0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 54.90521pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 26.4526pt\raise
48.36943pt\hbox{\hbox{\kern 0.0pt\raise
<EMAIL_ADDRESS>0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small 1}}$}}}}}{\hbox{\kern
54.65521pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small 2}}$}}}}}{\hbox{\kern
17.66682pt\raise 48.36943pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small
3}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.69052pt\raise 2.84526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.69052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.69052pt\raise-2.84526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
31.29787pt\raise 42.67891pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
27.02997pt\raise 42.10986pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
24.18471pt\raise 43.24794pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}
}$ (4.1)

Conversely, one can recover a skew-symmetric integer matrix $B$ from a quiver
without loops and 2-cycles by applying the above rule in the opposite
direction. It is clear that this correspondence is one-to-one.

For a quiver $Q$, the quiver obtained from $Q$ by changing all arrows of $Q$
to the ones in the opposite direction is called the _opposite quiver_ of $Q$,
and it is denoted by $Q^{\mathrm{op}}$. If a skew-symmetric integer matrix $B$
corresponds to a quiver $Q$, then $-B$ corresponds to $Q^{\mathrm{op}}$.

One can translate the matrix mutation (2.11) into the quiver mutation as
follows.

######  Definition 4.2 (Quiver mutation).

Let $Q$ be a quiver with $n$ vertices and without loops and 2-cycles. For each
$k=1$, …, $n$, we define a new quiver $Q^{\prime}=\mu_{k}(Q)$ by the following
operation:

  * •

For each pair $i$, $j$ ($i\neq j$) such that $i$, $j\neq k$, if there are
$p>0$ arrows from the vertex $i$ to the vertex $k$, and $q>0$ arrows from the
vertex $k$ to the vertex $j$, then add $pq$ arrows from the vertex $i$ to the
vertex $j$.

  * •

Remove the resulting 2-cycles as many as possible.

  * •

Invert all arrows into and out of the vertex $k$.

The quiver $Q^{\prime}$ is called the _mutation of $Q$ at the vertex $k$_.

Under the above identification, a $Y$-seed $(\mathbf{y},B)$ with a skew-
symmetric matrix $B$ is replaced with $(\mathbf{y},Q)$ for the corresponding
quiver $Q$. For the mutation
$(\mathbf{y}^{\prime},Q^{\prime})=\mu_{k}(\mathbf{y},Q)$ of a $Y$-seed, the
mutation of $y$-variables (2.10) is translated into the following simple
pictorial rule:

$\displaystyle y^{\prime}_{i}=\begin{cases}y_{k}^{-1}&i=k,\\\
y_{i},&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
2.0pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 25.9526pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.55031pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.96786pt\hbox{$\textstyle{\text{\small$i$}}$}}}}}{\hbox{\kern
25.96823pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-3.125pt\hbox{$\textstyle{\text{\small$k$}}$}}}}}}}}}\ \text{(no
arrow)},\\\ y_{i}(1\oplus y_{k})^{m}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
2.0pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 25.9526pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.55031pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.96786pt\hbox{$\textstyle{\text{\small$i$}}$}}}}}{\hbox{\kern
25.96823pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-3.125pt\hbox{$\textstyle{\text{\small$k$}}$}}}}}{\hbox{\kern
10.27525pt\raise 5.69052pt\hbox{\hbox{\kern
0.0pt\raise-1.9375pt\hbox{$\textstyle{\text{\small$m$}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.76208pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\
\text{($m$ arrows)},\\\ y_{i}(1\oplus y_{k}^{-1})^{-m}&\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 2.0pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 25.9526pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.55031pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.96786pt\hbox{$\textstyle{\text{\small$i$}}$}}}}}{\hbox{\kern
25.96823pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-3.125pt\hbox{$\textstyle{\text{\small$k$}}$}}}}}{\hbox{\kern
10.27525pt\raise 5.69052pt\hbox{\hbox{\kern
0.0pt\raise-1.9375pt\hbox{$\textstyle{\text{\small$m$}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
5.69052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\
\text{($m$ arrows)}.\end{cases}$ (4.2)

######  Example 4.3.

Let us consider the following quiver $Q$:

$Q=\ \lower 12.0pt\hbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
2.25pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
<EMAIL_ADDRESS>28.4526pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 26.4526pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern
26.4526pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 54.90521pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 54.90521pt\raise
28.4526pt\hbox{\hbox{\kern 0.0pt\raise
<EMAIL_ADDRESS>36.98839pt\hbox{\hbox{\kern 0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small
1}}$}}}}}{\hbox{\kern 26.2026pt\raise 36.98839pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small 2}}$}}}}}{\hbox{\kern
54.65521pt\raise 36.98839pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small
3}}$}}}}}{\hbox{\kern-2.25pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small 4}}$}}}}}{\hbox{\kern
26.2026pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small 5}}$}}}}}{\hbox{\kern
54.65521pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.9pt\hbox{$\textstyle{\text{\small
6}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
22.76208pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.14313pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
5.69052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
51.21469pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.4526pt\raise 5.69052pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.90521pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}
}$

We consider the sequence of mutations at the vertices $2$, $4$, $6$, $1$, $5$,
$3$ in this order. In the first half, the quivers mutate as follows, where a
black vertex is the one at which the mutation is going to be applied:

$\textstyle{\bullet}$1234562 $\textstyle{\bullet}$1234564
$\textstyle{\bullet}$1234566 123456

The last quiver is the opposite quiver $Q^{\mathrm{op}}$ of $Q$. It is easy to
see that the final result is independent of the order of the mutations at 2,
4, 6. Similarly, in the second half, the quivers mutate as follows:

$\textstyle{\bullet}$1234561 $\textstyle{\bullet}$1234563
$\textstyle{\bullet}$1234565 123456

The last quiver is the original quiver $Q$. Thus, we have a periodicity of
quivers. Again, it is easy to see that the final result is independent of the
order of the mutations at 1, 3, 5. For
$(\mathbf{y}^{\prime},Q^{\mathrm{op}})=\mu_{6}\mu_{4}\mu_{2}(\mathbf{y},Q)$,
by applying the pictorial rule (4.2), we have

$\displaystyle y^{\prime}_{i}=\begin{cases}y_{i}^{-1}&i=2,\,4,\,6,\\\
y_{1}(1\oplus y_{2})(1\oplus y_{4}^{-1})^{-1}&i=1,\\\ y_{3}(1\oplus
y_{2})(1\oplus y_{6}^{-1})^{-1}&i=3,\\\ y_{5}(1\oplus y_{4})(1\oplus
y_{6})(1\oplus y_{2}^{-1})^{-1}&i=5.\end{cases}$ (4.3)

Similarly, for
$(\mathbf{y}^{\prime\prime},Q)=\mu_{5}\mu_{3}\mu_{1}(\mathbf{y}^{\prime},Q^{\mathrm{op}})$,
we have

$\displaystyle
y^{\prime\prime}_{i}=\begin{cases}y^{\prime}_{i}{}^{-1}&i=1,\,3,\,5,\\\
y^{\prime}_{2}(1\oplus y^{\prime}_{1})(1\oplus y^{\prime}_{3})(1\oplus
y^{\prime}_{5}{}^{-1})^{-1}&i=2.\\\ y^{\prime}_{4}(1\oplus
y^{\prime}_{5})(1\oplus y^{\prime}_{1}{}^{-1})^{-1}&i=4,\\\
y^{\prime}_{6}(1\oplus y^{\prime}_{5})(1\oplus
y^{\prime}_{3}{}^{-1})^{-1}&i=6.\end{cases}$ (4.4)

####  4.2 $Y$-systems and $Y$-patterns

The readers may already notice the similarity between the mutation of
$y$-variables (4.3), (4.4) and $Y$-systems (1.38). Let us make this
observation more precise.

In general, mutations are not commutative, namely,
$\mu_{k}\mu_{\ell}(\Sigma)\neq\mu_{\ell}\mu_{k}(\Sigma)$. However, they
commute under the following condition.

######  Lemma 4.4 (e.g., [FZ07, §8]).

For a seed $\Sigma=(\mathbf{x},\mathbf{y},B)$ and a pair $k$, $\ell$
($k\neq\ell$), suppose that

$\displaystyle b_{k\ell}=b_{\ell k}=0$ (4.5)

holds. Then, for the seed
$\Sigma^{\prime}=(\mathbf{x}^{\prime},\mathbf{y}^{\prime},B^{\prime})=\mu_{k}(\Sigma)$,
we also have

$\displaystyle b^{\prime}_{k\ell}=b^{\prime}_{\ell k}=0.$ (4.6)

Moreover, we have

$\displaystyle\mu_{k}\mu_{\ell}(\Sigma)=\mu_{\ell}\mu_{k}(\Sigma).$ (4.7)

The equality (4.7) can be easily verified by a direct calculation (e.g.,
[Nak23, Prop. 2.9]).

Now we formulate the $Y$-system (1.38) of type $(X,X^{\prime})$ in terms of a
$Y$-pattern following Keller [Kel10]. We divide the formulation into three
steps.

Step 1. Quiver $Q(X,X^{\prime})$. At each vertex $a$ of $X$ we assign a sign
$\kappa_{a}\in\\{+,-\\}$ so that any adjacent vertices in $X$ have the
opposite sign. (There are two possible choices, and either will do.) We
conveniently identify the signs $\pm$ with $\pm 1$ as usual. We do the same
assignment of signs for $X^{\prime}$. Then, we consider the product graph
$T=X\times X^{\prime}$, where $X$ lies horizontally and $X^{\prime}$ lies
vertically. For each vertex $v=(a,a^{\prime})$ of $T$, we set a sign
$\kappa_{v}=\kappa_{a,a^{\prime}}:=\kappa_{a}\kappa_{a^{\prime}}\in\\{+,-\\}$.
Moreover, we set an orientation on each edge of $T$ by the following rule:

$\displaystyle\begin{matrix}(++)&\leftarrow&(-+)\\\ \downarrow&&\uparrow\\\
(+-)&\rightarrow&(--)\end{matrix}$ (4.8)

The resulting quiver is denoted by $Q(X,X^{\prime})$. See Figure 4.1 for
examples. The quiver $Q$ in Example 4.3 is also an example for
$Q(A_{3},A_{2})$.

$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$+$-$+$-$+$-$+(a)
$Q(A_{4},A_{3})$
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$+$-$+$-$$-$+$-$+(b)
$Q(D_{5},A_{3})$ Figure 4.1: Examples of quiver $Q(X,X^{\prime})$. Black and
white vertices belong to $V_{+}$ and $V_{-}$, respectively.

Step 2. Composite mutations $\mu_{+}$ and $\mu_{-}$. We consider the
$Y$-pattern $\mathbf{\Upsilon}=\\{\Upsilon_{t}\\}$ of a cluster pattern with
free coefficients at $t_{0}$, where the initial $Y$-seed is given by
$\Upsilon_{t_{0}}=\Upsilon=(\mathbf{y},Q)$ with $Q=Q(X,X^{\prime})$. Let
$V_{+}$ and $V_{-}$ be the sets of vertices $v$ of the quiver $Q$ with sign
$\kappa_{v}$ being $+$ and $-$, respectively. In Figure 4.1, the black and
white vertices belong to $V_{+}$ and $V_{-}$, respectively. Let $p=|V_{+}|$,
and let $v_{1}$, …, $v_{p}$ be all vertices in $V_{+}$. Similarly, let
$q=|V_{-}|$, and let $u_{1}$, …, $u_{q}$ be all vertices in $V_{-}$. Then, the
following facts hold:

  1. 1.

By the construction of $Q$, there is no arrow between the vertices $v$ and
$v^{\prime}$ with the same sign. Therefore, by Lemma 4.4, the sequence of
mutations

$\displaystyle\mu_{+}(\mathbf{y},Q):=\mu_{v_{p}}\cdots\mu_{v_{1}}(\mathbf{y},Q)$
(4.9)

is independent of the order of the product.

  2. 2.

Let $(\mathbf{y}^{\prime},Q^{\prime})=\mu_{+}(\mathbf{y},Q)$. Then,
$Q^{\prime}=Q^{\mathrm{op}}$ holds in the same way as the quiver
$Q^{\prime}=\mu_{6}\mu_{4}\mu_{2}(Q)$ in Example 4.3.

  3. 3.

For the above seed $(\mathbf{y}^{\prime},Q^{\mathrm{op}})$, we apply the
sequence of mutations

$\displaystyle\mu_{-}:=\mu_{u_{q}}\cdots\mu_{u_{1}}.$ (4.10)

Again, by Lemma 4.4, it is independent of the order of the product.

  4. 4.

Let
$(\mathbf{y}^{\prime\prime},Q^{\prime\prime})=\mu_{-}(\mathbf{y}^{\prime},Q^{\mathrm{op}})$.
Then, $Q^{\prime\prime}=Q$ holds in the same way as the quiver
$Q^{\prime\prime}=\mu_{5}\mu_{3}\mu_{1}(Q^{\mathrm{op}})$ in Example 4.3. Note
that this periodicity is only for quivers, and
$\mathbf{y}^{\prime\prime}=\mathbf{y}$ does not hold.

Step 3. Identification with $Y$-system. For the above $Y$-pattern, consider
the following sequence of (composite) mutations

$\displaystyle\begin{split}\cdots\ {\buildrel+\over{\rightarrow}}\
(\mathbf{y}(-1),Q^{\mathrm{op}})\ {\buildrel-\over{\rightarrow}}&\
(\mathbf{y}(0),Q)\ {\buildrel+\over{\rightarrow}}\
(\mathbf{y}(1),Q^{\mathrm{op}})\ {\buildrel-\over{\rightarrow}}\
(\mathbf{y}(2),Q)\ {\buildrel+\over{\rightarrow}}\ \cdots,\end{split}$ (4.11)

where $(\mathbf{y}(0),Q)=(\mathbf{y},Q)$ is the initial $Y$-seed, and
$\buildrel+\over{\rightarrow}$ and $\buildrel-\over{\rightarrow}$ represent
$\mu_{+}$ and $\mu_{-}$, respectively. Let us write $y$-variables as
$\mathbf{y}(s)=(y_{a,a^{\prime}}(s))_{(a,a^{\prime})\in I\times I^{\prime}}$.
Also, let us write the addition $\oplus$ in
$\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})$ as $+$ as in the previous chapter.
Then, in the same way as (4.3) and (4.4), we have, for even $s$,

$\displaystyle
y_{a,a^{\prime}}(s+1)=\begin{cases}y_{a,a^{\prime}}(s)^{-1}&\kappa_{a,a^{\prime}}=+,\\\
y_{a,a^{\prime}}(s)\frac{\displaystyle\prod_{b\in I:\,b\sim
a}(1+y_{b,a^{\prime}}(s))}{\displaystyle\prod_{b^{\prime}\in
I^{\prime}:\,b^{\prime}\sim
a^{\prime}}(1+y_{a,b^{\prime}}(s)^{-1})}&\kappa_{a,a^{\prime}}=-,\\\
\end{cases}$ (4.12)

and, for odd $s$,

$\displaystyle y_{a,a^{\prime}}(s+1)=\begin{cases}\displaystyle
y_{a,a^{\prime}}(s)\frac{\displaystyle\prod_{b\in I:\,b\sim
a}(1+y_{b,a^{\prime}}(s))}{\displaystyle\prod_{b^{\prime}\in
I^{\prime}:\,b^{\prime}\sim
a^{\prime}}(1+y_{a,b^{\prime}}(s)^{-1})}&\kappa_{a,a^{\prime}}=+,\\\
y_{a,a^{\prime}}(s)^{-1}&\kappa_{a,a^{\prime}}=-,\end{cases}$ (4.13)

where the symbol $\sim$ denotes the adjacency in $X$ in the numerator and in
$X^{\prime}$ in the denominator, respectively. Now we separate the
$y$-variables into the _even_ and the _odd sectors_ $\mathcal{Y}_{+}$,
$\mathcal{Y}_{-}$ by

$\displaystyle\mathcal{Y}_{\pm}$
$\displaystyle:=\\{y_{a,a^{\prime}}(s)\mid\kappa_{a,a^{\prime}}(-1)^{s}=\pm
1\\}.$ (4.14)

Then, by (4.12) and (4.13), the variables in $\mathcal{Y}_{+}$ and
$\mathcal{Y}_{-}$ are related as

$\displaystyle
y_{a,a^{\prime}}(s+1)=y_{a,a^{\prime}}(s)^{-1}\quad(y_{a,a^{\prime}}(s)\in\mathcal{Y}_{+},\
y_{a,a^{\prime}}(s+1)\in\mathcal{Y}_{-}).$ (4.15)

By using this relation, both relations (4.12) and (4.13) are summarized into a
single relation

$\displaystyle
y_{a,a^{\prime}}(s+1)y_{a,a^{\prime}}(s-1)=\frac{\displaystyle\prod_{b\in
I:\,b\sim a}(1+y_{b,a^{\prime}}(s))}{\displaystyle\prod_{b^{\prime}\in
I^{\prime}:\,b^{\prime}\sim a^{\prime}}(1+y_{a,b^{\prime}}(s)^{-1})},$ (4.16)

where all $y$-variables are in the even sector $\mathcal{Y}_{+}$. This is
exactly the $Y$-system (1.36) of type $(X,X^{\prime})$. On the other hand, by
(4.15) and (4.16), $y$-variables in the odd sector $\mathcal{Y}_{-}$ satisfy
the “dual” relations

$\displaystyle
y_{a,a^{\prime}}(s+1)y_{a,a^{\prime}}(s-1)=\frac{\displaystyle\prod_{b^{\prime}\in
I^{\prime}:\,b^{\prime}\sim
a^{\prime}}(1+y_{a,b^{\prime}}(s))}{\displaystyle\prod_{b\in I:\,b\sim
a}(1+y_{b,a^{\prime}}(s)^{-1})}.$ (4.17)

On the other hand, we observe that the $Y$-system (1.36) itself is decoupled
into two sectors. Namely, let us separate the variables $Y_{a,a^{\prime}}(u)$
therein into the even and the odd sectors $Y_{+}$, $Y_{-}$ by

$\displaystyle Y_{\pm}$
$\displaystyle:=\\{Y_{a,a^{\prime}}(u)\mid\kappa_{a,a^{\prime}}(-1)^{s}=\pm
1\\}.$ (4.18)

In contrast to (4.15), there is no relation between the variables in $Y_{+}$
and the ones in $Y_{-}$ in the $Y$-system (1.36). Therefore, the $Y$-system
can be restricted on each sector.

