DI with variables $\tilde{\mathbf{y}}$, we need to assume that $B$ is
nonsingular. On the other hand, we have already argued in the last section
that this assumption does not lose generality. So, from now on, _we assume
that the initial matrix $B$ is nonsingular_. Then, under this assumption, the
map (6.47) is injective and we may identify $\tilde{\mathbf{y}}(s)$ and
$\mathbf{y}(s)$.

Suppose that the mutation sequence (5.8) is $\nu$-periodic. Let us consider a
trajectory $\alpha=(\alpha_{t})_{t\in[0,P]}$ in the small phase space
$\tilde{M}_{0}$. By Proposition 6.10, $\alpha$ is periodic. Namely,

$\displaystyle\alpha_{0}=\alpha_{P}$ (6.85)

holds. The main idea of the proof of Theorem 3.5 is to consider the following
integral of the Lagrangian $\mathscr{L}$ given by (6.67) along the trajectory
$\alpha$:

$\displaystyle S[\alpha]:=\int_{0}^{P}\mathscr{L}(\alpha_{t})\,dt.$ (6.86)

We call (6.86) the _action integral_ along $\alpha$ following the convention
in classical mechanics. Recall that $\tilde{y}_{k_{s}}$ is constant in the
time span $[s,s+1]$. Thus, by (6.67), the action integral is reduced to a sum
of the modified Rogers dilogarithm

$\displaystyle
S[\alpha]=\sum_{s=0}^{P-1}{\varepsilon_{s}}{\delta_{k_{s}}}\tilde{L}(\tilde{y}_{k_{s}}(s)^{\varepsilon_{s}}).$
(6.87)

Observe that this is exactly the LHS of the DI (3.13).

Once the constancy of (6.87) is shown, the constant term is 0 by considering
the tropical limit $\mathbf{y}\rightarrow\mathbf{0}$ as we did when proving
Theorem 3.5. Thus, we are left to prove the constancy of (6.87). We formulate
it as the invariance of the action integral as follows.

######  Theorem 6.11 ([GNR17, Thm. 6.3]).

Under the periodicity (6.85), the action integral $S[\alpha]$ in (6.86) is
independent of a trajectory $\alpha$ in $\tilde{M}_{0}$.

The rest of the section is devoted to proving Theorem 6.11. We prove it by the
standard _variational calculus_ in classical mechanics. Consider an
infinitesimal variation $\alpha+\delta\alpha$ of a trajectory $\alpha$, where
we assume that $\alpha+\delta\alpha$ is also a trajectory in the small phase
space $\tilde{M}_{0}$. Let us concentrate on the time span $[s,s+1]$. Let
$(\overline{\mathbf{u}}(t),\overline{\mathbf{p}}(t))$ ($t\in[s,s+1]$) be the
point corresponding to $\alpha_{t}$ in the
$(\mathbf{u}(s),\mathbf{p}(s))$-coordinates. Due to the constraint (6.46), we
have

$\displaystyle\overline{u}_{i}(t)$
$\displaystyle\rightarrow\overline{u}_{i}(t)+\delta\overline{u}_{i}(t),$
(6.88) $\displaystyle\delta^{-1}_{i}\overline{p}_{i}(t)$
$\displaystyle\rightarrow\delta^{-1}_{i}\overline{p}_{i}(t)+\sum_{j=1}^{n}b_{ji}(s)\delta\overline{u}_{j}(t).$
(6.89)

Let

$\displaystyle S[\alpha;s]:=\int_{s}^{s+1}\mathscr{L}(s)(\alpha_{t})\,dt.$
(6.90)

The variation of $S[\alpha;s]$ is evaluated by the “boundary terms” as
follows.

