots,-1)$ in
the variety described in Example 2.5. The tangent space at this point is of
dimension 2 so that it is not a singular point of the variety (the variety is
actually smooth).

######  Conjecture 1.

The frieze of width 7 given in (3.15) is cluster-singular.

##  6\. Symplectic 2-friezes of positive integers

A major problem in the theory of friezes is the classification of the friezes
with positive integers. This problem usually leads to very nice combinatorial
interpretations. The main result in the domain is the theorem of Conway-
Coxeter [CC73] that classifies Coxeter friezes with triangulations of
polygons. There are many results along the same line for the classification of
other types of friezes, e.g. [BM09], [Tsc15], [FP16], [BHJ17].

There is also a nice extension of the Conway-Coxeter theorem in terms of
3d-dissections of polygons by allowing negative entries in the Coxeter friezes
[Ovs17].

######  Open problem 2.

Find a combinatorial model for the classification of the symplectic
$2$-friezes of positive integers.

Note that it is still an open problem in the case of classical $2$-friezes and
$\mathrm{SL}$-friezes.

An “easy” way to produce symplectic $2$-friezes of positive integers is to use
the maps $\phi_{\chi}$ given in Proposition 5.11 and evaluate on the point
$(z_{1},\ldots,z_{2w})=(1,\ldots,1)$. It was proved in [MG12] that when the
cluster algebra $\mathcal{A}$ is of infinite type, i.e. has infinitely many
clusters, this produces infinitely many friezes of positive integers. When the
cluster algebra $\mathcal{A}$ is of finite type this procedure is in general
not sufficient to produce all the friezes with positive integers (except in
the case of Coxeter friezes).

######  Proposition 6.1.

There exist exactly 6 symplectic 2-friezes of width 1 with positive integers.

###### Proof.

The formal frieze $F_{1}$, c.f. (5.26) contains the 6 cluster variables of
type $\mathrm{C}_{2}$. It is known that in this case there is only 6 ways to
get positive integers simultaneously for all the cluster variables [FP16]. ∎

The 6 friezes are all obtained from (2.4) under the action of the diedral
group $D_{6}$.

######  Conjecture 2.

There exist exactly 112 symplectic 2-friezes of width 2 with positive
integers.

This conjecture is based on two evidences. First, Michael Cuntz kindly provide
me with the 112 arrays of $\mathrm{SL}_{3}$-friezes which are symmetric with
respect to the middle line. By Theorem 2 they correspond to symplectic
2-friezes of width 2. The 112 $\mathrm{SL}_{3}$-friezes are grouped in 9
arrays modulo the action of the diedral group $D_{7}$. The second evidence is
that the entries of the symplectic $2$-friezes of width 2 are cluster
variables of type $\mathrm{F}_{4}$ (see Example 5.2) and it has been already
conjectured in [FP16] that there are only 112 evaluations that make all the
cluster variables positive integers. Note that in this case only 14 out of 28
cluster variables appear in the frieze, so that it was not necessary to have
all the cluster variables simultaneously positive integers. Note also that
there are 105 clusters of type $\mathrm{F}_{4}$. Which would mean that 7
friezes are not obtained as $\phi_{\chi}(1,\ldots,1)$.

######  Conjecture 3.

If $w\geq 3$ then there exist infinitely many symplectic $2$-friezes of
positive integers.

In other words we conjecture that the cluster algebra of type
$\mathrm{C}_{2}\square\mathrm{A}_{w}$ is of infinite type starting from $w=3$.