In summary, the $Y$-system (1.36) restricted on the even sector $Y_{+}$ is
“embedded” in the ambient $Y$-pattern, where the variables
$Y_{a,a^{\prime}}(u)\in Y_{+}$ for the $Y$-system are identified with the
$y$-variables $y_{a,a^{\prime}}(u)\in\mathcal{Y}_{+}$ in the sequence of
mutations (4.11).

####  4.3 DIs for $Y$-systems

For each simply-laced Dynkin diagram $X$, we introduce the following diagram
automorphism $\omega:I\rightarrow I$ of $X$,

for $A_{r}$ $\displaystyle\omega:a\leftrightarrow r+1-a,$ (4.19) for $D_{r}$
($r$: odd) $\displaystyle\omega:a\mapsto a\ (a\neq r-1,\ r),\
r-1\leftrightarrow r,$ for $E_{6}$ $\displaystyle\omega:4\mapsto 4,\
1\leftrightarrow 6,\ 2\leftrightarrow 5,$ otherwise
$\displaystyle\omega=\mathrm{id},$

where we used the labels of $I$ in Figure 1.4. The intrinsic meaning of
$\omega$ will be clarified soon in (4.30). Note that $\omega$ is an involution
for any $X$.

The following periodicity was proved for $(X,X^{\prime})=(X,A_{1})$ by Fomin-
Zelevinsky [FZ03b, FZ07], and in general by Keller [Kel10, Kel13b] (for full
periodicity) and Inoue-Iyama-Kuniba-Nakanishi-Suzuki [IIK+10] (for half
periodicity).

######  Theorem 4.5 ([FZ03b, Thm. 1.1], [Kel10, Thm. 8.2], [Kel13b, Thm. 2.3],
[IIK+10, Cor. 4.28]).

For the sequence of mutations (4.11), the following periodicities hold:

(half periodicity) $\displaystyle\quad y_{a,a^{\prime}}(s+h+h^{\prime})$
$\displaystyle=y_{\omega(a),\omega^{\prime}(a^{\prime})}(s),$ (4.20) (full
periodicity) $\displaystyle\quad y_{a,a^{\prime}}(s+2(h+h^{\prime}))$
$\displaystyle=y_{a,a^{\prime}}(s),$ (4.21)

where $h$, $h^{\prime}$ are the Coxeter numbers, and $\omega$,
$\omega^{\prime}$ are the diagram automorphisms (4.19) of $X$ and
$X^{\prime}$, respectively. (When $\omega=\omega^{\prime}=\mathrm{id}$, the
half periodicity (4.20) is already a full periodicity.)

The periodicity (4.21) is equivalent to the one for the $Y$-systems in
Conjecture 1.4 under the correspondence in the previous section. Temporarily
assume Theorem 4.5. Then, thanks to Theorem 2.31, or by $Q(2(h+h^{\prime}))=Q$
more directly, the periodicity (4.21) is lifted to the periodicity of
$Y$-seeds

$\displaystyle\Upsilon(s+2(h+h^{\prime}))=\Upsilon(s).$ (4.22)

Thus, by Theorem 3.5, there is a DI associated with this periodicity. Note
that $y_{k_{s}}(s)$ appearing in the DI are variables in the even sector
$\mathcal{Y}_{+}$. Also, we take $D=I$ because the $Y$-pattern is skew-
symmetric. After evaluating the constant term, we obtain the following DI.

######  Theorem 4.6 ([Nak11a, Thm. 2.8]).

Suppose that the initial $y$-variables $\mathbf{y}$ take values in
$\mathbb{R}_{>0}^{n}$. Then, the following DIs hold.

$\displaystyle\sum_{s=0}^{2(h+h^{\prime})-1}\sum_{\scriptstyle(a,a^{\prime})\in
I\times
I^{\prime}\atop\scriptstyle\kappa_{a,a^{\prime}}(-1)^{s}=1}\tilde{L}(y_{a,a^{\prime}}(s))=hrr^{\prime}\frac{\pi^{2}}{6}.$
(4.23)

Taking into account the following fine points, this is indeed the DI (1.40)
for the $Y$-system:

  * •

The constant term of (4.23) is _half_ of the one of (1.40) because the sum in
(4.23) is restricted to the even sector $\mathcal{Y}_{+}$, which corresponds
to the even sector $Y_{+}$ of the $Y$-system.

  * •

By (4.15), the initial $y$-variables $y_{a,a^{\prime}}(0)$ are identified with
the initial variables of the $Y$-system in the even sector $Y_{+}$ as

$\displaystyle
y_{a,a^{\prime}}(0)=\begin{cases}Y_{a,a^{\prime}}(0)&\kappa_{a,a^{\prime}}=+,\\\
Y_{a,a^{\prime}}(-1)^{-1}&\kappa_{a,a^{\prime}}=-.\end{cases}$ (4.24)

######  Corollary 4.7.

[Kel10, Kel13b, Nak11a] Conjecture 1.4 is true.

In the rest of the chapter, we present the proofs of Theorem 4.5 and the
counting of the constant term in Theorem 4.6. The main idea is to use the
tropicalization. By Theorem 2.30, to prove the periodicity of $y$-variables
$y_{a,a^{\prime}}(u)$, it is enough to prove the periodicity of _tropical
$y$-variables_ $[y_{a,a^{\prime}}(u)]$. Remarkably, the dynamics of tropical
$y$-variables is governed by _Coxeter elements_ of the root systems. This
mechanism was found for the $Y$-systems of type $(X,A_{1})$ in [FZ03b, FZ07],
and generalized to the $Y$-systems of type $(X,X^{\prime})$ in [Nak11a] .

####  4.4 Coxeter elements

In this section, we summarize the facts on the Coxeter elements we will use.
The presentation here is a minimal one. See [Bou02, Hum90] for further
information.

Let $X$ be a simply-laced Dynkin diagram of rank $r$. Let $\Phi=\Phi(X)\subset
V$ be the root system corresponding to $X$, where $V$ is the ambient real
vector space spanned by $\Phi$ with an inner product $(\cdot,\cdot)$. Let
$\alpha_{1}$, …, $\alpha_{r}\in\Phi$ be the simple roots parametrized by the
index set $I=\\{1,\,\dots,\,r\\}$ as in Figure 1.4. We normalize them as
$(\alpha_{a},\alpha_{a})=2$. Then, they satisfy, for $a\neq b$,

$\displaystyle(\alpha_{a},\alpha_{b})=\begin{cases}-1&a\sim b,\\\ 0&a\not\sim
b,\end{cases}$ (4.25)

where $\sim$ means the adjacency in the diagram $X$ as before. Let $\Phi_{+}$
and $\Phi_{-}$ be the sets of _positive_ and _negative_ roots, respectively,
so that we have $\Phi=\Phi_{+}\sqcup\Phi_{-}$. The explicit description of
$\Phi_{+}$ is found in [Bou02].

######  Example 4.8 ([Bou02, Plates]).

(1) Type $A_{r}$. The positive roots are given by

$\displaystyle[i]:=\alpha_{i}\ (1\leq i\leq
r),\quad[i,j]:=\alpha_{i}+\cdots+\alpha_{j}\ (1\leq i<j\leq r).$ (4.26)

Thus, we have $|\Phi_{+}|=r(r+1)/2$ and $|\Phi|=r(r+1)$.

(2) Type $D_{r}$. The positive roots are given by

$\displaystyle\begin{split}[i]&:=\alpha_{i}\ (1\leq i\leq r),\\\
[i,j]&:=\alpha_{i}+\cdots+\alpha_{j}\ (1\leq i<j\leq r-2),\\\
[i;r-1]&:=\alpha_{i}+\cdots+\alpha_{r-2}+\alpha_{r-1}\ (1\leq i\leq r-2),\\\
[i;r]&:=\alpha_{i}+\cdots+\alpha_{r-2}+\alpha_{r}\ (1\leq i\leq r-2),\\\
[i;r-1,r]&:=\alpha_{i}+\cdots+\alpha_{r-2}+\alpha_{r-1}+\alpha_{r}\ (1\leq
i\leq r-2),\\\
\\{i,j\\}&:=\alpha_{i}+\cdots+\alpha_{j-1}+2\alpha_{j}+\cdots+2\alpha_{r-2}+\alpha_{r-1}+\alpha_{r}\\\
&\hskip 150.0pt(1\leq i<j\leq r-2),\end{split}$ (4.27)

Thus, we have $|\Phi_{+}|=r(r-1)$ and $|\Phi|=2r(r-1)$.

(3) Types $E_{6}$, $E_{7}$, $E_{8}$. The explicit description of the positive
roots is case-by-case. For example, for type $E_{6}$, besides the ones from
subdiagrams of type $A$ and $D$, there are seven positive roots

(4.28)

Here we only record the fact $|\Phi|=72$, $126$, $240$ for $E_{6}$, $E_{7}$,
$E_{8}$, respectively.

To each simple root $\alpha_{a}$, we assign a linear map $s_{a}:V\rightarrow
V$ called a _simple reflection_ by (under the normalization
$(\alpha_{a},\alpha_{a})=2$)

$\displaystyle s_{a}(v)=v-(\alpha_{a},v)\alpha_{a}.$ (4.29)

The group $W=W(X)$ generated by all simple reflections is the Weyl (Coxeter)
group associated with the root system $\Phi=\Phi(X)$. We have $W(\Phi)=\Phi$;
it is indeed the defining condition of a root system.

Let $w_{0}$ be the _longest element_ of $W$; namely, it has a reduced
expression of the longest length in $W$. It is known that $w_{0}$ is unique,
and its length is $|\Phi_{+}|$ [Hum90, §1.8]. Explicitly, it is described as
follows [Bou02, Plates]:

$\displaystyle w_{0}(\alpha_{a})=-\alpha_{\omega(a)},$ (4.30)

where $\omega$ is the diagram automorphism defined in (4.19). (This gives an
intrinsic meaning of $\omega$.) We have $w_{0}^{2}=\mathrm{id}$.

Let us introduce some distinguished elements in $W$.

######  Definition 4.9.

A _Coxeter element_ $c\in W$ is a product of all simple reflections, where
each reflection appears exactly once and the order of the product is
arbitrary.

A Coxeter element $c$ _does_ depend on the order of the product. Thus, it is
not unique. However, all Coxeter elements are known to be conjugate to each
other in $W$ [Hum90, Prop. 3.16]. Therefore, the following definition makes
sense.

######  Definition 4.10.

The _Coxeter number_ $h=h(X)$ of the Dynkin diagram $X$ is the order of any
Coxeter element $c$.

The following fact is well-known and already mentioned in Conjecture 1.1.

######  Theorem 4.11 ([Hum90, Prop. 3.18]).

The following equality holds:

$\displaystyle|\Phi|=hr.$ (4.31)

Explicitly, the Coxeter number is given by $h=r+1$ for $A_{r}$, $2r-2$ for
$D_{r}$, and 12, 18, 30 for $E_{6}$, $E_{7}$, $E_{8}$. In particular, $h$ is
odd if and only if $X=A_{r}$ ($r$: even).

We are especially interested in the following choice of Coxeter elements. We
put the sign $\kappa_{a}$ to each vertex $a$ of $X$ so that any adjacent
vertices in $X$ have opposite signs as we did in Section 4.2. Then, for any
$a$ and $b$ with $\kappa_{a}=\kappa_{b}$, $s_{a}$ and $s_{b}$ commute. So, we
have the products $s_{+}$ and $s_{-}$,

$\displaystyle s_{\pm}=\prod_{a\in I;\ \kappa_{a}=\pm 1}s_{a},$ (4.32)

where both are independent of the order of the product. In the special case of
$X=A_{1}$, one of $s_{\pm}$ is the identity. By (4.25) and (4.29), we have

$\displaystyle s_{\kappa_{a}}(\alpha_{a})=-\alpha_{a},\quad
s_{-{\kappa_{a}}}(\alpha_{a})=\alpha_{a}+\sum_{b\in I:\,b\sim a}\alpha_{b}.$
(4.33)

Then, $s_{-}s_{+}$ and $s_{+}s_{-}$ are Coxeter elements. For any positive odd
integer $k$, we define the powers

$\displaystyle(s_{+}s_{-})^{k/2}:=\underbrace{s_{-}s_{+}\cdots
s_{-}}_{\text{$k$
terms}},\quad(s_{-}s_{+})^{k/2}:=\underbrace{s_{+}s_{-}\cdots
s_{+}}_{\text{$k$ terms}}.$ (4.34)

For any positive even integer $k$, the powers $(s_{+}s_{-})^{k/2}$ and
$(s_{-}s_{+})^{k/2}$ are defined in the ordinary way.

The following formula is regarded as the “half periodicity” of the Coxeter
elements $s_{-}s_{+}$ and $s_{+}s_{-}$.

######  Proposition 4.12 ([Bou02, V. Exercise §6.2], [Hum90, §3.19 Exercise
2]).

The following formula holds.

$\displaystyle(s_{-}s_{+})^{h/2}=(s_{+}s_{-})^{h/2}=w_{0}.$ (4.35)

Moreover, both products are reduced expressions of $w_{0}$.

Indeed, they are reduced expressions because the number of the simple
reflections in each product is $hr/2=|\Phi_{+}|$.

We define a family of roots $\alpha(a;k)\in\Phi$ ($a\in I,\ -1\leq k\leq h$)
by

$\displaystyle\alpha(a;-1)$ $\displaystyle=-\alpha_{a}.$ (4.36)
$\displaystyle\alpha(a;k)$
$\displaystyle=\begin{cases}(s_{+}s_{-})^{k/2}(\alpha_{a})&\kappa_{a}=+,\\\
(s_{-}s_{+})^{k/2}(\alpha_{a})&\kappa_{a}=-\end{cases}\quad(0\leq k\leq h).$
(4.37)

The following result due to [FZ03b, FZ07] is crucial to our problem. So, we
provide a proof. (Our definition of $\alpha(a;k)$ is slightly different from
the one in [FZ03b, FZ07].)

######  Proposition 4.13.

(a) [FZ03b, Prop. 2.5]. For $0\leq k\leq h-1$, $\alpha(a;k)$ is a positive
root.

(b).

$\displaystyle\alpha(a;h-1)=\alpha_{\omega(a)},\quad\alpha(a;h)=-\alpha_{\omega(a)}.$
(4.38)

(c) [FZ07, Eq. (10.9)]. For $0\leq k\leq h-1$, the following relation holds:

$\displaystyle\alpha(a;k+1)+\alpha(a;k-1)=\sum_{b\in I:\,b\sim a}\alpha(b;k).$
(4.39)

In particular, $\alpha(a;k)$ $(1\leq k\leq h)$ are uniquely determined by the
initial condition

$\displaystyle\alpha(a;-1)=-\alpha_{a},\quad\alpha(a;0)=\alpha_{a}.$ (4.40)

and the recursion relation (4.39).

###### Proof.

(a). Let us concentrate on the case $\kappa_{a}=+$. (The other case is
similar.) The following facts are known:

  * (i).

A positive root $\alpha$ is sent to a negative root by a simple reflection
$s_{a}$ if and only if $\alpha=\alpha_{a}$ [Hum90, Prop. 1.4].

  * (ii).

For $\alpha_{a}$ with $\kappa_{a}=+$, $s_{-}(\alpha_{a})$ is positive by
(4.33).

  * (iii).

For a given reduced expression $w=s_{a_{1}}\cdots s_{a_{m}}$, we do not have
any pair $1\leq i<j\leq m$ such that $\alpha_{a_{i}}=s_{a_{i+1}}\cdots
s_{a_{j-1}}(\alpha_{a_{j}})$ [Hum90, Thm. 1.7].

Now suppose that the statement (a) fails. Let $k$ be the smallest number such
that it fails. By (ii), we have $k\geq 2$. Suppose that $k$ is even. (The
other case is similar.) Let $s_{+}=s_{a_{p}}\cdots s_{a_{1}}$. Then, by (i),
there is some simple root in the sequence of roots

$\displaystyle\alpha=(s_{+}s_{-})^{(k-1)/2}(\alpha_{a})>0,\ s_{a_{1}}\alpha,\
s_{a_{2}}s_{a_{1}}\alpha,\ \dots,\ (s_{+}s_{-})^{k/2}(\alpha_{a})<0.$ (4.41)

By (iii), this contradicts the non-reducibility of $(s_{+}s_{-})^{h/2}$ in
Proposition 4.12.

(b). The second equality is true by Proposition 4.12. The first equality
follows from the second one. For example, if $h$ is even, then
$\kappa_{\omega(a)}=\kappa_{a}$. Thus,

$\displaystyle\alpha(a;h-1)=s_{\kappa_{a}}(\alpha(a;h))=s_{\kappa_{a}}(-\alpha_{\omega(a)})=\alpha_{\omega(a)}.$
(4.42)

(c). We prove it by the induction on $k$. The case $k=0$ is nothing but the
second equality in (4.33). Suppose that the relation (4.39) holds for some $k$
($1\leq k\leq h-1$). Suppose that $\kappa_{a}=+$ and $k$ is even. (The other
cases are similar.) Note that $\kappa_{b}=-$ for the one in the RHS of (4.33).
Then, we have

$\displaystyle\begin{split}s_{+}(\alpha(a;k+1))&=\alpha(a;k+2),\quad
s_{+}(\alpha(a;k-1))=\alpha(a;k),\\\
s_{+}(\alpha(b;k))&=\alpha(a;k+1).\end{split}$ (4.43)

Thus, we obtain the relation (4.39) for $k+1$ from the one for $k$. ∎

######  Example 4.14.

Let us give examples to illustrate Proposition 4.13. Let
$\buildrel+\over{\rightarrow}$ and $\buildrel-\over{\rightarrow}$ represent
$s_{+}$ and $s_{-}$, respectively. We choose the sign $\kappa_{1}=+$ for all
examples.