######  Lemma 6.12 ([GNR17, Lemma. 6.4]).

(a). In the $(\mathbf{u}(s),\mathbf{p}(s))$-coordinates, we have

$\displaystyle\delta S[\alpha;s]=R[\alpha;s+1]-R[\alpha;s],$ (6.91)

where

$\displaystyle
R[\alpha;t]:=\sum_{i=1}^{n}\overline{p}_{i}(t)\delta\overline{u}_{i}(t).$
(6.92)

(b). The above quantity $R[\alpha;t]$ is independent of the choice of
coordinates $(\mathbf{u}(s),\mathbf{p}(s))$ for $\tilde{M}_{0}$.

###### Proof.

(a). We regard $\mathscr{L}(\alpha_{t})$ as a function of
$(\overline{\mathbf{u}}(t),\dot{\overline{\mathbf{u}}}(t))$ as explained in
Section 6.4. We first recall that, by (6.30), we have

$\displaystyle\dot{\overline{u}}_{i}(t)$ $\displaystyle=0\quad(i\neq k_{s}).$
(6.93)

This implies that,

$\displaystyle\delta\dot{\overline{u}}_{i}(t)$ $\displaystyle=0\quad(i\neq
k_{s}).$ (6.94)

Thus, we decompose the variation $\delta S[\alpha;s]$ as

$\displaystyle\delta S[\alpha;s]=\delta S_{1}+\delta S_{2},$ (6.95)

where

$\displaystyle\delta S_{1}:=$ $\displaystyle\
\int_{s}^{s+1}\left\\{\frac{\partial\mathscr{L}(\alpha_{t})}{\partial\overline{u}_{k_{s}}(t)}\delta\overline{u}_{k_{s}}(t)+\frac{\partial\mathscr{L}(\alpha_{t})}{\partial\dot{\overline{u}}_{k_{s}}(t)}\delta\dot{\overline{u}}_{k_{s}}(t)\right\\}dt,$
(6.96) $\displaystyle\delta S_{2}:=$ $\displaystyle\ \sum_{\scriptstyle
i=1\atop\scriptstyle i\neq
k_{s}}^{n}\int_{s}^{s+1}\left\\{\frac{\partial\mathscr{L}(\alpha_{t})}{\partial\overline{u}_{i}(t)}\delta\overline{u}_{i}(t)\right\\}dt.$
(6.97)

As we see in Lemma 6.8, the equations (6.50) and (6.51) for $i=k_{s}$ are
reliable. By using them, we have

$\displaystyle\begin{split}\delta
S_{1}&=\int_{s}^{s+1}\left\\{\dot{\overline{p}}_{k_{s}}(t)\delta\overline{u}_{k_{s}}(t)+\overline{p}_{k_{s}}(t)\delta\dot{\overline{u}}_{k_{s}}(t)\right\\}dt\\\
&=\int_{s}^{s+1}\frac{d}{dt}\left\\{{\overline{p}}_{k_{s}}(t)\delta\overline{u}_{k_{s}}(t)\right\\}dt\\\
&=\overline{p}_{k_{s}}(s+1)\delta\overline{u}_{k_{s}}(s+1)-\overline{p}_{k_{s}}(s)\delta\overline{u}_{k_{s}}(s).\end{split}$
(6.98)

To evaluate $\delta S_{2}$, we have to be careful because the equations (6.50)
and (6.51) for $i\neq k_{s}$ are false. By (6.65), we have

$\displaystyle\delta S_{2}=\sum_{\scriptstyle i=1\atop\scriptstyle i\neq
k_{s}}^{n}\int_{s}^{s+1}\left\\{-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{s}i}(s)\log(1+\overline{y}_{k_{s}}(s)^{\varepsilon_{s}})\delta\overline{u}_{i}(t)\right\\}dt,$
(6.99)

where

$\displaystyle\overline{y}_{i}(t)=\exp\left(\sqrt{2}\delta_{i}^{-1}\overline{p}_{i}(t)\right)=\exp\left(\sqrt{2}\overline{w}_{i}(t)\right),\quad\overline{w}_{i}(t):=\sum_{j=1}^{n}b_{ji}(s)\overline{u}_{j}(t).$
(6.100)

in $\tilde{M}_{0}$. By (6.94), the integrand is constant on $[s,s+1]$. So, we
have