##  Appendix A Desnanot-Jacobi identity

The Desnanot-Jacobi identity, or Dodgson formula, is a classical formula
involving the determinant of an $(n\times n)-matrix$ and its minors of order
an $n-1$ and $n-2$ obtained by erasing the first and/or last row/column. The
formula can be pictured as follows

$\begin{vmatrix}*&*&*&*\\\ *&*&*&*\\\ *&*&*&*\\\ *&*&*&*\\\
\end{vmatrix}\begin{vmatrix}&&&\\\ \;&*&*&\,\\\ \;&*&*&\,\\\ \;&&&\,\\\
\end{vmatrix}=\begin{vmatrix}\;*&*&*&\;\;\\\ \;*&*&*&\;\;\\\ \;*&*&*&\;\;\\\
&&&\;\;\\\ \end{vmatrix}\begin{vmatrix}\;&\;&&\\\ \;&\;*&*&*\;\;\\\
\;&\;*&*&*\;\;\\\ \;&\;*&*&*\;\;\\\
\end{vmatrix}-\begin{vmatrix}&\;*&*&*\;\;\\\ &\;*&*&*\;\;\\\ &\;*&*&*\;\;\\\
&\;&&\\\ \end{vmatrix}\begin{vmatrix}&&&\;\\\ \;*&*&*&\;\;\\\ \;*&*&*&\;\;\\\
\;*&*&*&\;\;\\\ \end{vmatrix}.$

where the deleted columns/rows are left blank.

It is a key identity to establish Proposition 3.2.

##  Appendix B Properties of $\mathrm{SL}_{k+1}$-friezes

###  B.1. Definition of tame $\mathrm{SL}_{k+1}$-friezes

An $\mathrm{SL}_{k+1}$-frieze pattern is an array of numbers consisting in a
finite number of infinite rows:

(B.30) $\begin{array}[]{ccccccccccccc}&&&&\vdots&&&&\vdots&&&\\\
&0&&0&&0&&0&&0&&\ldots\\\\[6.0pt] \ldots&&1&&1&&1&&1&&1&\\\\[6.0pt]
&\ldots&&\;d_{0,w-1}&&\;d_{1,w}&&\;d_{2,w+1}&&\ldots&&\ldots\\\
&&&\\!\iddots&&\iddots&&\iddots&&&&\\\
\ldots&&d_{0,1}&&d_{1,2}&&d_{2,3}&&d_{3,4}&&d_{4,5}&\\\\[6.0pt]
&d_{0,0}&&d_{1,1}&&d_{2,2}&&d_{3,3}&&d_{4,4}&&\ldots\\\\[6.0pt]
\ldots&&1&&1&&1&&1&&1&\\\\[6.0pt] &0&&0&&0&&0&&0&&\ldots\\\
&&&&\vdots&&&&\vdots&&&\\\ \end{array}$

where the strip is bounded by $k$ rows of 0’s at the top, and at the bottom,
and where every “diamond” $(k+1)\times(k+1)$-subarray forms an element of
$\mathrm{SL}_{k+1}$. The number of rows between the bounding rows of $1$’s is
called the width and is denoted by $w$.

More precisely, the entries in the array are denoted by $(d_{i,j})$, with
$i,j\in\mathbb{Z}$ such that

$i-k-1\leq{}j\leq{}i+w+k.$

The array is a tame $\mathrm{SL}_{k+1}$-frieze of width $w$ when it satisfies:

  * •

“boundary conditions”

$\left\\{\begin{array}[]{rccccl}d_{i,i-1}&=&d_{i,i+w}&=&1&\hbox{for
all}\;i,\\\\[4.0pt] d_{i,j}&=&0&&&\hbox{for}\;i-k-1\leq
j<i-1\;\hbox{or}\;i+w<j\leq i+w+k.\end{array}\right.$

  * •

“$\mathrm{SL}_{k+1}$-conditions” on the $(k+1)\times(k+1)$-adjacent minors

$\left|\begin{array}[]{llll}d_{i,j}&d_{i,j+1}&\ldots&d_{i,j+k}\\\\[4.0pt]
d_{i+1,j}&d_{i+1,j+1}&\ldots&d_{i+1,j+k}\\\\[4.0pt] \ldots&\ldots&&\ldots\\\
d_{i+k,j}&d_{i+k,j+1}&\ldots&d_{i+k,j+k}\end{array}\right|=1,$

for all $(i,j)$ in the index set.