(1) Type $A_{5}$. $h=6$. We have the following sequences.

$\displaystyle\begin{split}&[1]\buildrel-\over{\rightarrow}[1,2]\buildrel+\over{\rightarrow}[2,3]\buildrel-\over{\rightarrow}[3,4]\buildrel+\over{\rightarrow}[4,5]\buildrel-\over{\rightarrow}[5],\\\
&[2]\buildrel+\over{\rightarrow}[1,3]\buildrel-\over{\rightarrow}[1,4]\buildrel+\over{\rightarrow}[2,5]\buildrel-\over{\rightarrow}[3,5]\buildrel+\over{\rightarrow}[4],\\\
&[3]\buildrel-\over{\rightarrow}[2,4]\buildrel+\over{\rightarrow}[1,5]\buildrel-\over{\rightarrow}[1,5]\buildrel+\over{\rightarrow}[2,4]\buildrel-\over{\rightarrow}[3].\end{split}$
(4.44)

The sequences for $[4]$ and $[5]$ are also found in the above by reading them
in reverse. As an example of the relation (4.39), for $a=2$ and $k=3$, we have
$[3,5]+[1,4]=[3,4]+[1,5]$.

(2) Type $A_{6}$. $h=7$. We have the following sequences.

$\displaystyle\begin{split}&[1]\buildrel-\over{\rightarrow}[1,2]\buildrel+\over{\rightarrow}[2,3]\buildrel-\over{\rightarrow}[3,4]\buildrel+\over{\rightarrow}[4,5]\buildrel-\over{\rightarrow}[5,6]\buildrel+\over{\rightarrow}[6],\\\
&[2]\buildrel+\over{\rightarrow}[1,3]\buildrel-\over{\rightarrow}[1,4]\buildrel+\over{\rightarrow}[2,5]\buildrel-\over{\rightarrow}[3,6]\buildrel+\over{\rightarrow}[4,6]\buildrel-\over{\rightarrow}[5],\\\
&[3]\buildrel-\over{\rightarrow}[2,4]\buildrel+\over{\rightarrow}[1,5]\buildrel-\over{\rightarrow}[1,6]\buildrel+\over{\rightarrow}[2,6]\buildrel-\over{\rightarrow}[3,5]\buildrel+\over{\rightarrow}[4].\end{split}$
(4.45)

(3) Type $D_{5}$. $h=8$. We have the following sequences.

$\displaystyle\begin{split}&[1]\buildrel-\over{\rightarrow}[1,2]\buildrel+\over{\rightarrow}[2,3]\buildrel-\over{\rightarrow}[3;4,5]\buildrel+\over{\rightarrow}[3;4,5]\buildrel-\over{\rightarrow}[2,3]\buildrel+\over{\rightarrow}[1,2]\buildrel-\over{\rightarrow}[1],\\\
&[2]\buildrel+\over{\rightarrow}[1,3]\buildrel-\over{\rightarrow}[1;4,5]\buildrel+\over{\rightarrow}\\{2,3\\}\buildrel-\over{\rightarrow}\\{2,3\\}\buildrel+\over{\rightarrow}[1;4,5]\buildrel-\over{\rightarrow}[1,3]\buildrel+\over{\rightarrow}[2],\\\
&[3]\buildrel-\over{\rightarrow}[2;4,5]\buildrel+\over{\rightarrow}\\{1,3\\}\buildrel-\over{\rightarrow}\\{1,2\\}\buildrel+\over{\rightarrow}\\{1,2\\}\buildrel-\over{\rightarrow}\\{1,3\\}\buildrel+\over{\rightarrow}[2;4,5]\buildrel-\over{\rightarrow}[3],\\\
&[4]\buildrel+\over{\rightarrow}[3;4]\buildrel-\over{\rightarrow}[2;5]\buildrel+\over{\rightarrow}[1;5]\buildrel-\over{\rightarrow}[1;4]\buildrel+\over{\rightarrow}[2;4]\buildrel-\over{\rightarrow}[3;5]\buildrel+\over{\rightarrow}[5].\end{split}$
(4.46)

The automorphism $\omega$ is not trivial as given in (4.19). As an example of
the relation (4.39), for $a=3$ and $k=3$, we have
$\\{1,2\\}+\\{1,3\\}=\\{2,3\\}+[1;5]+[1;4]$.

(4) Type $D_{6}$. $h=10$. We have the following sequences.

$\displaystyle\begin{split}&[1]\buildrel-\over{\rightarrow}[1,2]\buildrel+\over{\rightarrow}[2,3]\buildrel-\over{\rightarrow}[3,4]\buildrel+\over{\rightarrow}[4;5,6]\buildrel-\over{\rightarrow}[4;5,6]\buildrel+\over{\rightarrow}\cdots\buildrel-\over{\rightarrow}[1],\\\
&[2]\buildrel+\over{\rightarrow}[1,3]\buildrel-\over{\rightarrow}[1,4]\buildrel+\over{\rightarrow}[2;5,6]\buildrel-\over{\rightarrow}\\{3,4\\}\buildrel+\over{\rightarrow}\\{3,4\\}\buildrel-\over{\rightarrow}\cdots\buildrel+\over{\rightarrow}[2],\\\
&[3]\buildrel-\over{\rightarrow}[2,4]\buildrel+\over{\rightarrow}[1;5,6]\buildrel-\over{\rightarrow}\\{1,4\\}\buildrel+\over{\rightarrow}\\{2,3\\}\buildrel-\over{\rightarrow}\\{2,3\\}\buildrel+\over{\rightarrow}\cdots\buildrel-\over{\rightarrow}[3],\\\
&[4]\buildrel+\over{\rightarrow}[3;5,6]\buildrel-\over{\rightarrow}\\{2,4\\}\buildrel+\over{\rightarrow}\\{1,3\\}\buildrel-\over{\rightarrow}\\{1,2\\}\buildrel+\over{\rightarrow}\\{1,2\\}\buildrel-\over{\rightarrow}\cdots\buildrel+\over{\rightarrow}[4],\\\
&[5]\buildrel-\over{\rightarrow}[4;5]\buildrel+\over{\rightarrow}[3;6]\buildrel-\over{\rightarrow}[2;6]\buildrel+\over{\rightarrow}[1;5]\buildrel-\over{\rightarrow}[1;5]\buildrel+\over{\rightarrow}\cdots\buildrel-\over{\rightarrow}[5],\\\
&[6]\buildrel-\over{\rightarrow}[4;6]\buildrel+\over{\rightarrow}[3;5]\buildrel-\over{\rightarrow}[2;5]\buildrel+\over{\rightarrow}[1;6]\buildrel-\over{\rightarrow}[1;6]\buildrel+\over{\rightarrow}\cdots\buildrel-\over{\rightarrow}[6].\end{split}$
(4.47)

The automorphism $\omega$ is trivial as given in (4.19).

(5) Type $E_{6}$. $h=12$. We have the following sequences.

$\displaystyle\begin{split}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
&\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\\\
&\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\ ,\\\
\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 0}
\put(5.0,0.0){\small 1} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
&\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 2}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 2}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
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\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\\\
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\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\ ,\\\
\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 0}
\put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
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\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
&\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 2} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 2}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 2} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
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\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 2} \end{picture}\\\ &\hskip
137.0pt\buildrel-\over{\rightarrow}\cdots\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\ ,\\\
\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 0}
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\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
&\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
1} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
\buildrel-\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\\\ &\hskip
137.0pt\buildrel+\over{\rightarrow}\cdots\
\buildrel+\over{\rightarrow}\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small
0} \put(5.0,0.0){\small 0} \put(10.0,0.0){\small 0} \put(15.0,0.0){\small 0}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\ .\end{split}$
(4.48)

As an example of the relation (4.39), for $a=3$ and $k=3$, we have

$\displaystyle\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 1}
\put(5.0,0.0){\small 2} \put(10.0,0.0){\small 3} \put(15.0,0.0){\small 2}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
+\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 1}
\put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
=\,\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 0}
\put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 1} \end{picture}\
+\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 1}
\put(5.0,0.0){\small 1} \put(10.0,0.0){\small 2} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 0} \put(10.0,8.0){\small 1} \end{picture}\
+\begin{picture}(22.0,12.0)(0.0,4.0)\put(0.0,0.0){\small 1}
\put(5.0,0.0){\small 1} \put(10.0,0.0){\small 1} \put(15.0,0.0){\small 1}
\put(20.0,0.0){\small 1} \put(10.0,8.0){\small 0} \end{picture}\ .$ (4.49)

####  4.5 Dynamics of tropical $Y$-systems

We prove Theorems 4.5 and 4.6 simultaneously by the _tropicalization method_.
Here, we continue to use the notation $[y_{a,a^{\prime}}(u)]$ for the tropical
$y$-variables in Chapter 3.

By applying the tropicalization (2.69), the $Y$-system (4.16) has the form

$\displaystyle[y_{a,a^{\prime}}(s+1)][y_{a,a^{\prime}}(s-1)]=\frac{\displaystyle\prod_{b\in
I:\,b\sim a}[1+y_{b,a^{\prime}}(s)]}{\displaystyle\prod_{b^{\prime}\in
I^{\prime}:\,b^{\prime}\sim a^{\prime}}[1+y_{a,b^{\prime}}(s)^{-1}]},$ (4.50)

which we call the _tropical $Y$-system_. Even though it is in the same form as
(4.16), it is drastically simplified as explained below. As we see in
Proposition 2.22, each tropical $y$-variable $[y_{a,a^{\prime}}(s)]$ is a
monomial in the initial $y$-variables $\mathbf{y}$ whose powers are given by
the corresponding $c$-vector $\mathbf{c}_{a,a^{\prime}}(s)$. Let us write it
simply as

$\displaystyle[y_{a,a^{\prime}}(s)]=y^{\mathbf{c}_{a,a^{\prime}}(s)}.$ (4.51)

Recall the sign-coherence property of $\mathbf{c}_{a,a^{\prime}}(s)$ in
Theorem 2.24. Let $\varepsilon_{a,a^{\prime}}(s)$ be the tropical sign of
$\mathbf{c}_{a,a^{\prime}}(s)$. Then,

$\displaystyle[1+y_{a,a^{\prime}}(s)]$
$\displaystyle=\begin{cases}1&\varepsilon_{a,a^{\prime}}(s)=1,\\\
y^{\mathbf{c}_{a,a^{\prime}}(s)}&\varepsilon_{a,a^{\prime}}(s)=-1,\end{cases}$
(4.52) $\displaystyle[1+y_{a,a^{\prime}}(s)^{-1}]^{-1}$
$\displaystyle=\begin{cases}y^{\mathbf{c}_{a,a^{\prime}}(s)}&\varepsilon_{a,a^{\prime}}(s)=1,\\\
1&\varepsilon_{a,a^{\prime}}(s)=-1.\end{cases}$ (4.53)

Thus, the tropical $Y$-system (4.16) is equivalent to the following
_piecewise-linear_ system for $c$-vectors:

$\displaystyle\mathbf{c}_{a,a^{\prime}}(s+1)+\mathbf{c}_{a,a^{\prime}}(s-1)=\sum_{\scriptstyle
b\in I:\,b\sim
a\atop\scriptstyle\varepsilon_{b,a^{\prime}}(s)=-1}\mathbf{c}_{b,a^{\prime}}(s)+\sum_{\scriptstyle
b^{\prime}\in I^{\prime}:\,b^{\prime}\sim
a^{\prime}\atop\scriptstyle\varepsilon_{a,b^{\prime}}(s)=1}\mathbf{c}_{a,b^{\prime}}(s),$
(4.54)

where the sum is zero if it it empty. It is piecewise-linear because it
involves the signs of $c$-vectors. Do not confuse the tropical sign
$\varepsilon_{a,a^{\prime}}(s)$ with the sign $\kappa_{a,a^{\prime}}$ for the
graph $X\times X^{\prime}$. Observe the similarity to the relation (4.39).

We will solve the tropical $Y$-system (4.54) with the help of Proposition
4.13. Recall that we concentrate on $c$-vectors in the even sector. Let
$\mathbf{e}_{a,a^{\prime}}\in\mathbb{Z}^{I\times I^{\prime}}$ be the unit
vector whose component is 1 only for the index $(a,a^{\prime})$ and 0
otherwise.

Step I: The initial condition at $s=-1$, $0$.

We have $y_{a,a^{\prime}}(-1)=y_{a,a^{\prime}}(0)^{-1}$ for $\kappa_{a,a}=-$.
Thus, the initial conditions for (4.54) is given by

$\displaystyle\mathbf{c}_{a,a^{\prime}}(-1)$
$\displaystyle=-\mathbf{e}_{a,a^{\prime}}\quad(\kappa_{a,a}=-),\quad\mathbf{c}_{a,a^{\prime}}(0)=\mathbf{e}_{a,a^{\prime}}\quad(\kappa_{a,a}=+).$
(4.55)

Let us compare it with (4.40).

Step II: The positive region $0\leq s\leq h^{\prime}-1$.

We determine $\mathbf{c}_{a,a^{\prime}}(1)$ by (4.54) with the initial
condition (4.55). Since $\mathbf{c}_{a,a^{\prime}}(0)$ is positive, the first
term in the RHS of (4.54) vanishes. Then, it has the same form as the
recursion relation (4.39) of $\alpha(a;k)$ for the vertical diagram
$X^{\prime}$. Let $\alpha^{\prime}_{a^{\prime}}$ and
$\alpha^{\prime}(a^{\prime};k)$ be the corresponding ones in $X^{\prime}$ to
$\alpha_{a}$ and $\alpha(a;k)$ in $X$. Then, we have

$\displaystyle\mathbf{c}_{a,a^{\prime}}(1)=\sum_{b^{\prime}\in
I^{\prime}}c_{b^{\prime}}\mathbf{e}_{a,b^{\prime}},\quad\text{where
$\alpha^{\prime}(a^{\prime};1)=\sum_{b^{\prime}\in
I^{\prime}}c_{b^{\prime}}\alpha^{\prime}_{b^{\prime}}$}.$ (4.56)

By (4.33), $\mathbf{c}_{a,a^{\prime}}(1)$ is positive. So, the same formula
holds for $\mathbf{c}_{a,a^{\prime}}(2)$ by replacing
$\alpha^{\prime}(a^{\prime};1)$ with $\alpha^{\prime}(a^{\prime};2)$.
Moreover, by Proposition 4.13 (a), $\mathbf{c}_{a,a^{\prime}}(2)$ is positive
if $h^{\prime}\geq 3$. The same process continues until $s=h^{\prime}-1$,
where

$\displaystyle\mathbf{c}_{a,a^{\prime}}(h^{\prime}-1)=\mathbf{e}_{a,\omega^{\prime}(a^{\prime})},\quad\mathbf{c}_{a,a^{\prime}}(h^{\prime})=-\mathbf{e}_{a,\omega^{\prime}(a^{\prime})}$
(4.57)

by (4.38).

Step III: The negative region $h^{\prime}\leq s\leq h+h-1$.

We determine $\mathbf{c}_{a,a^{\prime}}(h^{\prime}+1)$ by (4.54) and with the
initial condition (4.57). Since $\mathbf{c}_{a,a^{\prime}}(h^{\prime})$ is
negative, the second term in the RHS of (4.54) vanishes. Then, it has the same
form as the recursion relation (4.39) of $\alpha(a;k)$ for the horizontal
diagram $X$. Thus, we have

$\displaystyle\mathbf{c}_{a,a^{\prime}}(h^{\prime}+1)=-\sum_{b\in
I}c_{b}\mathbf{e}_{b,a^{\prime}},\quad\text{where $\alpha(a;1)=\sum_{b\in
I}c_{b}\alpha_{b}$}.$ (4.58)

Again, the process continues until $s=h^{\prime}+h-1$, where

$\displaystyle\mathbf{c}_{a,a^{\prime}}(h^{\prime}+h-1)=-\mathbf{e}_{\omega(a),\omega^{\prime}(a^{\prime})},\quad\mathbf{c}_{a,a^{\prime}}(h^{\prime}+h)=\mathbf{e}_{\omega(a),\omega^{\prime}(a^{\prime})}.$
(4.59)

Let us collect the desired results obtained in the above process.

  * •

We have the periodicity

$\displaystyle\mathbf{c}_{a,a^{\prime}}(h^{\prime}+h)=\mathbf{c}_{\omega(a),\omega^{\prime}(a^{\prime})}(0).$
(4.60)

  * •

The number of the positive and negative $c$-vectors $c_{a,a^{\prime}}(s)$ (in
the even sector) for $0\leq s\leq h^{\prime}+h-1$ are counted as

$\displaystyle N_{+}=\frac{1}{2}h^{\prime}rr^{\prime},\quad
N_{-}=\frac{1}{2}hrr^{\prime}.$ (4.61)

This is clear when $h^{\prime}$ and $h$ are even. When $h^{\prime}$ is odd,
for example, note that $r^{\prime}$ is even. Then, the formula still holds.

This completes the proof of Theorems 4.5 and 4.6.