$\displaystyle\delta S_{2}=\sum_{\scriptstyle i=1\atop\scriptstyle i\neq
k_{s}}^{n}\left(-\frac{1}{\sqrt{2}}\delta_{i}b_{k_{s}i}(s)\log(1+\overline{y}_{k_{s}}(s)^{\varepsilon_{s}})\delta\overline{u}_{i}(s)\right).$
(6.101)

Moreover, we have

$\displaystyle\begin{split}\delta
S_{2}\overset{\eqref{eq:rhop1}}{}&{=}\sum_{\scriptstyle i=1\atop\scriptstyle
i\neq
k_{s}}^{n}\left\\{\overline{p}_{i}(s+1)-\overline{p}_{i}(s)\right\\}\delta\overline{u}_{i}(s)\\\
\overset{\eqref{eq:du1}}{}&=\sum_{\scriptstyle i=1\atop\scriptstyle i\neq
k_{s}}^{n}\left\\{\overline{p}_{i}(s+1)\delta\overline{u}_{i}(s+1)-\overline{p}_{i}(s)\delta\overline{u}_{i}(s)\right\\}.\end{split}$
(6.102)

By summing up (6.98) and (6.102), we obtain (6.91).

(b). Since $\tau^{*}(s)$ acts on $\overline{u}_{i}(t)$ as a linear
transformation, it also acts on the variation $\delta\overline{u}_{i}(t)$ in
the same way. Then, the claim is proved in the same way as Proposition 6.4
(b). ∎

Let us finish the proof of Theorem 6.11. By Lemma 6.12 (a) and (b), we have

$\displaystyle\begin{split}\delta
S[\alpha]&=\sum_{s=0}^{P-1}(R[\alpha;s+1]-R[\alpha;s])\\\
&=R[\alpha;P]-R[\alpha;0].\end{split}$ (6.103)

By Lemma 6.12 (b) again, we may evaluate both $R[\alpha;P]$ and $R[\alpha;0]$
in the initial coordinates $(\mathbf{u},\mathbf{p})$. Finally and most
importantly, by the periodicity (6.85) of $\alpha$, we have

$\displaystyle R[\alpha;P]=R[\alpha;0].$ (6.104)

Therefore, $\delta S[\alpha]=0$. This completes the proof of Theorem 6.11 and
also Theorem 3.5.

The method of evaluating $\delta S[\alpha;s]$ in the proof of Lemma 6.12 is
essentially the same one for celebrated _Noether’s theorem_ (e.g., [Tak08,
Thm. 1.3]) in classical mechanics. Thus, we may regard Theorem 6.11 as a
variant of Noether’s theorem. This gives us a conceptual understanding of the
constancy of the DI in Theorem 3.5.

####  6.7 Classical mechanical proof of DIs (II): In global picture

The proof of Theorem 3.5 in the previous section uses the canonical
coordinates $(\mathbf{u}(s),\mathbf{p}(s))$ for each time span $[s,s+1]$. Here
we present the second proof of Theorem 3.5 based on the same classical
mechanical method, but using only the initial canonical coordinates. The proof
is less intrinsic than the first one. On the other hand, it is applicable in a
more general context. The main reason to present this proof is that we use it
later in Chapter 9.

Again, we assume that $B$ is nonsingular, and we identify $\tilde{\mathbf{y}}$
and $\mathbf{y}$ by the map (6.25). First, we consider a little more general
Hamiltonian (see (5.55))

$\displaystyle\mathscr{H}_{\mathbf{c},a}=a\mathrm{Li}_{2}(-\tilde{y}^{\mathbf{c}})$
(6.105)

for the time span $[0,1]$ and a trajectory $\alpha=(\alpha_{t})_{t\in[0,1]}$
in $\tilde{M}_{0}$ by $\mathscr{H}_{\mathbf{c},a}$, where
$\alpha_{0}=(\overline{\mathbf{u}},\overline{\mathbf{p}})$ and
$\alpha_{1}=(\overline{\mathbf{u}}^{\prime},\overline{\mathbf{p}}^{\prime})$.
Let