  * •

“tameness conditions” on the $(k+2)\times(k+2)$-adjacent minors

$\left|\begin{array}[]{llll}d_{i,j}&d_{i,j+1}&\ldots&d_{i,j+k+1}\\\\[4.0pt]
d_{i+1,j}&d_{i+1,j+1}&\ldots&d_{i+1,j+k+1}\\\\[4.0pt] \ldots&\ldots&&\ldots\\\
d_{i+k+1,j}&d_{i+k+1,j+1}&\ldots&d_{i+k+1,j+k+1}\end{array}\right|=0,$

for all $(i,j)$ in the index set.

###  B.2. Tame $\mathrm{SL}_{k+1}$-friezes and difference equations

An $n$-superperiodic difference equation of order $k+1$ is a system

(B.31)
$V_{i}=a_{i}^{1}V_{i-1}-a_{i}^{2}V_{i-2}+\cdots+(-1)^{k-1}a_{i}^{k}V_{i-k}+(-1)^{k}V_{i-k-1},$

where $a_{i}^{j}\in\mathbb{C}$, with $i\in\mathbb{Z}$ and $1\leq j\leq k$, are
given coefficients (note that the superscript $j$ is an index, not a power)
and $V_{i}$ are unknowns, such that

  * •

the coefficients are $n$-periodic i.e. $a_{i+n}^{j}=a_{i}^{j}$, for all $i,j$.

  * •

every solution $(V_{i})_{i\in\mathbb{Z}}$ is $n$-(anti)periodic i.e.
$V_{i+n}=(-1)^{k}\,V_{i}$, for all $i\in\mathbb{Z}$.

######  Proposition B.1 ([MGOST14]).

An array $(d_{i,j})$ as (B.30) forms a tame $\mathrm{SL}_{k+1}$ friezes if and
only if every diagonal

$d_{i,\bullet}=(0,\ldots,0,1,d_{i,i},d_{i,i+1},\ldots,d_{i,i+w-1},1)$

gives a solution of a same $(w+k+2)$-superperiodic equation of the form (B.31)
with initial conditions

$(V_{i-k-1},V_{i-k},\ldots,V_{i-1})=(0,0,\ldots,0,1).$

###  B.3. Coefficients of the difference equations from the entries of the
frieze and vice versa

We collect some formulas given in [MGOST14, §5.2, §5.3].

Let (B.31) be the associated equation associated to a tame
$\mathrm{SL}_{k+1}$-frieze (B.30). The coefficient $a_{i-1}^{k-j}$ of the
equation can be expressed as a $(j+1)\times(j+1)$-minor of adjacent entries in
the frieze:

(B.32) $a_{i-1}^{k-j}=\left|\begin{array}[]{cccc}d_{i+1,i+w}&1&\\\\[8.0pt]
\vdots&\ddots&\;1\\\\[6.0pt]
d_{i+j+1,i+w}&\cdots&d_{i+j+1,i+j+w}\end{array}\right|.$

Conversely, the entries of the frieze can be computed as determinant of
matrices involving the coefficients of the equation. The entry $d_{i,i+j}$ can
be computed using $(j+1)\times(j+1)$-determinants

(B.39) $\displaystyle d_{i,i+j}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccclcc}a_{i}^{1}&1&\\\\[8.0pt]
\vdots&a_{i+1}^{1}&1&\\\\[4.0pt] a_{i+k-1}^{k}&&\ddots&\ddots\\\\[6.0pt]
1&&&\ddots&\ddots\\\\[4.0pt] &\ddots&&&a_{i+j-1}^{1}&1\\\\[8.0pt]
&&1&a_{i+j}^{k}&\ldots&a_{i+j}^{1}\end{array}\right|,$