######  Example 4.15.

Consider the case $(X,X^{\prime})=(A_{3},A_{2})$ in Example 4.3. We have $h=4$
and $h^{\prime}=3$. Let $Q$ be the quiver therein with the following
assignment of signs:

$Q=\ \lower 12.0pt\hbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.88104pt\hbox{{\hbox{\kern-2.0pt\raise 28.4526pt\hbox{\hbox{\kern
0.0pt\raise<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern
25.9526pt\raise 28.4526pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern
26.4526pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 54.90521pt\raise
28.4526pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern
54.40521pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.5pt\raise
36.98839pt\hbox{\hbox{\kern
0.0pt\raise-1.9375pt\hbox{$\textstyle{\text{\small$-$}}$}}}}}{\hbox{\kern
24.9526pt\raise 36.98839pt\hbox{\hbox{\kern
0.0pt\raise-2.25pt\hbox{$\textstyle{\text{\small+}}$}}}}}{\hbox{\kern
55.40521pt\raise 36.98839pt\hbox{\hbox{\kern
0.0pt\raise-1.9375pt\hbox{$\textstyle{\text{\small$-$}}$}}}}}{\hbox{\kern-12.88104pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9375pt\hbox{$\textstyle{\text{\small$-$}}$}}}}}{\hbox{\kern-14.88104pt\raise
28.4526pt\hbox{\hbox{\kern
0.0pt\raise-2.25pt\hbox{$\textstyle{\text{\small$+$}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
22.76208pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.14313pt\raise 28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
5.69052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
51.21469pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.4526pt\raise 5.69052pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.90521pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}
}$

$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(5,35)(5,55) 3(13,35)(13,55) $\scriptstyle 0$$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(45,35)(45,55) 3(53,35)(53,55) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(85,35)(85,55) 3(93,35)(93,55) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$3(5,-5)(5,15) 3(13,-5)(13,15) $\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$3(45,-5)(45,15)
3(53,-5)(53,15) $\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$3(85,-5)(85,15) 3(93,-5)(93,15)
$C(0)$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$3(5,35)(5,55) 3(13,35)(13,55) $\scriptstyle
0$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(45,35)(45,55) 3(53,35)(53,55) $\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$3(85,35)(85,55) 3(93,35)(93,55) $\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$3(5,-5)(5,15) 3(13,-5)(13,15) $\scriptstyle 0$$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle
0$3(45,-5)(45,15) 3(53,-5)(53,15) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle
1$3(85,-5)(85,15) 3(93,-5)(93,15)
$C(1)$$\rightarrow$$\scriptstyle+$-$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$3(5,35)(5,55) 3(13,35)(13,55) $\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$3(45,35)(45,55)
3(53,35)(53,55) $\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$-$\scriptstyle 1$3(85,35)(85,55) 3(93,35)(93,55)
$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(5,-5)(5,15) 3(13,-5)(13,15) $\scriptstyle 0$-$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle
0$3(45,-5)(45,15) 3(53,-5)(53,15) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(85,-5)(85,15) 3(93,-5)(93,15)
$C(2)$$\rightarrow$$\scriptstyle-$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$3(0,45)(20,45)$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$-$\scriptstyle 1$$\scriptstyle 0$3(40,45)(60,45) $\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$-$\scriptstyle 1$3(80,45)(100,45) -$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$3(0,5)(20,5)
$\scriptstyle 0$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(40,5)(60,5) $\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$3(80,5)(100,5)
$C(3)$$\rightarrow$$\scriptstyle+$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$3(0,45)(20,45)$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle
1$-$\scriptstyle 1$-$\scriptstyle 1$3(40,45)(60,45) $\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$3(80,45)(100,45) -$\scriptstyle 1$-$\scriptstyle
1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$3(0,5)(20,5)
$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(40,5)(60,5) $\scriptstyle 0$-$\scriptstyle
1$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(80,5)(100,5)$C(4)$$\rightarrow$$\scriptstyle-$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle
1$-$\scriptstyle 1$3(0,45)(20,45)$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$$\scriptstyle 1$$\scriptstyle 1$3(40,45)(60,45)
$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$-$\scriptstyle
1$-$\scriptstyle 1$$\scriptstyle 0$3(80,45)(100,45) $\scriptstyle
1$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$3(0,5)(20,5) -$\scriptstyle 1$-$\scriptstyle
1$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(40,5)(60,5) $\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$3(80,5)(100,5)
$C(5)$$\rightarrow$$\scriptstyle+$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle
1$3(0,45)(20,45)$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$-$\scriptstyle 1$$\scriptstyle 0$3(40,45)(60,45) $\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle
1$$\scriptstyle 0$3(80,45)(100,45) $\scriptstyle 0$$\scriptstyle
0$-$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(0,5)(20,5) $\scriptstyle 1$$\scriptstyle 1$$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$3(40,5)(60,5) -$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(80,5)(100,5)$C(6)$$\rightarrow$$\scriptstyle-$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$3(5,35)(5,55) 3(13,35)(13,55) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle
0$3(45,35)(45,55) 3(53,35)(53,55) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle
0$3(85,35)(85,55) 3(93,35)(93,55) $\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$3(5,-5)(5,15) 3(13,-5)(13,15) $\scriptstyle 0$$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$3(45,-5)(45,15)
3(53,-5)(53,15) $\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle 0$$\scriptstyle 0$3(85,-5)(85,15) 3(93,-5)(93,15)
$C(7)$$\rightarrow$$\scriptstyle+$ Figure 4.2: Tropical $Y$-system for
$(X,X^{\prime})=(A_{3},A_{2})$, where $h+h^{\prime}=7$. The framed ones are
the $c$-vectors in the even sector.

We take $Q$ as the initial quiver for the corresponding $Y$-pattern. We
arrange each $c$-vector in a matrix form with horizontal and vertical indices
in $I\times I^{\prime}$; for example, the matrix

$\displaystyle\begin{matrix}0&0&0\\\ 0&1&1\end{matrix}$ (4.62)

represents the $c$-vector $\mathbf{c}=\mathbf{e}_{2,2}+\mathbf{e}_{3,2}$. Here
we calculate the mutation of $c$-vectors directly by the mutation rule (2.50)
in the ambient $Y$-pattern, rather than by the tropical $Y$-system (4.54). The
rule (2.50) is translated into the following pictorial rule, which is very
efficient:

$\displaystyle\mathbf{c}^{\prime}_{k}$ $\displaystyle=-\mathbf{c}_{k},$ (4.63)
$\displaystyle\mathbf{c}^{\prime}_{i}$
$\displaystyle=\begin{cases}\mathbf{c}_{i}+\mathbf{c}_{k}&\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 2.0pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 25.9526pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.55031pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.96786pt\hbox{$\textstyle{\text{\small$i$}}$}}}}}{\hbox{\kern
25.96823pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-3.125pt\hbox{$\textstyle{\text{\small$k$}}$}}}}}{\hbox{\kern
45.35603pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25pt\hbox{$\textstyle{\text{\small and
$\varepsilon_{k}=-$}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.76208pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}},\\\
\mathbf{c}_{i}+\mathbf{c}_{k}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
2.0pt\hbox{{\hbox{\kern-2.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}{\hbox{\kern 25.9526pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-1.55031pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.96786pt\hbox{$\textstyle{\text{\small$i$}}$}}}}}{\hbox{\kern
25.96823pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-3.125pt\hbox{$\textstyle{\text{\small$k$}}$}}}}}{\hbox{\kern
43.35602pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25pt\hbox{$\textstyle{\text{\small and
$\varepsilon_{k}=+$}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
5.69052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}},\\\
\mathbf{c}_{i}&\text{othterwise}\end{cases}\quad\text{($i\neq k$)}.$ (4.64)

All $c$-vectors from $s=0$ to $s=7$ are presented in Figure 4.2. The desired
periodicity is found at $s=h+h^{\prime}=7$.

As we see in the above example, and it is also clear from the analysis in the
general case, in the _positive region_ $0\leq s\leq h^{\prime}-1$, a
nontrivial mutation occurs only “in the vertical direction” by the
coordination of tropical signs and arrows. (This is indicated in the dotted
vertical lines in Figure 4.2.) Similarly, in the _negative region_
$h^{\prime}\leq s\leq h^{\prime}+h-1$, a nontrivial mutation occurs only “in
the horizontal direction”. We call this factorization into the vertical and
horizontal directions the _factorization property_ of the tropical $Y$-system
(4.50). Such a simplification does not occur in the nontropical $Y$-system
(4.16).

In summary, we use the formulation of the $Y$-systems by $Y$-patterns,
together with the tropicalization, to prove the periodicity and the DIs for
$Y$-systems. The tropicalization reveals that Coxeter elements are behind the
scene and govern the dynamics of the $Y$-systems. This is why the Coxeter
number is relevant for the periodicity.

Notes

The cluster algebraic structure for the $Y$-systems (1.36) of type $(X,A_{1})$
was recognized first by [FZ03b] and formulated by $Y$-patterns by [FZ07]. It
was generalized to the type $(X,X^{\prime})$ by [Kel10] using the quiver
$Q(X,X^{\prime})$.

There is a “twin” system to $Y$-systems called _$T$ -systems_, which were also
introduced and studied together with $Y$-systems in B.C. [KNS94]. $T$-systems
are satisfied by $x$-variables for the same sequence of mutations (4.11)
[DK09, IIK+10].

After the results by [GT96, FS95] in B.C. mentioned in Chapter 1, the
periodicity of $Y$-systems was proved by [FZ03b] for type $(X,A_{1})$ by a
cluster algebraic method. The method consists of the tropicalization and the
construction of $F$-polynomials (with the assistance of a computer). The role
of the Coxeter elements was also clarified. The periodicity of $Y$-systems for
type $(X,X^{\prime})$ was proved by [Kel10, Kel13b] based on the
_categorification_ of cluster patterns by triangulated categories. The
refinement to the half periodicity was given in [IIK+10], where the
periodicity of associated $T$-systems was also proved. Using the ideas and
results of [CGT99, FZ03b, FZ07, Cha05, Kel10], the DIs for the $Y$-systems of
type $(X,X^{\prime})$ were proved by [Nak11a].

By further developing the above methods and ideas, together with the
detropicalization by cluster categories by [Pla11], the periodicity and DIs
for more complicated $Y$-systems were proved. For the $Y$-systems of
nonsimply-laced types they were proved by [IIK+13a, IIK+13b]. For the sine-
Gordon and reduced sine-Gordon $Y$-systems they were proved by [NT10] in a
special case and by [NS16] in full generality.

Meanwhile, the periodicity of $Y$-systems of type $(A_{r},A_{r^{\prime}})$ was
also proved by [Vol07] with the determinant solution and by [Sze09] using the
flatness of a graph connection. Thus, the type $A$ is the special case where
several ideas in mathematics intersect as in Lie theory.

##  Part II Classical mechanical method

###  Chapter 5 Fock-Goncharov decomposition and dilogarithm

We introduce two new ideas for $Y$-patterns. The first one is the
decomposition of a mutation of $Y$-seeds into tropical and nontropical parts
based on the idea of Fock-Goncharov. The second one is the mutation-compatible
Poisson bracket introduced by Gekhtman-Shapiro-Vainshtein. By combining two
ideas, the Euler dilogarithm emerges as the Hamiltonian for the nontropical
part of each mutation.

####  5.1 Fock-Goncharov decomposition

Let us introduce the following notion.

######  Definition 5.1 (Skew-symmetric decomposition).

For a skew-symmetrizable (integer) matrix $B=(b_{ij})_{i,j=1}^{n}$, we
consider a decomposition

$\displaystyle B=\Delta\Omega,$ (5.1)

where $\Delta=\mathrm{diag}(\delta_{1},\dots,\delta_{n})$ is a diagonal matrix
with positive integer diagonals and $\Omega$ is a skew-symmetric rational
matrix. We call it a _skew-symmetric decomposition_ of $B$.

For a skew-symmetrizer $D$ of the form (3.8) for $B$, we set

$\displaystyle\Delta=D^{-1},\quad\Omega=DB.$ (5.2)

Then, we have a skew-symmetric decomposition of $B$. Conversely, for a skew-
symmetric decomposition of $B$, $D=\Delta^{-1}$ is a skew-symmetrizer of $B$
of the form (3.8) such that $DB=\Omega$. Thus, choosing a skew-symmetrizer of
the form (3.8) is equivalent to choosing a skew-symmetric decomposition.

We start by observing that there is some flexibility in writing the mutation
(2.9)–(2.11). Namely, Let $\varepsilon\in\\{1,-1\\}$. Then, by (2.8), one can
easily check that the RHSs of the following expressions are independent of
$\varepsilon$:

$\displaystyle x^{\prime}_{i}$ $\displaystyle=\begin{cases}\displaystyle
x_{k}^{-1}\Biggl{(}\,\prod_{j=1}^{n}x_{j}^{[-\varepsilon
b_{jk}]_{+}}\Biggr{)}\frac{1+\hat{y}_{k}^{\varepsilon}}{1\oplus
y_{k}^{\varepsilon}}&i=k,\\\ x_{i}&i\neq k,\end{cases}$ (5.3) $\displaystyle
y^{\prime}_{i}$ $\displaystyle=\begin{cases}\displaystyle y_{k}^{-1}&i=k,\\\
y_{i}y_{k}^{[\varepsilon b_{ki}]_{+}}(1\oplus
y_{k}^{\varepsilon})^{-b_{ki}}&i\neq k,\end{cases}$ (5.4) $\displaystyle
b^{\prime}_{ij}$ $\displaystyle=\begin{cases}-b_{ij}&\text{$i=k$ or $j=k$,}\\\
b_{ij}+b_{ik}[\varepsilon b_{kj}]_{+}+[-\varepsilon b_{ik}]_{+}b_{kj}&i,j\neq
k.\end{cases}$ (5.5)

The case $\varepsilon=1$ yields the original mutation formulas. We call them
the _$\varepsilon$ -expressions_ of the mutation. Similarly, the mutations of
$C$\- and $G$-matrices in Section 2.5 have the following
$\varepsilon$-expressions:

$\displaystyle c_{ij;t^{\prime}}$
$\displaystyle=\begin{cases}-c_{ik;t}&j=k,\\\ c_{ij;t}+c_{ik;t}[\varepsilon
b_{kj;t}]_{+}+[-\varepsilon c_{ik;t}]_{+}b_{kj;t}&j\neq k,\end{cases}$ (5.6)
$\displaystyle g_{ij;t^{\prime}}$ $\displaystyle=\begin{cases}\displaystyle-
g_{ik;t}+\sum_{\ell=1}^{n}g_{i\ell;t}[-\varepsilon b_{\ell
k;t}]_{+}-\sum_{\ell=1}^{n}b_{i\ell;t_{0}}[-\varepsilon c_{\ell
k;t}]_{+}&j=k,\\\ g_{ij;t}&j\neq k.\end{cases}$ (5.7)

To check the $\varepsilon$-independence of (5.7), we also use the formula
(2.55).

Now we are back in the situation in Section 3.1. Namely, we consider a cluster
pattern $\mathbf{\Sigma}$ with _free coefficients_ at $t_{0}$. Again, we
concentrate on the $Y$-pattern only, and we write $\oplus$ in
$\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})$ as $+$. Consider a sequence of
mutations therein

$\displaystyle\Upsilon=\Upsilon(0)\ {\buildrel{k_{0}}\over{\rightarrow}}\
\Upsilon(1)\ {\buildrel{k_{1}}\over{\rightarrow}}\ \cdots\
{\buildrel{k_{P-1}}\over{\rightarrow}}\ \Upsilon(P),$ (5.8)

where $\Upsilon(s)=(\mathbf{y}(s),B(s))$. Here, we especially assume that _the
$Y$-seed $\Upsilon(0)$ coincides with the initial $Y$-seed
$\Upsilon_{t_{0}}=\Upsilon=(\mathbf{y},B)$_. The $\varepsilon$-expression of
the mutation $\mu_{k_{s}}$ for $y$-variables is given by

$\displaystyle y_{i}(s+1)$ $\displaystyle=\begin{cases}\displaystyle
y_{k_{s}}(s)^{-1}&i=k_{s},\\\ y_{i}(s)y_{k_{s}}(s)^{[\varepsilon
b_{k_{s}i}(s)]_{+}}(1+y_{k_{s}}(s)^{\varepsilon})^{-b_{k_{s}i}(s)}&i\neq
k_{s}.\end{cases}$ (5.9)

Originally, all $y$-variables $y_{i}(s)$ ($s=0$, …, $P$) belong to the common
semifield $\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})$ for the initial $y$-variables
$\mathbf{y}$, and the formula (5.9) gives relations among them. Here, we
introduce an alternative veiwpoint. For each $s=0$, …, $P$, we regard the
$y$-variables $\mathbf{y}(s)=(y_{1}(s),\dots,y_{n}(s))$ as formal variables
and consider the universal semifield
$\mathbb{P}(s):=\mathbb{Q}_{\mathrm{sf}}(\mathbf{y}(s))$. Then, we interpret
the mutation (5.9) as a semifield isomorphism ($s=0$, …, $P-1$)

$\displaystyle\begin{matrix}\mu{(s)}\colon&\mathbb{P}(s+1)&\longrightarrow&\mathbb{P}(s)\\\
&y_{i}(s+1)&\mapsto&\begin{cases}\displaystyle y_{k_{s}}(s)^{-1}&i=k_{s},\\\
y_{i}(s)y_{k_{s}}(s)^{[\varepsilon
b_{k_{s}i}(s)]_{+}}(1+y_{k_{s}}(s)^{\varepsilon})^{-b_{k_{s}i}(s)}&i\neq
k_{s}.\end{cases}\end{matrix}$ (5.10)

Now taking advantage of the $\varepsilon$-expression, we set $\varepsilon$
therein to the tropical sign $\varepsilon_{s}:=\varepsilon_{k_{s}}(s)$ in
Section 3.3. Accordingly, we introduce a semifield isomorphism ($s=0$, …,
$P-1$)

$\displaystyle\begin{matrix}\tau{(s)}\colon&\mathbb{P}(s+1)&\longrightarrow&\mathbb{P}(s)\\\
&y_{i}(s+1)&\mapsto&\begin{cases}\displaystyle y_{k_{s}}(s)^{-1}&i=k_{s},\\\
y_{i}(s)y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}&i\neq
k_{s}\end{cases}\end{matrix}$ (5.11)

and a semifield automorphism ($s=0$, …, $P-1$)

$\displaystyle\begin{matrix}\rho{(s)}\colon&\mathbb{P}(s)&\longrightarrow&\mathbb{P}(s)\\\
&y_{i}(s)&\mapsto&y_{i}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}.\end{matrix}$
(5.12)

Then, we have a decomposition of the mutation $\mu{(s)}$

$\displaystyle\mu{(s)}=\rho{(s)}\circ\tau{(s)}.$ (5.13)

We call it the _Fock-Goncharov decomposition_ $\mu{(s)}$. Note that this
decomposition fully depends on the choice of the initial vertex $t_{0}$,
because the tropical sign $\varepsilon_{s}$ depends on $t_{0}$. Also, it
crucially relies on the sign-coherence property in Theorem 2.24.