$\displaystyle\overline{y}_{i}:=\exp\left({\sqrt{2}}\delta^{-1}_{i}\overline{p}_{i}\right)=\exp\left(\sqrt{2}\overline{w}_{i}\right),\quad\overline{w}_{i}:=\sum_{j=1}^{n}b_{ji}\overline{u}_{j}.$
(6.106)

Then, we have

$\displaystyle\overline{u}_{i}^{\prime}$
$\displaystyle=\overline{u}_{i}-\frac{a}{\sqrt{2}}c_{i}\delta^{-1}_{i}\log(1+\overline{y}^{\mathbf{c}}),$
(6.107) $\displaystyle\overline{p}_{i}^{\prime}$
$\displaystyle=\overline{p}_{i}+\frac{a}{\sqrt{2}}\biggl{(}\sum_{j=1}^{n}c_{j}b_{ij}\biggr{)}\log(1+\overline{y}^{\mathbf{c}}),$
(6.108) $\displaystyle\log\overline{y}_{i}^{\prime}$
$\displaystyle=\log\overline{y}_{i}-a\biggl{(}\sum_{j=1}^{n}c_{j}\omega_{ji}\biggr{)}\log(1+\overline{y}^{\mathbf{c}}).$
(6.109)

Consider the corresponding Lagrangian

$\displaystyle\mathscr{L}_{\mathbf{c},a}=a\tilde{L}(\tilde{y}^{\mathbf{c}}),$
(6.110)

and the action integral along the above trajectory

$\displaystyle
S_{\mathbf{c},a}[\alpha]:=\int_{0}^{1}\mathscr{L}_{\mathbf{c},a}(\alpha_{t})\,dt.$
(6.111)

The integrand is constant because
$\\{\mathscr{H}_{\mathbf{c},a},\tilde{y}^{\mathbf{c}}\\}=0$. Thus, we have

$\displaystyle
S_{\mathbf{c},a}[\alpha]=\mathscr{L}_{\mathbf{c},a}(\alpha_{0}).$ (6.112)

Then, we consider an infinitesimal variation $\alpha+\delta\alpha$ of a
trajectory $\alpha$ in $\tilde{M}_{0}$.

The following result formally coincides with Lemma 6.12. However, it is more
general than Lemma 6.12 because it is applicable to any integer vector
$\mathbf{c}$ other than the $c$-vectors $\mathbf{c}^{+}_{k_{s}}(s)$.

######  Proposition 6.13 ([Nak24, Lemma 3.5]).

The infinitesimal variation of $S_{\mathbf{c},a}[\alpha]$ in (6.112) is given
by

$\displaystyle\delta
S_{\mathbf{c},a}[\alpha]=\sum_{i=1}^{n}\overline{p}^{\prime}_{i}\delta\overline{u}^{\prime}_{i}-\sum_{i=1}^{n}\overline{p}_{i}\delta\overline{u}_{i}.$
(6.113)

###### Proof.

Here, we compute and compare both sides explicitly. Thanks to (6.106) and
(1.25), the LHS of (6.113) is given by

$\displaystyle\delta\mathscr{L}_{\mathbf{c},a}(\alpha_{0})=\frac{a}{\sqrt{2}}\biggl{(}\,\sum_{i,j=1}^{n}c_{j}b_{ij}\delta\overline{u}_{i}\biggr{)}\biggl{(}\log(1+\overline{y}^{\mathbf{c}})-\frac{\overline{y}^{\mathbf{c}}\log\overline{y}^{\mathbf{c}}}{1+\overline{y}^{\mathbf{c}}}\biggr{)}.$
(6.114)