or alternatively, using $(w-j)\times(w-j)$-determinants

(B.46) $\displaystyle d_{i,i+j}=$
$\displaystyle\left|\begin{array}[]{ccllcccc}a_{i-w+j-1}^{k}&\ldots&a_{i-w+j-1}^{1}&\;\;1&&\\\\[8.0pt]
1&a_{i-w+j}^{k}&\ldots&a_{i-w+j}^{1}&\\!\\!1&\\\\[8.0pt]
&1&\ldots&&\ddots&\\!\\!1\\\\[8.0pt] &&\ddots&&\ddots&\vdots\\\\[6.0pt]
&&&\;1&a_{i-3}^{k}&a_{i-3}^{k-1}\\\\[8.0pt]
&&&&\\!\\!\\!\\!1&a_{i-2}^{k}\end{array}\right|.$

###  B.4. Projective duality

Given a tame $\mathrm{SL}_{k+1}$-frieze $(d_{i,j})$, its projective dual is
defined as the array consisting of $k\times k$ adjacent minors:

$d_{i,j}^{*}=\left|\begin{array}[]{llll}d_{i,j}&d_{i,j+1}&\ldots&d_{i,j+k-1}\\\\[4.0pt]
d_{i+1,j}&d_{i+1,j+1}&\ldots&d_{i+1,j+k-1}\\\\[4.0pt]
\vdots&&&\vdots\\\\[6.0pt]
d_{i+k-1,j}&d_{i+k-1,j+1}&\ldots&d_{i+k-1,j+k-1}\end{array}\right|.$

######  Proposition B.2 ([MGOST14]).

(i) The projective dual frieze to a tame $\mathrm{SL}_{k+1}$-frieze is a tame
$\mathrm{SL}_{k+1}$-frieze.

(ii) If $a_{i}^{j}$ are the coefficients of the equation (B.31) associated to
a $\mathrm{SL}_{k+1}$-frieze, then

(B.47)
$V^{*}_{i}=a_{i+k-1}^{k}V^{*}_{i-1}-a_{i+k-2}^{k-1}V^{*}_{i-2}+\cdots+(-1)^{k-1}a_{i}^{1}V^{*}_{i-k}+(-1)^{k}V^{*}_{i-k-1}$

is the superperiodic equation associated to the projective dual.

(iii) The projective dual to a tame $\mathrm{SL}_{k+1}$-frieze is just the
symmetric array with respect to the median horizontal axis.

Let us make the above statement number (iii) more precise.

######  Proposition B.3.

A tame $\mathrm{SL}_{k+1}$-frieze of width $w$ satisfies

(B.48)
$d_{i,j}=\left|\begin{array}[]{ccc}d_{j-w-k,i-k-1}&\ldots&d_{j-w-k,i-2}\\\\[6.0pt]
\vdots&&\vdots\\\\[6.0pt]
d_{j-w-1,i-k-1}&\ldots&d_{j-w-1,i-2}\end{array}\right|=d^{*}_{j-w-k,i-k-1}$

for all $i,j$.

###### Proof.

First, note the diagonals

$d_{\bullet,i}=(0,\ldots,0,1,d_{i,i},d_{i-1,i},\ldots,d_{i-w+1,i},1)$

satisfy the dual equation (B.47). Let us consider two sequences
$V=(V_{\ell})_{1\leq\ell\leq j-i+k+2}$ and $W=(W_{\ell})_{1\leq\ell\leq
j-i+k+2}$ forming blocks of size $(k+1)\times(j-i+k+2)$ in the frieze that
differ by a shift along the North-West diagonal:

$\blockarray{ccccccccccccc}V_{1}&V_{2}\cdots V_{k+1}V_{k+2}\cdots\qquad
V_{j-i+1}V_{j-i+k+2}\\\
\block{(ccccccccccc)cc}d_{j-w-k,i-k-1}d_{j-w-k,i-k}\cdots\cdots
d_{j-w-k,i}\cdots 100\\\ \vdots\vdots\vdots\cdots 1=V\\\
\vdots\vdots\vdots\cdots\ddots 0\\\ d_{j-w,i-k-1}d_{j-w,i-k}\cdots\cdots
d_{j-w,i}\cdots\cdots 1\\\ \vdots\vdots\vdots\vdots\\\
\block{(ccccccccccc)cc}1d_{i-k,i-k}\cdots\cdots d_{i-k,i}\cdots\cdots
d_{i-k,j}\\\ 01\ddots\vdots\vdots=W\\\ \vdots\vdots\ddots\ddots\vdots\vdots\\\
00\cdots 1d_{i,i}\cdots\cdots d_{i,j}\\\ W_{1}W_{2}\cdots
W_{k+1}W_{k+2}\cdots\qquad\cdots W_{j-i+k+2}\\\ $

To simplify the picture we have rotated the frieze by 45 degree so that the
shift between the blocks is now seen vertically.

Due to the recurrence relation in the frieze, the matrix $V$ can be obtained
from $W$ by multiplying on the left by a product of matrices of the form

$(-1)^{k}\left(\begin{array}[]{cccccc}(-1)^{k}a^{k}_{\ell+k-1}&\ldots&-a_{\ell}^{1}&1\\\
1&&\\\ &\ddots&\\\ &&1&0\\\ \end{array}\right).$

So that we have $V=TW$ with $V\in\mathrm{SL}_{k+1}$. This implies that the
minors of order $(k+1)$ in $V$ and $W$ involving the same columns are equal.
In particular, we have:

(B.49)
$\begin{array}[]{rcl}d_{i,j}&=&\det(W_{1},\ldots,W_{k},W_{j-i+k+2})\\\\[4.0pt]
&=&\det(V_{1},\ldots,V_{k},V_{j-i+k+2})\\\\[4.0pt]
&=&\left|\begin{array}[]{ccc}d_{j-w-k,i-k-1}&\ldots&d_{j-w-k,i-2}\\\\[6.0pt]
\vdots&&\vdots\\\\[6.0pt]
d_{j-w-1,i-k-1}&\ldots&d_{j-w-1,i-2}\end{array}\right|.\end{array}$

∎

###  B.5. Gale duality

The Gale duality is a one-to-one correspondence between
$\mathrm{SL}_{k+1}$-frieze of width $w$ and $\mathrm{SL}_{w+1}$-frieze pattern
of width $k$. The correpsondence is given by the following result.

######  Proposition B.4.

[MGOST14, Proposition 4.1.3] Consider a $\mathrm{SL}_{k+1}$-frieze of width
$w$, and its associated equation B.31. The following array formed by the
coeffcients of the equation

$\begin{array}[]{ccccccccccccccccccccccccccc}&\ldots&1&&1&&1&&1&&1&&1\\\\[4.0pt]
&&&\ldots&&a^{1}_{n}&&a^{1}_{1}&&a^{1}_{2}&&\ldots&&a^{1}_{n}&\\\\[4.0pt]
&&&&a^{2}_{n}&&a^{2}_{1}&&a^{2}_{2}&&&&a^{2}_{n}&\\\
&\ldots&&\iddots&&\iddots&&\iddots&&&&\iddots&&\ldots\\\
&&a^{k}_{n}&&a^{k}_{1}&&a^{k}_{2}&&\ldots&&a^{k}_{n}&&\ldots\\\\[4.0pt]
&1&&1&&1&&1&&1&&1&&\ldots\\\ \end{array}$

is an $\mathrm{SL}_{w+1}$-frieze pattern of width $k$.

Acknowledgements. I am deeply grateful to Valentin Ovsienko for sharing with
me ideas and the results of [CO]. I am also grateful to Michael Cuntz for
computer calculations. This work is supported by the ANR project SC3A,
ANR-15-CE40-0004-01.

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