Let us clarify the meaning of this decomposition. Let $C(s)$ be the $C$-matrix
for $\Upsilon(s)$, and let $\mathbf{c}_{i}(s)$ be its $i$th $c$-vector. We do
the same specialization $\varepsilon=\varepsilon_{s}$ for the $C$-matrix
mutation (5.6). By the sign-coherence property, we have

$\displaystyle[-\varepsilon_{s}c_{\ell k_{s}}(s)]_{+}=0\quad(1\leq\ell\leq
n).$ (5.14)

Thus, the mutation (5.6) is simplified as

$\displaystyle\mathbf{c}_{i}(s+1)$
$\displaystyle=\begin{cases}-\mathbf{c}_{k_{s}}(s)&i=k_{s},\\\
\mathbf{c}_{i}(s)+[\varepsilon_{s}b_{k_{s}i}(s)]_{+}\mathbf{c}_{k_{s}}(s)&i\neq
k_{s}.\end{cases}$ (5.15)

We recognize it as the log-version of (5.11). By this reason, we call
$\tau{(s)}$ and $\rho{(s)}$ the _tropical part_ and the _nontropical part_ of
$\mu{(s)}$, respectively.

Next, we introduce the composite mutation $(s=0,\,\dots,\,P-1)$

$\displaystyle\mu(s;0)$
$\displaystyle:=\mu(0)\circ\mu(1)\circ\cdots\circ\mu(s)\colon\mathbb{P}(s+1)\longrightarrow\mathbb{P}(0).$
(5.16)

Let us define its Fock-Goncharov decomposition.

Consider a semifield isomorphism ($s=0,\,\dots,\,P-1$)

$\displaystyle\tau(s;0)$
$\displaystyle:=\tau(0)\circ\tau(1)\circ\cdots\circ\tau(s)\colon\mathbb{P}(s+1)\longrightarrow\mathbb{P}(0).$
(5.17)

The map $\tau{(s;0)}$ is identified with the _tropical part_ of $\mu{(s;0)}$
as follows.

######  Proposition 5.2 (e.g., [Nak23, Prop. II.4.3]).

The following formula holds:

$\displaystyle\tau(s;0)(y_{i}(s+1))=y^{\mathbf{c}_{i}(s+1)}.$ (5.18)

###### Proof.

We prove it by the induction on $s$. For $s=0$, by (5.15), we have

$\displaystyle\begin{matrix}\tau{(0)}:&\mathbb{P}(1)&\longrightarrow&\mathbb{P}(0)\\\
&y_{i}(1)&\mapsto&\begin{cases}\displaystyle
y_{k_{0}}^{-1}=y^{\mathbf{c}_{k_{0}}(1)}&i=k_{0},\\\
y_{i}y_{k_{0}}^{[\varepsilon_{0}b_{k_{0}i}(0)]_{+}}=y^{\mathbf{c}_{i}(1)}&i\neq
k_{0}.\end{cases}\end{matrix}$ (5.19)

Similarly, assuming (5.18) for $s-1$, we obtain

$\displaystyle\quad\
\tau{(s;0)}(y_{i}(s+1))=(\tau{(s-1;0)}\circ\tau(s))(y_{i}(s+1))$
$\displaystyle=\begin{cases}\displaystyle
y^{-\mathbf{c}_{k_{s}}(s)}&i=k_{s},\\\
y^{\mathbf{c}_{i}(s)}y^{\mathbf{c}_{k_{s}}(s)}{}^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}&i\neq
k_{s}\end{cases}$ $\displaystyle=y^{\mathbf{c}_{i}(s+1)},$

where we used (5.15) again in the last equality. ∎

Below, we consider the nontropical part of $\mu{(s;0)}$. Let us fix a skew-
symmetric decomposition

$\displaystyle B(s)=\Delta\Omega(s),$ (5.20)

where $\Delta=\mathrm{diag}(\delta_{1},\dots,\delta_{n})$ is common for $B(s)$
($s=0$, …, $P$). Let $\Omega=\Omega(0)$. We introduce a skew-symmetric
bilinear form $\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}$ on
$\mathbb{Z}^{n}$ by

$\displaystyle\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}:=\mathbf{n}^{T}\Omega\mathbf{n}^{\prime}.$
(5.21)

Also, for each $c$-vector $\mathbf{c}_{i}(s)$ and its tropical sign
$\varepsilon_{i}(s)$, we set

$\displaystyle\mathbf{c}^{+}_{i}(s):=\varepsilon_{i}(s)\mathbf{c}_{i}(s).$
(5.22)

This is a positive vector, and we call it a _$c^{+}$ -vector_. Then, we
introduce a semifield automorphism ($s=0,\,\dots,\,P-1$)

$\displaystyle\begin{matrix}\mathfrak{q}{(s)}\colon&\mathbb{P}(0)&\longrightarrow&\mathbb{P}(0)\\\
&y^{\mathbf{n}}&\mapsto&y^{\mathbf{n}}(1+y^{\mathbf{c}^{+}_{k_{s}}(s)})^{-\\{\delta_{k_{s}}\mathbf{c}_{k_{s}}(s),\mathbf{n}\\}_{\Omega}}.\end{matrix}$
(5.23)

Note that $\mathfrak{q}{(0)}=\rho(0)$.

######  Proposition 5.3 ([Nak23, Prop.II.4.5]).

(a). The following formulas hold:

$\displaystyle\\{\delta_{i}\mathbf{c}_{i}(s),\mathbf{c}_{j}(s)\\}_{\Omega}=b_{ij}(s),$
(5.24)
$\displaystyle\mathfrak{q}{(s)}(y^{\mathbf{c}_{i}(s)})=y^{\mathbf{c}_{i}(s)}(1+y^{\mathbf{c}^{+}_{k_{s}}(s)})^{-b_{k_{s}i}(s)}.$
(5.25)

(b). The following commutative diagram holds $(s=1,\,\dots,\,P-1)$:

$\textstyle{\mathbb{P}(s)}$$\textstyle{\mathbb{P}(0)}$$\textstyle{\mathbb{P}(s)}$$\textstyle{\mathbb{P}(0)}$$\tau(s-1;0)$$\tau(s-1;0)$$\rho(s)$$\mathfrak{q}(s)$
(5.26)

###### Proof.

(a). The equality (5.24) is written in the matrix form

$\displaystyle\Delta C(s)^{T}\Omega C(s)=\Delta\Omega(s).$ (5.27)

This is equivalent to the equality (2.78). The equality (5.25) follows from
(5.23) and (5.24). (b). By Proposition 5.2, (5.12), and (5.25), we have

$\displaystyle\begin{split}y_{i}(s)&\buildrel\tau(s-1;0)\over{\mapsto}y^{\mathbf{c}_{i}(s)}\buildrel\mathfrak{q}(s)\over{\mapsto}y^{\mathbf{c}_{i}(s)}(1+y^{\mathbf{c}^{+}_{k_{s}}(s)})^{-b_{k_{s}i}(s)},\\\
y_{i}(s)&\buildrel\rho(s)\over{\mapsto}y_{i}(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}\buildrel\tau(s-1;0)\over{\mapsto}y^{\mathbf{c}_{i}(s)}(1+y^{\mathbf{c}^{+}_{k_{s}}(s)})^{-b_{k_{s}i}(s)}.\end{split}$
(5.28)

∎

All ingredients introduced so far are summarized in a single commutative
diagram as follows:

$\textstyle{\mathbb{P}(P)}$$\textstyle{\mathbb{P}(P-1)}$$\textstyle{\mathbb{P}(P-1)}$$\textstyle{\mathbb{P}(P-2)}$$\textstyle{\mathbb{P}(2)}$$\textstyle{\mathbb{P}(1)}$$\textstyle{\mathbb{P}(1)}$$\textstyle{\mathbb{P}(0)}$$\textstyle{\mathbb{P}(0)}$$\textstyle{\mathbb{P}(0)}$$\textstyle{\mathbb{P}(0)}$$\textstyle{\mathbb{P}(0)}$$\tau(P-1)$$\tau(P-2;0)$$\tau(P-3;0)$$\tau(P-2)$$\tau(1)$$\tau(0)$$\tau(0)$$\mu(P-1)$$\mu(1)$$\mu(0)$$\rho(P-1)$$\rho(1)$$\mathfrak{q}(P-1)$$\mathfrak{q}(1)$$\mathfrak{q}(0)$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\ddots}$
(5.29)

Finally, we introduce a semifield automorphism $(s=0,\,\dots,\,P-1)$

$\displaystyle\mathfrak{q}(s;0)$
$\displaystyle:=\mathfrak{q}(0)\circ\mathfrak{q}(1)\circ\cdots\circ\mathfrak{q}(s)\colon\mathbb{P}(0)\longrightarrow\mathbb{P}(0).$
(5.30)

From the diagram (5.29), we immediately obtain the decomposition

$\displaystyle\mu(s;0)=\mathfrak{q}(s;0)\circ\tau(s;0).$ (5.31)

We call it the _Fock-Goncharov decomposition_ of the composite mutation
$\mu(s;0)$, where $\mathfrak{q}{(s;0)}$ is the _nontropical part_ of
$\mu{(s;0)}$.

####  5.2 Mutation-compatible Poisson bracket

Let us recall the most basic notion in Poisson geometry (e.g., [Wei83]).

######  Definition 5.4 (Poisson bracket).

Let $\mathcal{C}(M)$ be the commutative algebra of all smooth real-valued
functions on a smooth manifold $M$. A bilinear map
$\\{\cdot,\cdot\\}\colon\mathcal{C}(M)\times\mathcal{C}(M)\longrightarrow\mathcal{C}(M)$
is a _Poisson bracket_ on $M$ (or $\mathcal{C}(M)$) if the following
conditions are satisfied.

  * •

skew-symmetry: $\\{f,g\\}=-\\{g,f\\}$.

  * •

Leibniz rule: $\\{f,gh\\}=\\{f,g\\}h+g\\{f,h\\}.$

  * •

Jacobi identity: $\\{\\{f,g\\},h\\}+\\{\\{g,h\\},f\\}+\\{\\{h,f\\},g\\}=0.$

Equipped with a Poisson bracket, the manifold $M$ is called a _Poisson
manifold_ , and the algebra $\mathcal{C}(M)$ is called a _Poisson algebra_.

For simplicity, consider a smooth manifold $M=\mathbb{R}^{n}$ with global
coordinates $\mathbf{y}=(y_{1},\dots,y_{n})$. Then, $\mathcal{C}(M)$ is
identified with the algebra $\mathcal{C}(\mathbf{y})$ of all smooth functions
of $\mathbf{y}$. The following fact is due to Lie.

######  Proposition 5.5 (e.g., [Wei83]).

(a). Let $\\{\cdot,\cdot\\}$ be any poisson bracket on $\mathcal{C}(M)$. Then,
the following formula holds:

$\displaystyle\\{f,g\\}=\sum_{i,j=1}^{n}\frac{\partial f}{\partial
y_{i}}\frac{\partial g}{\partial y_{j}}\\{y_{i},y_{j}\\}.$ (5.32)

(b). Conversely, for any data $w_{ij}\in\mathcal{C}(M)$
$(i,\,j=1,\,\dots,\,r)$ satisfying

$\displaystyle w_{ij}=-w_{ji},$ (5.33)
$\displaystyle\sum_{\ell=1}^{n}\left(\frac{\partial w_{ij}}{\partial
y_{\ell}}w_{\ell k}+\frac{\partial w_{jk}}{\partial y_{\ell}}w_{\ell
i}+\frac{\partial w_{ki}}{\partial y_{\ell}}w_{\ell j}\right)=0,$ (5.34)

the bracket $\\{\cdot,\cdot\\}$ defined by $\\{y_{i},y_{j}\\}:=w_{ij}$ and
(5.32) is a Poisson bracket on $\mathcal{C}(\mathbf{y})$, Moreover, (5.34) is
written as

$\displaystyle\\{\\{y_{i},y_{j}\\},y_{k}\\}+\\{\\{y_{j},y_{k}\\},y_{i}\\}+\\{\\{y_{k},y_{i}\\},y_{j}\\}=0.$
(5.35)

###### Proof.

For the reader’s convenience, we present an outline of the proof.

(a). For polynomial functions $f$, $g$ in $\mathbf{y}$, the formula (5.32) is
deduced from the Leibniz rule. Then, it is uniquely extended to smooth
functions by the Weierstrass approximation theorem.

(b). The only nontrivial condition to be checked is the Jacobi identity. Let
us use the abbreviations $\partial f/\partial y_{i}=f_{i}$,
$\partial^{2}f/\partial y_{i}\partial y_{j}=f_{ij}$. Then, we have

$\displaystyle\begin{split}&\quad\
\\{\\{f,g\\},h\\}=\biggl{\\{}\sum_{i,j}f_{i}g_{j}w_{ij},h\biggr{\\}}\\\
&=\sum_{i,j,k,\ell}(f_{i}g_{j})_{k}h_{\ell}w_{ij}w_{k\ell}+\sum_{i,j,k,\ell}f_{i}g_{j}h_{\ell}(w_{ij})_{k}w_{k\ell}\\\
&=\sum_{i,j,k,\ell}(f_{ik}g_{j}h_{\ell}-g_{ik}h_{j}f_{\ell})w_{ij}w_{k\ell},+\sum_{i,j,k,\ell}f_{i}g_{j}h_{\ell}(w_{ij})_{k}w_{k\ell},\end{split}$
(5.36)

where we used the skew-symmetry (5.33) in the last equality. The Jacobi
identity follows from the last expression and the condition (5.34). ∎

We are interested in the following _quadratic_ Poisson brackets.

######  Lemma 5.6.

Let $\Omega=(\omega_{ij})_{i,j=1}^{n}$ be any skew-symmetric real matrix.
Define the data $w_{ij}\in\mathcal{C}(\mathbf{y})$ $(i,\,j=1,\,\dots,\,n)$ by

$\displaystyle w_{ij}=\\{y_{i},y_{j}\\}:=\omega_{ij}y_{i}y_{j}.$ (5.37)

Then, they satisfy the conditions (5.33) and (5.35). Thus, they define a
Poisson bracket on $\mathcal{C}(\mathbf{y})$.

###### Proof.

The condition (5.33) is clear. The condition (5.35) is verified by the cyclic
expression

$\displaystyle\\{\\{y_{i},y_{j}\\},y_{k}\\}=(\omega_{ij}\omega_{ik}-\omega_{jk}\omega_{ji})y_{i}y_{j}y_{k}.$
(5.38)

∎

The variables $\mathbf{y}$ in the Poisson bracket (5.37) are called _log-
canonical variables_ by [GSV03] because of the following property:

$\displaystyle\\{\log y_{i},\log y_{j}\\}=\omega_{ij}.$ (5.39)

(The name might be a little confusing because $\log y_{i}$ are not canonical
in the terminology for classical mechanics.)

Now we come back to the situation in the previous section. Let
$\mathcal{C}_{+}(\mathbf{y}(s))$ be the algebra of all smooth functions of
$\mathbf{y}(s)$ defined on $\mathbb{R}_{>0}^{n}$ with coordinates
$\mathbf{y}(s)$. By regarding the semifield isomorphism (5.10)–(5.12) as
_changes of variables_ , we have the algebra isomorphisms (denoted by the same
symbols)

$\displaystyle\mu(s)\colon$ $\displaystyle\
\mathcal{C}_{+}(\mathbf{y}(s+1))\rightarrow\mathcal{C}_{+}(\mathbf{y}(s)),$
(5.40) $\displaystyle\tau(s)\colon$ $\displaystyle\
\mathcal{C}_{+}(\mathbf{y}(s+1))\rightarrow\mathcal{C}_{+}(\mathbf{y}(s)),$
(5.41) $\displaystyle\rho(s)\colon$ $\displaystyle\
\mathcal{C}_{+}(\mathbf{y}(s))\rightarrow\mathcal{C}_{+}(\mathbf{y}(s)).$
(5.42)

Now, we consider a smooth manifold $M\simeq\mathbb{R}_{>0}^{n}$ equipped with
$P+1$ global positive coordinates $\mathbf{y}(s)$ ($s=0,\,\dots,\,P$) such
that $\mathbf{y}(s+1)$ and $\mathbf{y}(s)$ are glued by the tropical
transformation $\tau(s)$ (_not by the ordinary mutation $\mu(s)$_). The
algebra $\mathcal{C}(M)$ is identified with $\mathcal{C}_{+}(\mathbf{y}(s))$
in the coordinates $\mathbf{y}(s)$ for each $s$.

Let $\Omega(s)$ be the rational skew-symmetric matrix in (5.20). By (5.5), we
have the mutation formula

$\displaystyle\omega_{ij}(s+1)$
$\displaystyle=\begin{cases}-\omega_{ij}(s)&\text{$i=k$ or $j=k$,}\\\
\omega_{ij}(s)+[\varepsilon b_{kj}(s)]_{+}\omega_{ik}(s)+[\varepsilon
b_{ki}(s)]_{+}\omega_{kj}(s)&i,\,j\neq k.\end{cases}$ (5.43)

Following Gekhtman-Shapiro-Vainshtein [GSV03], we define a Poisson bracket on
each $\mathcal{C}_{+}(\mathbf{y}(s))$ ($s=0$, …, $P-1$) by

$\displaystyle\\{y_{i}(s),y_{j}(s)\\}_{s}:=\omega_{ij}(s)y_{i}(s)y_{j}(s),$
(5.44)

so that $y_{i}(s)$ are log-canonical variables. They are _mutation-compatible_
in the following sense.

######  Theorem 5.7 ([GSV03, Thm. 1.4]).

The maps $\mu(s)$, $\tau(s)$, and $\rho(s)$ in (5.40)–(5.42) commute with the
Poisson brackets (5.44). Namely, we have

$\displaystyle\\{\mu(s)(f),\mu(s)(g)\\}_{s}$
$\displaystyle=\mu(s)(\\{f,g\\}_{s+1})\quad(f,g\in\mathcal{C}_{+}(\mathbf{y}(s+1))),$
(5.45) $\displaystyle\\{\tau(s)(f),\tau(s)(g)\\}_{s}$
$\displaystyle=\tau(s)(\\{f,g\\}_{s+1})\quad(f,g\in\mathcal{C}_{+}(\mathbf{y}(s+1))),$
(5.46) $\displaystyle\\{\rho(s)(f),\rho(s)(g)\\}_{s}$
$\displaystyle=\rho(s)(\\{f,g\\}_{s})\quad(f,g\in\mathcal{C}_{+}(\mathbf{y}(s))).$
(5.47)

###### Proof.

Let us first prove (5.47). By Proposition 5.5 (a), it is enough to prove it
for $f(\mathbf{y})=y_{i}(s)$ and $g(\mathbf{y})=y_{j}(s)$. We have

$\displaystyle\quad\ \\{\rho(s)(y_{i}(s)),\rho(s)(y_{j}(s))\\}_{s}$
$\displaystyle=\\{y_{i}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)},y_{j}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}j}(s)}\\}_{s}$
$\displaystyle=(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}j}(s)}\\{y_{i}(s),y_{j}(s)\\}_{s}$
$\displaystyle\qquad+y_{i}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}j}(s)}\\{(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)},y_{j}(s)\\}_{s}$
$\displaystyle\qquad+(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}y_{j}(s)\\{y_{i}(s),(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}j}(s)}\\}_{s}$
$\displaystyle{=}\omega_{ij}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}j}(s)}y_{i}(s)y_{j}(s)$
$\displaystyle=\omega_{ij}(s)\rho(s)(y_{i}(s))\rho(s)(y_{j}(s))=\rho(s)(\\{y_{i}(s),y_{j}(s)\\}_{s}).$

The equality (5.46) is proved similarly. For example, for $i,\,j\neq k$, we
have

$\displaystyle\quad\ \\{\tau(s)(y_{i}(s+1)),\tau(s)(y_{j}(s+1))\\}_{s}$
$\displaystyle=\\{y_{i}(s)y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}},y_{j}(s)y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}\\}_{s}$
$\displaystyle=y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}j}(s)]_{+}}\\{y_{i}(s),y_{j}(s)\\}_{s}$
$\displaystyle\qquad+y_{i}(s)y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}j}(s)]_{+}}\\{y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}},y_{j}(s)\\}_{s}$
$\displaystyle\qquad+y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}y_{j}(s)\\{y_{i}(s),y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}j}(s)]_{+}}\\}_{s}$
$\displaystyle=(\omega_{ij}(s)+[\varepsilon_{s}b_{k_{s}i}(s)]_{+}\omega_{k_{s}j}(s)+[\varepsilon_{s}b_{k_{s}j}(s)]_{+}\omega_{ik_{s}}(s))$
$\displaystyle\qquad\times
y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}y_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}j}(s)]_{+}}y_{i}(s)y_{j}(s)$
$\displaystyle{=}\omega_{ij}(s+1)\tau(s)(y_{i}(s+1))\tau(s)(y_{j}(s+1))$
$\displaystyle=\tau(s)(\\{y_{i}(s+1),y_{j}(s+1)\\}_{s+1}).$

Other cases are similar and easier. Lastly, the equality (5.45) follows from
(5.46) and (5.47). ∎

####  5.3 Euler dilogarithm as Hamiltonian

Let $\rho(s)$ be the automorphism of $\mathcal{C}_{+}(\mathbf{y}(s))$ in
(5.42). Fock-Goncharov [FG09c, §1.3] made the following remarkable observation
(translated in the present context): _With the Poisson bracket ( 5.44), the
automorphism $\rho(s)$ is described by the time-one flow for the Hamiltonian
given by the Euler dilogarithm._ Below, we explain the meaning of this
statement.

In general, for an algebra of functions $\mathcal{C}(M)$ with a Poisson
bracket $\\{\cdot,\cdot\\}$, let $f_{t}\in\mathcal{C}$ ($t\in\mathbb{R}$) be a
one-parameter family of functions, where $t$ is the _continuous time_. We
specify a function $\mathscr{H}\in\mathcal{C}(M)$ called the _Hamiltonian_
that governs the time development of $f_{t}$ by the following _equation of
motion_ :

$\displaystyle\dot{f}_{t}=\\{\mathscr{H},f_{t}\\},\quad\dot{f}_{t}:=\frac{d}{dt}f_{t}.$
(5.48)

We apply this idea to our Poisson bracket (5.44) to describe the automorphism
$\rho(s)$. We choose the Hamiltonian
$\mathscr{H}(s)\in\mathcal{C}(\mathbf{y}(s))$ that is given by the Euler
dilogarithm as follows:

$\displaystyle\mathscr{H}(s):={\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-y_{k_{s}}(s)^{\varepsilon_{s}})=-{\varepsilon_{s}}{\delta_{k_{s}}}\int_{0}^{y_{k_{s}}(s)^{\varepsilon_{s}}}\frac{\log(1+y)}{y}\,dy.$
(5.49)

We have

$\displaystyle\begin{split}\\{\mathscr{H}(s),y_{i}(s)\\}_{s}&={\varepsilon_{s}}{\delta_{k_{s}}}\\{\mathrm{Li}_{2}(-y_{k_{s}}(s)^{\varepsilon_{s}}),y_{i}(s)\\}_{s}\\\
&=-{\delta_{k_{s}}}\frac{\log(1+y_{k_{s}}(s)^{\varepsilon_{s}})}{y_{k_{s}}(s)^{\varepsilon_{s}}}\omega_{k_{s}i}(s)y_{k_{s}}(s)^{\varepsilon_{s}}y_{i}(s)\\\
&=y_{i}(s)\log(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}.\end{split}$
(5.50)

In particular, we have

$\displaystyle\\{\mathscr{H}(s),y_{k_{s}}(s)\\}_{s}=0.$ (5.51)

Thus, we have a solution of the equation of motion (5.48)

$\displaystyle
f_{t}=y_{i}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)t}.$ (5.52)

By setting $t=0$ and $t=1$, we obtain

$\displaystyle f_{0}=y_{i}(s),\quad
f_{1}=y_{i}(s)(1+y_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}=\rho(s)(y_{i}(s)).$
(5.53)

This explains the meaning of “the automorphism $\rho(s)$ is described by the
time-one flow of $\mathscr{H}(s)$”.

By the formula (5.18) and the compatibility (5.46), the automorphism
$\mathfrak{q}(s)$ in (5.23) is described by the time-one flow of the
Hamiltonian

$\displaystyle\mathscr{H}_{0}(s):=\tau(s-1;0)(\mathscr{H}(s))={\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-y^{\mathbf{c}^{+}_{k_{s}}(s)})$
(5.54)

in the initial $y$-coordinates $\mathbf{y}=\mathbf{y}(0)$ with the Poisson
bracket $\\{\cdot,\cdot\\}=\\{\cdot,\cdot\\}_{0}$.

More generally, for any integer vector $\mathbf{c}=(c_{i})\in\mathbb{Z}^{n}$
and any real number $a$, we may consider a Hamiltonian

$\displaystyle\mathscr{H}_{\mathbf{c},a}=a\mathrm{Li}_{2}(-y^{\mathbf{c}})$
(5.55)

for the Poisson bracket $\\{\cdot,\cdot\\}=\\{\cdot,\cdot\\}_{0}$.

######  Proposition 5.8.

The time-one flow of the Hamiltonian $\mathscr{H}_{\mathbf{c},a}$ in (5.55) is
given by

$\displaystyle y^{\mathbf{n}}\mapsto
y^{\mathbf{n}}(1+y^{\mathbf{c}})^{-a\\{\mathbf{c},\mathbf{n}\\}_{\Omega}},$
(5.56)

where $\\{\mathbf{c},\mathbf{n}\\}_{\Omega}$ is the one in (5.21).

###### Proof.

In the same way as (5.50), we have

$\displaystyle\\{\mathscr{H}_{\mathbf{c},a},y_{i}\\}=-a\biggl{(}\sum_{j=1}^{n}c_{j}\omega_{ji}\biggr{)}y_{i}\log(1+y^{\mathbf{c}}).$
(5.57)

Then, by the Leibniz rule, we have

$\displaystyle\\{\mathscr{H}_{\mathbf{c},a},y^{\mathbf{n}}\\}=-a\\{\mathbf{c},\mathbf{n}\\}_{\Omega}y^{\mathbf{n}}\log(1+y^{\mathbf{c}}).$
(5.58)

Thus, we obtain the claim. ∎

In summary, _the Euler dilogarithm is built into cluster algebra theory_ as
the Hamiltonian for the nontropical part of each mutation in a $Y$-pattern.
This gives an _intrinsic connection_ between the dilogarithm and cluster
algebra theory.

Notes

The $\varepsilon$-expressions and the connection to the tropicalization are
gradually recognized by [BFZ05, Nak11b, NZ12, Nak12]. The Fock-Goncharov
decomposition was introduced by [FG09a, FG09c] for $\varepsilon=1$ in studying
the cluster varieties and their quantizations. The signed version first
appeared in [Nag13, Kel11] in the study of the Donaldson-Thomas invariants.
The presentation here follows [Nak23]. A parallel decomposition is also
available for $x$-variables [FG09a, Nak23].

The mutation-compatible Poisson bracket was introduced by [GSV03]. A more
extensive treatment is given in [GSV10]. For Poisson geometry, see the reviews
[Wei83, Wei98] and references therein.

The appearance of the dilogarithm as the Hamiltonian naturally guided Fock-
Goncharov [FG09a, FG09c] to quantize cluster algebras using the _quantum
dilogarithm_ , which we treat in Part IV.

###  Chapter 6 Classical mechanical method for dilogarithm identities

We reformulate the Hamiltonian system for mutations with _canonical
coordinates_. The Legendre transformation in classical mechanics turns the
_Hamiltonian_ given by the Euler dilogarithm into the _Lagrangian_ given by
the modified Rogers dilogarithm. Based on this perspective, we give an
alternative and independent proof of Theorem 3.5. This proof provides a more
intrinsic understanding of the DIs for $Y$-patterns, where the constancy is
regarded as a variant of Noether’s theorem.

####  6.1 Hamiltonian formulation of mutations

In the previous chapter, we observed that the Euler dilogarithm is built into
a $Y$-pattern as the Hamiltonian (5.49) for the nontropical part of each
mutation. We then ask the following natural questions:

  * Q1.

Is there any relation between this fact and the DIs in Theorem 3.5?

  * Q2.

If so, how and why does the Euler dilogarithm turn into the modified Rogers
dilogarithm in these DIs?

It was shown by Gekhtman-Nakanishi-Rupel [GNR17] that the above questions are
naturally answered if we reformulate the Hamiltonian system with _canonical
coordinates_ and transform the Hamiltonian into the _Lagrangian_ by the
_Legendre transformation_ in classical mechanics.

To this end, let us recall and make it precise how the sequence of mutations
(5.8) is formulated as a Hamiltonian system with _continuous time_ based on
the setting and results in Chapter 5.

  1. 1.

We consider a smooth manifold $M\simeq\mathbb{R}_{>0}^{n}$ equipped with $P+1$
global positive coordinates by $\mathbf{y}(s)$ ($s=0,\,\dots,\,P$) that are
glued by the tropical transformations $\tau(s)$ in (5.41) (not by the ordinary
mutation $\mu(s)$). We call $M$ the _phase space_.

  2. 2.

As usual, the change of coordinates (variables)
$\tau(s):\mathcal{C}_{+}(s+1)\rightarrow\mathcal{C}_{+}(s)$ induces the
bijection $\tau^{*}(s):\mathbb{R}_{>0}^{n}\rightarrow\mathbb{R}_{>0}^{n}$ in
the _opposite direction_. Namely, $\tau^{*}(s)$ maps a point
$(a_{1},\dots,a_{n})$ in the $\mathbf{y}(s)$-coordinates to the point
$(a^{\prime}_{1},\dots,a^{\prime}_{n})$ in the $\mathbf{y}(s+1)$-coordinates,
where

$\displaystyle a^{\prime}_{i}=\begin{cases}\displaystyle
a_{k_{s}}(s)^{-1}&i=k_{s},\\\
a_{i}a_{k_{s}}^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}&i\neq k_{s}.\end{cases}$
(6.1)

The same applies to other maps $\mu(s)$, $\rho(s)$, and $\mathfrak{q}(s)$ and
induces the maps $\mu^{*}(s)$, $\rho^{*}(s)$, and $\mathfrak{q}^{*}(s)$ all
from $\mathbb{R}_{>0}^{n}$ to itself.

  3. 3.

The phase space $M$ is equipped with the mutation-compatible Poisson bracket
in (5.44).

  4. 4.

We consider the time span $[0,P]$ for the continuous time $t$. In each time
span $[s,s+1]$ ($s=0$, …, $P$), the Hamiltonian $\mathscr{H}(s)$ is given in
the $\mathbf{y}(s)$-coordinates by

$\displaystyle\mathscr{H}(s)={\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-y_{k_{s}}(s)^{\varepsilon_{s}})$
(6.2)

as in (5.49). In the initial $\mathbf{y}$-coordinates, the Hamiltonian is
given by

$\displaystyle\mathscr{H}_{0}(s)={\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-y^{\mathbf{c}^{+}_{k_{s}}(s)})$
(6.3)

  5. 5.

By (5.52), in each time span $[s,s+1]$, the trajectory (the Hamiltonian flow)
of a point in the phase space $M$ along time $t$ is described in the
$\mathbf{y}(s)$-coordinates as follows: Let
$\mathbf{a}_{s}=(a_{1},\cdots,a_{n})\in\mathbb{R}_{>0}^{n}$ be the position of
the point at $t=s$. Then, the position
$\mathbf{a}_{t}=(a^{\prime}_{1},\cdots,a^{\prime}_{n})$ at $t\in[s,s+1]$ is
given by

$\displaystyle a^{\prime}_{i}=a_{i}(1+a_{k_{s}})^{-b_{k_{s}i}(s)(t-s)}.$ (6.4)

In particular, $a^{\prime}_{k_{s}}=a_{k_{s}}$ for any $t\in[s,s+1]$.

  6. 6.

For a trajectory in $M$ in the full time span $[0,P]$, if the initial position
in the initial coordinates $\mathbf{y}=\mathbf{y}(0)$ is $\mathbf{a}_{0}$, the
final position in the $\mathbf{y}(P)$-coordinates is given by
$\mu^{*}(P-1;0)(\mathbf{a}_{0})$, while the final position in the initial
coordinates $\mathbf{y}$ is given by
$\mathfrak{q}^{*}(P-1;0)(\mathbf{a}_{0})$.

See Figure 6.1 for the summary of the above formulation.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$0123$P$$M$$\mathbf{y}$$\mathbf{y}(1)$$\mathbf{y}(2)$$\mathbf{y}(3)$$\mathbf{y}(P)$$\tau^{*}(1)$$\tau^{*}(2)$$\tau^{*}(3)$$\tau^{*}(P-1)$$\mathbf{a}_{0}$$\mathfrak{q}^{*}(P-1;0)(\mathbf{a}_{0})$$\mu^{*}(P-1;0)(\mathbf{a}_{0})$
Figure 6.1: Schematic diagram of Hamilton formulation of mutations.

####  6.2 Canonical coordinates

We will “embed” the above Hamiltonian system into the one with canonical
coordinates. Let us first concentrate on the initial coordinates $\mathbf{y}$,
for simplicity. Let $\mathbf{u}=(u_{1},\dots,u_{n})$ and
$\mathbf{p}=(p_{1},\dots,p_{n})$ be variables. We regard the pair
$(\mathbf{u},\mathbf{p})$ as global coordinates of a manifold
$\tilde{M}\simeq\mathbb{R}^{2n}$. Let $\mathcal{C}(\mathbf{u},\mathbf{p})$ be
the set of all smooth functions of $(\mathbf{u},\mathbf{p})$ defined on
$\mathbb{R}^{2n}$. We consider the following standard Poisson bracket on
$\tilde{M}$:

$\displaystyle\\{f,g\\}:=\sum_{i=1}^{n}\left(\frac{\partial f}{\partial
p_{i}}\frac{\partial g}{\partial u_{i}}-\frac{\partial g}{\partial
p_{i}}\frac{\partial f}{\partial
u_{i}}\right)\quad(f,g\in\mathcal{C}(\mathbf{u},\mathbf{p})).$ (6.5)

Thus, $(\mathbf{u},\mathbf{p})$ are _canonical coordinates (variables)_ in
classical mechanics (also called _Darboux coordinates_)). The following
properties hold.

$\displaystyle\\{p_{i},u_{j}\\}=\delta_{ij},\quad\\{u_{i},u_{j}\\}=\\{p_{i},p_{j}\\}=0,$
(6.6) $\displaystyle\\{f,u_{i}\\}=\frac{\partial f}{\partial
p_{i}},\quad\\{f,p_{i}\\}=-\frac{\partial f}{\partial
u_{i}}\quad(f\in\mathcal{C}(\mathbf{u},\mathbf{p})).$ (6.7)

(Here and below, we use both the Kronecker delta $\delta_{ij}$ and the symbol
$\delta_{i}$ for a component of $\Delta$. We ask the reader to distinguish
them carefully.)

We set

$\displaystyle\tilde{y}_{i}:=\exp\left(\frac{1}{\sqrt{2}}(\delta^{-1}_{i}p_{i}+w_{i})\right),\quad
w_{i}:=\sum_{j=1}^{n}b_{ji}u_{j}.$ (6.8)

######  Proposition 6.1 ([GNR17, Prop. 3.3]).

The following formula holds.

$\displaystyle\\{\tilde{y}_{i},\tilde{y}_{j}\\}=\omega_{ij}\tilde{y}_{i}\tilde{y}_{j}.$
(6.9)

###### Proof.

As noted in (5.39), the claim is equivalent to

$\displaystyle\\{\log\tilde{y}_{i},\log\tilde{y}_{j}\\}=\omega_{ij}.$ (6.10)

Indeed, we have

$\displaystyle\begin{split}\\{\log\tilde{y}_{i},\log\tilde{y}_{j}\\}&=\frac{1}{2}\\{\delta^{-1}_{i}p_{i},w_{j}\\}+\frac{1}{2}\\{\delta^{-1}_{j}p_{j},w_{i}\\}\\\
&=\frac{1}{2}\omega_{ij}-\frac{1}{2}\omega_{ji}=\omega_{ij}.\end{split}$
(6.11)

∎

The bracket (6.9) coincides with the one (5.44) with $s=0$. Moreover, the
functions (variables) $\tilde{y}_{1}$, …, $\tilde{y}_{n}$ are algebraically
independent. Thus, we have an embedding of the Poisson algebra

$\displaystyle\begin{matrix}\eta\colon&\mathcal{C}_{+}(\mathbf{y})&\longrightarrow&\mathcal{C}(\mathbf{u},\mathbf{p})\\\
&y_{i}&\mapsto&\tilde{y}_{i}.\end{matrix}$ (6.12)

Let us concentrate on the time span $[0,1]$. We consdier the Hamiltonian

$\displaystyle\mathscr{H}={\delta_{k_{0}}}\mathrm{Li}_{2}(-\tilde{y}_{k_{0}})$
(6.13)

corresponding to $\mathscr{H}(0)$ in (6.2). We have

$\displaystyle\\{\mathscr{H},\tilde{y}_{k_{0}}\\}=0.$ (6.14)

Therefore, the function $\tilde{y}_{k_{0}}$ is constant along the time
development. The equations of motion for the canonical coordinates are given
by

$\displaystyle\dot{u}_{i}$
$\displaystyle=\\{\mathscr{H},u_{i}\\}=\frac{\partial\mathscr{H}}{\partial
p_{i}}=-\frac{1}{\sqrt{2}}\delta_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),$ (6.15)
$\displaystyle\dot{p}_{i}$
$\displaystyle=\\{\mathscr{H},p_{i}\\}=-\frac{\partial\mathscr{H}}{\partial
u_{i}}=-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),$
(6.16)

where we used $b_{ik_{0}}\delta_{k_{0}}=-b_{k_{0}i}\delta_{i}$ in the second
formula. Thus, for $t\in[0,1]$, the functions $u_{i}$, $p_{i}$, $w_{i}$,
$\delta^{-1}_{i}p_{i}$ develop linearly as

$\displaystyle u_{i}(t)$
$\displaystyle=u_{i}-\left(\frac{1}{\sqrt{2}}\delta_{k_{0}i}\log(1+\tilde{y}_{k_{0}})\right)t,$
(6.17) $\displaystyle p_{i}(t)$
$\displaystyle=p_{i}-\left(\frac{1}{\sqrt{2}}\delta_{i}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}})\right)t,$
(6.18) $\displaystyle w_{i}(t)$
$\displaystyle=w_{i}-\left(\frac{1}{\sqrt{2}}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}})\right)t,$
(6.19) $\displaystyle\delta^{-1}_{i}p_{i}(t)$
$\displaystyle=\delta^{-1}_{i}p_{i}-\left(\frac{1}{\sqrt{2}}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}})\right)t,$
(6.20) $\displaystyle\tilde{y}_{i}$
$\displaystyle=\tilde{y}_{i}(1+\tilde{y}_{k_{0}})^{-b_{k_{0}i}t}.$ (6.21)

The last one agrees with the result (5.52) as expected. Therefore, the time-
one flow of the Hamiltonian $\mathscr{H}$ induces the following automorphism

$\displaystyle\begin{matrix}\rho(0)\colon&\mathcal{C}(\mathbf{u},\mathbf{p})&\longrightarrow&\mathcal{C}(\mathbf{u},\mathbf{p})\\\
&u_{i}&\mapsto&\displaystyle
u_{i}-\frac{1}{\sqrt{2}}\delta_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),\\\
&p_{i}&\mapsto&\displaystyle\rule{0.0pt}{18.0pt}p_{i}-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),\\\
&w_{i}&\mapsto&\displaystyle\rule{0.0pt}{18.0pt}w_{i}-\frac{1}{\sqrt{2}}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),\\\
&\delta^{-1}_{i}p_{i}&\mapsto&\displaystyle\rule{0.0pt}{18.0pt}\delta^{-1}_{i}p_{i}-\frac{1}{\sqrt{2}}b_{k_{0}i}\log(1+\tilde{y}_{k_{0}}),\\\
&\tilde{y}_{i}&\mapsto&\displaystyle\tilde{y}_{i}(1+\tilde{y}_{k_{0}})^{-b_{k_{0}i}}.\end{matrix}$
(6.22)

Again, the last one is compatible with the maps $\rho(0)=\mathfrak{q}(0)$ in
(5.42).

######  Proposition 6.2.

The map $\rho(0)$ preserves the Poisson bracket in (6.6).

###### Proof.

This is a special case of a well-known fact that the Hamiltonian flow
preserves the Poisson bracket (e.g., [AM78, Pop. 3.3.4]). Alternatively, one
can directly check it by (6.22). ∎

A drawback of this formulation is that we have $2n$ variables
$(\mathbf{u},\mathbf{p})$, which are twice of the original $y$-variables
$\mathbf{y}$. To eliminate this redundancy, let us introduce the subset
$\tilde{M}_{0}$ of $\tilde{M}$ consisting of the points such that in the
$(\mathbf{u},\mathbf{p})$-coordinates the following condition holds:

$\displaystyle p_{i}=\delta_{i}w_{i}\quad(i=1,\,\dots,\,n).$ (6.23)

######  Lemma 6.3.

The subset $\tilde{M}_{0}$ is preserved under the time-development by the
Hamiltonian $\mathscr{H}$ in (6.13).

###### Proof.

This is clear from (6.19) and (6.20). ∎

We call $\tilde{M}_{0}$ the _small phase space_. The algebra
$\mathcal{C}(\tilde{M}_{0})$ of the smooth functions on $\tilde{M}_{0}$ is
identified with the quotient

$\displaystyle\mathcal{C}(\mathbf{u},\mathbf{p})_{0}:=\mathcal{C}(\mathbf{u},\mathbf{p})/I_{0},$
(6.24)

where $I_{0}$ is the ideal of $\mathcal{C}(\mathbf{u},\mathbf{p})$ consisting
of the functions vanishing on $\tilde{M}_{0}$.

This causes, however, a new problem. Due to the definition (6.8) of
$\tilde{y}$, the map

$\displaystyle\begin{matrix}\eta_{0}\colon&\mathcal{C}_{+}(\mathbf{y})&\longrightarrow&\mathcal{C}(\mathbf{u},\mathbf{p})_{0}\\\
&y_{i}&\mapsto&\tilde{y}_{i}.\end{matrix}$ (6.25)

is injective (and also bijective) if and only if the matrix $B$ is
nonsingular. Equivalently, $\tilde{y}_{i}$’s are algebraically independent in
$\mathcal{C}(\mathbf{u},\mathbf{p})_{0}$ if and only if the matrix $B$ is
nonsingular. For example, in the extreme case $B=0$, $\tilde{y}_{i}=1$ for any
$i$. We will come back to this issue later. At this moment we do not require
that $B$ is nonsingular.

####  6.3 Tropical transformations of canonical coordinates

The canonical coordinates for the time span $[0,1]$ in the previous section
can be generalized for each time span $[s,s+1]$ ($s=0,\,\dots,\,P-1$) just by
repeating the same construction. So, we quickly summarize the notation and
results.

For each $s=0,\,\dots,\,P-1$, we introduce variables
$\mathbf{u}(s)=(u_{i}(s))_{i=1}^{n}$ and $\mathbf{p}(s)=(p_{i}(s))_{i=1}^{n}$,
and regard the pair $(\mathbf{u}(s),\mathbf{p}(s))$ as global coordinates of
$\tilde{M}\simeq\mathbb{R}^{2n}$. The Poisson bracket $\\{\cdot,\cdot\\}_{s}$
on $\tilde{M}$ is defined as in (6.5) by replacing $u_{i}$ and $p_{i}$ therein
with $u_{i}(s)$ and $p_{i}(s)$. We set

$\displaystyle\tilde{y}_{i}(s):=\exp\left(\frac{1}{\sqrt{2}}(\delta^{-1}_{i}p_{i}(s)+w_{i}(s))\right),\quad
w_{i}(s):=\sum_{j=1}^{n}b_{ji}(s)u_{j}(s).$ (6.26)

Then, we have

$\displaystyle\\{\tilde{y}_{i}(s),\tilde{y}_{j}(s)\\}_{s}=\omega_{ij}(s)\tilde{y}_{i}(s)\tilde{y}_{j}(s),$
(6.27)

which agrees with (5.44). So, we have an embedding

$\displaystyle\begin{matrix}\eta(s)\colon&\mathcal{C}_{+}(\mathbf{y}(s))&\longrightarrow&\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))\\\
&y_{i}(s)&\mapsto&\tilde{y}_{i}(s).\end{matrix}$ (6.28)

The Hamiltonian $\mathscr{H}(s)$ is given by

$\displaystyle\mathscr{H}(s)={\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.29)

The equations of motion for the canonical coordinates are given as

$\displaystyle\dot{u}_{i}(s)$
$\displaystyle=-\frac{1}{\sqrt{2}}\delta_{k_{s}i}\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}),$
(6.30) $\displaystyle\dot{p}_{i}(s)$
$\displaystyle=-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{s}i}(s)\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.31)

The time-one flow of the Hamiltonian $\mathscr{H}(s)$ induces the automorphism

$\displaystyle\rho(s)\colon\mathcal{C}(\mathbf{u}(s),$
$\displaystyle\,\mathbf{p}(s))\longrightarrow\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))$
$\displaystyle u_{i}(s)$ $\displaystyle\mapsto\displaystyle
u_{i}(s)-\frac{1}{\sqrt{2}}\delta_{k_{s}i}\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}),$
(6.32) $\displaystyle p_{i}(s)$ $\displaystyle\mapsto\displaystyle
p_{i}(s)-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{s}i}(s)\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}),$
(6.33) $\displaystyle w_{i}(s)$ $\displaystyle\mapsto\displaystyle
w_{i}(s)-\left(\frac{1}{\sqrt{2}}b_{k_{s}i}(s)\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})\right),$
(6.34) $\displaystyle\delta^{-1}_{i}p_{i}(s)$
$\displaystyle\mapsto\displaystyle\delta^{-1}_{i}p_{i}(s)-\left(\frac{1}{\sqrt{2}}b_{k_{s}i}(s)\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})\right),$
(6.35) $\displaystyle\tilde{y}_{i}(s)$
$\displaystyle\mapsto\tilde{y}_{i}(s)(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})^{-b_{k_{s}i}(s)}.$
(6.36)

Moreover, it preserves the Poisson bracket $\\{\cdot,\cdot\\}_{s}$.

Now we glue the canonical coordinates $(\mathbf{u}(s+1),\mathbf{p}(s+1))$ and
$(\mathbf{u}(s),\mathbf{p}(s))$ by the following linear transformation, which
is the counterpart of the tropical transformation $\tau(s)$ in (5.41).

$\displaystyle\tau{(s)}\colon\mathcal{C}(\mathbf{u}(s+1),$
$\displaystyle\,\mathbf{p}(s+1)))\longrightarrow\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s)),$
$\displaystyle u_{i}(s+1)$ $\displaystyle\mapsto\begin{cases}\displaystyle-
u_{k_{s}}(s)+\sum_{j=1}^{n}[-\varepsilon_{s}b_{jk_{s}}(s)]_{+}u_{j}(s)&i=k_{s},\\\
u_{i}(s)&i\neq k_{s},\end{cases}$ (6.37) $\displaystyle p_{i}(s+1)$
$\displaystyle\mapsto\begin{cases}\displaystyle-p_{k_{s}}(s)&i=k_{s},\\\
p_{i}(s)+[-\varepsilon_{s}b_{ik_{s}}(s)]_{+}p_{k_{s}}(s)&i\neq
k_{s},\end{cases}$ (6.38) $\displaystyle w_{i}(s+1)$
$\displaystyle\mapsto\begin{cases}\displaystyle-w_{i}(s)&i=k_{s},\\\
w_{i}(s)+[\varepsilon_{s}b_{k_{s}i}(s)]_{+}w_{k_{s}}(s)&i\neq
k_{s},\end{cases}$ (6.39) $\displaystyle\delta^{-1}_{i}p_{i}(s+1)$
$\displaystyle\mapsto\begin{cases}\displaystyle-\delta^{-1}_{k_{s}}p_{k_{s}}(s)&i=k_{s},\\\
\delta^{-1}_{i}p_{i}(s)+[\varepsilon_{s}b_{k_{s}i}(s)]_{+}\delta^{-1}_{k_{s}}p_{k_{s}}(s)&i\neq
k_{s},\end{cases}$ (6.40) $\displaystyle\tilde{y}_{i}(s+1)$
$\displaystyle\mapsto\begin{cases}\displaystyle\tilde{y}_{k_{s}}(s)^{-1}&i=k_{s},\\\
\tilde{y}_{i}(s)\tilde{y}_{k_{s}}(s)^{[\varepsilon_{s}b_{k_{s}i}(s)]_{+}}&i\neq
k_{s}.\end{cases}$ (6.41)

By (6.41), we have a commutative diagram:

$\textstyle{\mathcal{C}(\mathbf{y}(s+1))}$$\textstyle{\mathcal{C}(\mathbf{u}(s+1),\mathbf{p}(s+1))}$$\textstyle{\mathcal{C}(\mathbf{y}(s))}$$\textstyle{\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))}$$\eta(s+1)$$\eta(s)$$\tau(s)$$\tau(s)$
(6.42)

######  Proposition 6.4 ([GNR17, Prop. 3.9]).

(a). The map $\tau(s)$ commutes with the Poisson brackets. Namely, we have

$\displaystyle\\{\tau(s)(f),\tau(s)(g)\\}_{s}$
$\displaystyle=\tau(s)(\\{f,g\\}_{s+1})\quad(f,\,g\in\mathcal{C}(\mathbf{u}(s+1),\mathbf{p}(s+1))).$
(6.43)

(b). The following property holds:

$\displaystyle\tau(s)\left(\sum_{i=1}^{n}u_{i}(s+1)p_{i}(s+1)\right)=\sum_{i=1}^{n}u_{i}(s)p_{i}(s).$
(6.44)

###### Proof.

(a). It is enough to check it for $f=p_{i}(s+1)$ and $g=u_{j}(s+1)$. So, let
us show $\\{\tau(s)(p_{i}(s+1)),\tau(s)(u_{j}(s+1))\\}_{s}=\delta_{ij}$. By
(6.37) and (6.38), the map $\tau(s)$ acts on $\mathbf{u}(s+1)$ and
$\mathbf{p}(s+1)$ separately and linearly. Let $A$ and $B$ be their matrix
representations with respect to the bases $\mathbf{u}(s+1)$, $\mathbf{u}(s)$
and $\mathbf{p}(s+1)$, $\mathbf{p}(s)$, respectively. Then, it is easy to
check that $B^{T}=A=A^{-1}$ holds. Thus, the equality (6.43) holds. (b). The
same proof is applicable. ∎

Define the mutation $\mu(s)$ by the composition

$\displaystyle\mu(s):=\rho(s)\circ\tau(s)\colon\mathcal{C}(\mathbf{u}(s+1),\mathbf{p}(s+1))\rightarrow\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s)).$
(6.45)

We also define $\mu(s;0)$, $\tau(s;0)$, $\mathfrak{q}(s)$, and
$\mathfrak{q}(s;0)$ in a parallel way with the ones in Section 5.1.

Let us turn to the small phase space $\tilde{M}_{0}$ in the previous section.
Let $\tilde{M}_{0}(s)$ be set of the points in the phase space $\tilde{M}$
such that in the $(\mathbf{u}(s),\mathbf{p}(s))$-coordinates the following
condition holds:

$\displaystyle p_{i}(s)=\delta_{i}w_{i}(s)\quad(i=1,\,\dots,\,n).$ (6.46)

######  Lemma 6.5.

The subset $\tilde{M}_{0}(s)$ is preserved under the time-development by the
Hamiltonian $\mathscr{H}(s)$. Moreover, the set $\tilde{M}_{0}(s)$ is
independent of $s$, namely, $\tilde{M}_{0}(s)=\tilde{M}_{0}$.

###### Proof.

The first claim is proved in the same way as Lemma 6.3. Also, by (6.39) and
(6.40), the condition (6.46) is preserved by $\tau(s)$. ∎

By Lemma 6.5, we can restrict our Hamiltonian system on the small phase space
$\tilde{M}_{0}$ through the full time-span $[0,P]$ whenever necessary.

Again, we have a map

$\displaystyle\begin{matrix}\eta_{0}(s)\colon&\mathcal{C}(\mathbf{y}(s))&\longrightarrow&\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))_{0}:=\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))/I_{0}(s)\\\
&y_{i}(s)&\mapsto&\tilde{y}_{i}(s),\end{matrix}$ (6.47)

where $I_{0}(s)$ is the ideal of $\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))$
consisting of the functions vanishing on $\tilde{M}_{0}$. It is injective
(also bijective) if and only if $B(s)$ is nonsingular.

####  6.4 Modified Rogers dilogarithm as Lagrangian

Let us recall basic facts on the Legendre transformation and the Lagrangian.
See, for example, [AM78] for more information.

For simplicity, consider a Hamiltonian $\mathscr{H}$ on the phase space
$M\simeq\mathbb{R}^{2n}$ with canonical coordinates $(\mathbf{u},\mathbf{p})$
as global coordinates. The space $M$ is identified with the contangent bundle
$\pi:T^{*}\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ so that
$\pi(\mathbf{u},\mathbf{p})\mapsto\mathbf{u}$. Then, the Hamiltonian
$\mathscr{H}$ induces the following fiber-preserving map:

$\displaystyle\begin{matrix}F_{\mathscr{H}}\colon&T^{*}\mathbb{R}^{n}&\longrightarrow&T^{*}\mathbb{R}^{n}\\\
&(\mathbf{u},\mathbf{p})&\mapsto&(\mathbf{u},\dot{\mathbf{u}}),&\quad\dot{u}_{i}=\\{\mathscr{H},u_{i}\\}.\end{matrix}$
(6.48)

######  Definition 6.6.

A Hamiltonian $\mathscr{H}$ is _regular_ (resp.  _singular_) if the map
$F_{\mathscr{H}}$ is a diffeomorphism (resp. otherwise).

Suppose that a Hamiltonian $\mathscr{H}$ is regular. The _Lagrangian_
$\mathscr{L}$ for $\mathscr{H}$ is defined by the _Legendre transformation_

$\displaystyle\mathscr{L}(\mathbf{u},\dot{\mathbf{u}})=\sum_{i=1}^{n}\dot{u}_{i}p_{i}-\mathscr{H}(\mathbf{u},\mathbf{p}).$
(6.49)

Here and below the symbol $\mathscr{L}$ should not be confused with the Rogers
dilogarithm $L(x)$. The RHS of (6.49) is a function of
$(\mathbf{u},\mathbf{p})$. However, by converting the map (6.48), we may
regard it as a function of $(\mathbf{u},\dot{\mathbf{u}})$. The equations of
motion for $\mathscr{H}$ are equivalent to the following equations:

$\displaystyle p_{i}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial\dot{u}_{i}},$ (6.50)
$\displaystyle\frac{d}{dt}\left(\frac{\partial\mathscr{L}}{\partial\dot{u}_{i}}\right)$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial u_{i}}\quad\text{(the
\emph{Euler-Lagrange equations})}.$ (6.51)

Moreover, the Lagrangian $\mathscr{L}$ is regular in the following sense
[AM78, §3.6].

######  Definition 6.7.

A Lagrangian $\mathscr{L}$ is _regular_ (resp. singular) if the map

$\displaystyle\begin{matrix}F_{\mathscr{L}}\colon&T^{*}\mathbb{R}^{n}&\longrightarrow&T^{*}\mathbb{R}^{n}\\\
&(\mathbf{u},\dot{\mathbf{u}})&\mapsto&(\mathbf{u},\mathbf{p}),&\quad\displaystyle
p_{i}=\frac{\partial\mathscr{L}}{\partial\dot{u}_{i}}(\mathbf{u},\dot{\mathbf{u}}).\end{matrix}$
(6.52)

is a diffeomorphism (resp. otherwise).

Conversely, starting from a regular Lagrangian $\mathscr{L}$, we obtain a
regular Hamiltonian $\mathscr{H}$ by reversing the procedure. Moreover, the
two systems are equivalent again.

Coming back to the current situation, for the Hamiltonian $\mathscr{H}(s)$ in
(6.29), the map (6.48) is given by

$\displaystyle
F_{\mathscr{H}(s)}\colon(\mathbf{u}(s),\mathbf{p}(s))\mapsto(\mathbf{u}(s),\dot{\mathbf{u}}(s)),$
(6.53)
$\displaystyle\dot{u}_{i}(s)=\displaystyle-\frac{1}{\sqrt{2}}\delta_{k_{s}i}\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.54)

Unfortunately, this is not surjective at all. Thus, the Hamiltonian
$\mathscr{H}(s)$ is singular. Nevertheless, let us apply the Legendre
transformation to $\mathscr{H}(s)$ to see what we get. We have

$\displaystyle\mathscr{L}(s)=\dot{u}_{k_{s}}(s)p_{k_{s}}(s)-{\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.55)

At this moment, this is a function of $(\mathbf{u}(s),\mathbf{p}(s))$. Observe
that $\mathscr{L}(s)$ depends on $\mathbf{p}(s)$ only through $p_{k_{s}}(s)$,
which is written by

$\displaystyle
p_{k_{s}}(s)={\delta_{k_{s}}}(\sqrt{2}\log\tilde{y}_{k_{s}}(s)-w_{k_{s}}(s)).$
(6.56)

Also, we convert (6.54) as

$\displaystyle\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}=\exp(-\sqrt{2}\dot{u}_{k_{s}}(s))-1.$
(6.57)

Thus, the Lagrangian (6.55) is written in the form

$\displaystyle\mathscr{L}(s)={\delta_{k_{s}}}\dot{u}_{k_{s}}(s)(\sqrt{2}\log\tilde{y}_{k_{s}}(s)-w_{k_{s}}(s))-{\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}),$
(6.58) $\displaystyle
w_{i}(s)=\sum_{j=1}^{n}b_{ji}(s)u_{j}(s),\quad\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}=\exp(-\sqrt{2}\dot{u}_{k_{s}}(s))-1,$
(6.59)

which is a function of $(\mathbf{u}(s),\dot{\mathbf{u}}(s))$. The Lagrangian
$\mathscr{L}(s)$ is singular, because
$\partial\mathscr{L}(s)/\partial\dot{u}_{i}(s)=0$ for $i\neq k_{s}$.

Since both $\mathscr{H}(s)$ and $\mathscr{L}(s)$ are singular, we expect that
the equations (6.50) and (6.51) for $\mathscr{L}(s)$ are _not_ equivalent to
the equations of motion for $\mathscr{H}(s)$, which are (6.30) and (6.31).
Indeed, we have the following result.

######  Lemma 6.8 ([GNR17, Prop. 4.3]).

For the Lagrangian $\mathscr{L}(s)$ in (6.58), the equations (6.50) and (6.51)
are explicitly written as follows: For $i=k_{s}$,

$\displaystyle
p_{k_{s}}(s)={\delta_{k_{s}}}(\sqrt{2}\log\tilde{y}_{k_{s}}(s)-w_{k_{s}}(s)),\quad\dot{p}_{k_{s}}(s)=0,$
(6.60)

and, for $i\neq k_{s}$,

$\displaystyle
p_{i}(s)=0,\quad(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})^{b_{k_{s}i}(s)}=1.$
(6.61)

###### Proof.

They follow from the following calculations:

$\displaystyle\begin{split}\frac{\partial\mathscr{L}(s)}{\partial\dot{u}_{k_{s}}(s)}&={\delta_{k_{s}}}(\sqrt{2}\log\tilde{y}_{k_{s}}(s)-w_{k_{s}}(s))\\\
&\qquad-2{\varepsilon_{k_{s}}}{\delta_{k_{s}}}\frac{\dot{u}_{k_{s}}(s)}{\tilde{y}_{k_{s}}(s)}-\sqrt{2}{\varepsilon_{k_{s}}}{\delta_{k_{s}}}\frac{\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})}{\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}}\\\
\overset{\eqref{eq:yu1}}{}&{=}{\delta_{k_{s}}}(\sqrt{2}\log\tilde{y}_{k_{s}}(s)-w_{k_{s}}(s)),\end{split}$
(6.62) $\displaystyle\frac{\partial\mathscr{L}(s)}{\partial u_{k_{s}}}$
$\displaystyle=0,$ (6.63)

and, for $i\neq k_{s}$,

$\displaystyle\frac{\partial\mathscr{L}(s)}{\partial\dot{u}_{i}(s)}$
$\displaystyle=0,$ (6.64) $\displaystyle\frac{\partial\mathscr{L}(s)}{\partial
u_{i}(s)}$
$\displaystyle=-{\delta_{k_{s}}}b_{ik_{s}}(s)\dot{u}_{k_{s}}(s)=-\frac{\delta_{i}}{\sqrt{2}}b_{k_{s}i}(s)\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.65)

∎

We observe that the equations (6.60) coincide with (6.26) and (6.31) for
$i=k_{s}$. On the other hand, the equations (6.61) impose unwanted
constraints. Therefore, they are false for our dynamical system.

Next, let us restrict the system in the small phase space $\tilde{M}_{0}$ and
set $p_{k_{s}}(s)=\delta_{k_{s}}w_{k_{s}}(s)$ in (6.60). Then, we have

$\displaystyle w_{k_{s}}(s)=\frac{1}{\sqrt{2}}\log\tilde{y}_{k_{s}}(s).$
(6.66)

By (6.57) and (6.66), the function (6.58) is rewritten entirely as a function
of $\tilde{y}_{k_{s}}(s)$ (therefore, as a function of $\dot{u}_{k_{s}}(s)$),
as

$\displaystyle\begin{split}\mathscr{L}(s)&=-\frac{1}{2}\varepsilon_{s}\delta_{k_{s}}\log(1+\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})\log\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}-{\varepsilon_{s}}{\delta_{k_{s}}}\mathrm{Li}_{2}(-\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}})\\\
&={\varepsilon_{s}}{\delta_{k_{s}}}\tilde{L}(\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).\end{split}$
(6.67)

We see that the modified Rogers dilogarithm emerges as the (false) Lagrangian.

Having obtained the exact function appearing in the DIs in Theorem 3.5, our
strategy for the rest of the chapter is as follows: We continue to rely on the
equations of motion for the Hamiltonian $\mathscr{H}(s)$ for our dynamical
system. On the other hand, we also use the function $\mathscr{L}(s)$ and its
intrinsic property to prove Theorem 3.5.

####  6.5 Periodicity conditions

Let us return to the sequence (5.8) for a $Y$-pattern. Our next task is to
clarify the relation between the periodicity of $y$-variables therein and the
one for the dynamical system with canonical coordinates
$(\mathbf{u},\mathbf{p})$ in the previous section.

Before that, we resolve the issue concerning the nonsingularity of $B$. For
the initial matrix $B$ of (5.8) with a skew-symmetrizer $D$, we introduce an
$2n\times 2n$ matrix

$\displaystyle\overline{B}=\begin{pmatrix}B&-I\\\ I&O\\\ \end{pmatrix}$ (6.68)

called the _principal extension of $B$_. The matrix $\overline{B}$ is skew-
symmetrizable with a skew-symmetrizer $D\oplus D$. Moreover, $\overline{B}$ is
nonsingular whether $B$ is singular or not. Starting from the initial extended
$Y$-seed
$\overline{\Upsilon}(0)=\overline{\Upsilon}(\overline{\mathbf{y}},\overline{B})$
with $\overline{\mathbf{y}}=(y_{1},\dots,y_{2n})$, we have a lift of the
sequence (5.8) in the extended $Y$-pattern of rank $2n$,

$\displaystyle\overline{\Upsilon}(0)\ {\buildrel{k_{0}}\over{\rightarrow}}\
\overline{\Upsilon}(1)\ {\buildrel{k_{1}}\over{\rightarrow}}\ \cdots\
{\buildrel{k_{P-1}}\over{\rightarrow}}\ \overline{\Upsilon}(P),$ (6.69)

where $k_{s}\in\\{1,\,\dots,\,n\\}$.

######  Proposition 6.9 ([Nak11b, Thm. 4.3]).

Let $\nu\in S_{n}$ be a permutation of the first $n$ indices $1$, …, $n$ of
extended $Y$-seeds in (6.69). Then, the sequence (6.69) is $\nu$-periodic if
and only if the sequence (5.8) is $\nu$-periodic.

###### Proof.

Let $\overline{C}(s)$ be the $C$-matrix for the extended $Y$-seed
$\overline{\Upsilon}(s)$. Then, the following fact is known ([FG19, Thm.
4.2]):

$\displaystyle\overline{C}(s)=\begin{pmatrix}C(s)&O\\\ O&I\\\ \end{pmatrix}.$
(6.70)

(The proof requires the sign-coherence property in Theorem 2.24. Thus, it is
considered as an advanced result.) Thus, $\overline{C}(P)=\nu\overline{C}(0)$
holds if and only if $C(P)=\nu C(0)$ holds. Therefore, by Theorem 2.31,
$\overline{\Upsilon}(P)=\nu\overline{\Upsilon}(0)$ holds if and only if
$\Upsilon(P)=\nu\Upsilon(0)$ holds. ∎

Observe that the DI for the sequence (5.8) coincides with the one for the
extended sequence (6.69) due to (6.70). Thus, thanks to Proposition 6.9, to
prove the DI for $B$ in Theorem 3.5, we may prove the one for $\overline{B}$.
Therefore, in proving Theorem 3.5, we may assume that $B$ is nonsingular
without loss of generality.

Now, let us formulate the periodicity condition for the sequence (5.8) in the
current setting. In a parallel way with (5.10), for any permutation $\nu\in
S_{n}$, we define a semifield automorphism (denoted by the same symbol)

$\displaystyle\begin{matrix}\nu\colon&\mathbb{P}(P)&\longrightarrow&\mathbb{P}=\mathbb{P}(0)\\\
&y_{i}(P)&\mapsto&\displaystyle y_{\nu^{-1}(i)}.\end{matrix}$ (6.71)

Then, the $\nu$-periodicity for the sequence (5.8) is rephrased as the
equality

$\displaystyle\mu(P-1;0)=\nu\quad\text{for}\
\mathbb{P}(P)\longrightarrow\mathbb{P}.$ (6.72)

The $\nu$-periodicity of (5.8) also implies the same periodicity for tropical
$y$-variables. Thus, by (5.18), we also have the following equality:

$\displaystyle\tau(P-1;0)=\nu\quad\text{for}\
\mathbb{P}(P)\longrightarrow\mathbb{P}.$ (6.73)

Therefore, by the decomposition (5.31), we have

$\displaystyle\mathfrak{q}(P-1;0)=\mathrm{id}\quad\text{for}\
\mathbb{P}\longrightarrow\mathbb{P}.$ (6.74)

Note that $\nu$ disappeared in this equality. This further induces the
equality

$\displaystyle\mathfrak{q}(P-1;0)=\mathrm{id}\quad\text{for}\
\mathcal{C}(\mathbf{y})\rightarrow\mathcal{C}(\mathbf{y}).$ (6.75)

In particular, any trajectory $\alpha=(\alpha_{t})_{t\in[0,P]}$ in the phase
space $M$ in the full time span $[0,P]$ is periodic. (See Figure 6.1.).
Namely, we have

$\displaystyle\alpha_{0}=\alpha_{P}.$ (6.76)

Let us formulate a parallel result for the canonical coordinates
$(\mathbf{u},\mathbf{p})$. For any permutation $\nu\in S_{n}$, define

$\displaystyle\begin{matrix}\nu\colon&\mathbf{C}(\mathbf{u}(P),\mathbf{p}(P))&\longrightarrow&\mathbf{C}(\mathbf{u},\mathbf{p})=\mathbf{C}(\mathbf{u}(0),\mathbf{p}(0))\\\
&u_{i}(P)&\mapsto&\displaystyle u_{\nu^{-1}(i)},\\\
&p_{i}(P)&\mapsto&\displaystyle p_{\nu^{-1}(i)}.\end{matrix}$ (6.77)

Remarkably, the same periodicity still holds in the phase space $\tilde{M}$,
which is larger than $M$, if $B$ is nonsingular.

######  Proposition 6.10 ([GNR17, Prop. 5.11]).

Suppose that that the sequence (5.8) is $\nu$-periodic and the initial
exchange matrix $B$ is nonsingular. Then, the following periodicities hold:

$\displaystyle\mu(P-1;0)$ $\displaystyle=\nu\quad\text{for}\
\mathcal{C}(\mathbf{u}(P),\mathbf{p}(P))\rightarrow\mathcal{C}(\mathbf{u},\mathbf{p}),$
(6.78) $\displaystyle\tau(P-1;0)$ $\displaystyle=\nu\quad\text{for}\
\mathcal{C}(\mathbf{u}(P),\mathbf{p}(P))\rightarrow\mathcal{C}(\mathbf{u},\mathbf{p}),$
(6.79) $\displaystyle\mathfrak{q}(P-1;0)$
$\displaystyle=\mathrm{id}\quad\text{for}\
\mathcal{C}(\mathbf{u},\mathbf{p})\rightarrow\mathcal{C}(\mathbf{u},\mathbf{p}).$
(6.80)

In particular, any trajectory $\alpha=(\alpha_{t})_{t\in[0,P]}$ in the phase
space $\tilde{M}$ in the full time span $[0,P]$ is periodic; namely,

$\displaystyle\alpha_{0}=\alpha_{P}.$ (6.81)

###### Proof.

The variables $y_{i}(s)$ in $\mathcal{C}(\mathbf{y}(s))$ and
$\tilde{y}_{i}(s)$ in $\mathcal{C}(\mathbf{u}(s),\mathbf{p}(s))$ obey the same
transformations under $\rho(s)$ and $\tau(s)$. Thus, they also obey the same
transformations under $\mathfrak{q}(s)$ and $\mathfrak{q}(s;0)$. Therefore,
the periodicities (6.72)–(6.75) imply the periodicities for
$\tilde{y}$-variables

$\displaystyle\mu(P-1;0):$ $\displaystyle\
\tilde{y}_{i}(P)\mapsto\tilde{y}_{\nu^{-1}(i)},$ (6.82)
$\displaystyle\tau(P-1;0):$ $\displaystyle\
\tilde{y}_{i}(P)\mapsto\tilde{y}_{\nu^{-1}(i)},$ (6.83)
$\displaystyle\mathfrak{q}(P-1;0):$ $\displaystyle\
\tilde{y}_{i}\mapsto\tilde{y}_{i}.$ (6.84)

Let us show that they imply the same periodicities for $\mathbf{u}(s)$ and
$\mathbf{p}(s)$ as well. We set $z_{i}(s)=\log\tilde{y}_{i}(s)$. Then, by
(6.35), (6.36), (6.40), (6.41), we see that $w_{i}(s)$,
$\sqrt{2}\delta_{i}^{-1}p_{i}(s)$, and $z_{i}(s)$ obey the same affine
transformations under $\rho(s)$ and $\tau(s)$. It follows that $\mathbf{w}(s)$
and $\mathbf{p}(s)$ obey the same periodicities in (6.82)–(6.84) for
$\tilde{\mathbf{y}}(s)$. Moreover, since $B$ is nonsingular, one can invert
the relation between $\mathbf{u}(s)$ and $\mathbf{w}(s)$ in (6.26). Thus,
$\mathbf{u}(s)$ also obeys the same periodicity. ∎

####  6.6 Classical mechanical proof of DIs (I): In local picture

Let us give an alternative proof of Theorem 3.5 based on the classical
mechanical formulation presented in this chapter. Actually, we will give two
alternative proofs, and this is the first one.

We prove the DI (3.13) with variables $\tilde{\mathbf{y}}$. However, variables
$\tilde{\mathbf{y}}$ are algebraically independent on $\tilde{M}_{0}$ only
when the matrix $B$ is nonsingular. So, to restore the DI in (3.13) from the