Let us calculate the RHS of (6.113). By (6.107), we have

$\displaystyle\delta\overline{u}_{i}^{\prime}$
$\displaystyle=\delta\overline{u}_{i}-\frac{a}{\sqrt{2}}c_{i}\delta^{-1}_{i}\biggl{(}\,\sum_{j,k=1}^{n}c_{j}b_{kj}\delta\overline{u}_{k}\biggr{)}\frac{\overline{y}^{\mathbf{c}}}{1+\overline{y}^{\mathbf{c}}}.$
(6.115)

Let us write (6.108) and (6.115) as
$\overline{p}_{i}^{\prime}=\overline{p}_{i}+A_{i}$ and
$\delta\overline{u}_{i}^{\prime}=\delta\overline{u}_{i}-B_{i}$. Then, the RHS
of (6.113) is given by

$\displaystyle\sum_{i=1}^{n}A_{i}\delta\overline{u}_{i}-\sum_{i=1}^{n}\overline{p}_{i}B_{i}-\sum_{i=1}^{n}A_{i}B_{i}.$
(6.116)

They are calculated as

$\displaystyle\sum_{i=1}^{n}A_{i}\delta\overline{u}_{i}$
$\displaystyle=\frac{a}{\sqrt{2}}\biggl{(}\,\sum_{i,j=1}^{n}c_{j}b_{ij}\delta\overline{u}_{i}\biggr{)}\log(1+\overline{y}^{\mathbf{c}}),$
(6.117) $\displaystyle\sum_{i=1}^{n}\overline{p}_{i}B_{i}$
$\displaystyle=\frac{a}{\sqrt{2}}\biggl{(}\,\sum_{i,j=1}^{n}c_{j}b_{ij}\delta\overline{u}_{i}\biggr{)}\frac{\overline{y}^{\mathbf{c}}\log\overline{y}^{\mathbf{c}}}{1+\overline{y}^{\mathbf{c}}},$
(6.118) $\displaystyle\sum_{i=1}^{n}A_{i}B_{i}$
$\displaystyle=\frac{a^{2}}{{2}}\\{\mathbf{c},\mathbf{c}\\}_{\Omega}\biggl{(}\,\sum_{i,j=1}^{n}c_{j}b_{ij}\delta\overline{u}_{i}\biggr{)}\frac{\overline{y}^{\mathbf{c}}\log(1+\overline{y}^{\mathbf{c}})}{1+\overline{y}^{\mathbf{c}}}=0,$
(6.119)

where the last equality is due to the skew-symmetry of
$\\{\cdot,\cdot\\}_{\Omega}$ in (5.21). Then, we see that (6.116) coincides
with (6.114). ∎

The rest of the proof of Theorem 6.11 is the same as before. Assume that that
$B$ is nonsingular, and suppose that
$\bar{\mathbf{y}}_{t=P}=\bar{\mathbf{y}}_{t=0}$. Then, by (6.107)–(6.109), we
have $\bar{\mathbf{u}}_{t=P}=\bar{\mathbf{u}}_{t=0}$ and
$\bar{\mathbf{p}}_{t=P}=\bar{\mathbf{p}}_{t=0}$. Therefore, for the action
integral

$\displaystyle
S[a]:=\int_{0}^{P}\mathscr{L}(\alpha_{t})\,dt=\sum_{s=0}^{P-1}{\varepsilon_{s}}{\delta_{k_{s}}}\tilde{L}(y^{\mathbf{c}^{+}_{k_{s}}(s)}),$
(6.120)

we have

$\displaystyle\begin{split}\delta
S[\alpha]&=\sum_{s=0}^{P-1}(R[\alpha;s]_{+}-R[\alpha;s]_{-})=R[\alpha;P]-R[\alpha;0]=0,\end{split}$
(6.121)

where we used Proposition 6.13 for the first equality.

Notes

The content of this section is mostly based on [GNR17] except for Section 6.7,
which appeared in [Nak24]. The idea of representing $y$-variables with
canonical coordinates (variables) originates in the work of [FG09c, FG09b,
KN11], where the quantum version was considered. The appearance of the
modified Rogers dilogarithm as the Lagrangian was expected by the semi-
classical calculation of quantum dilogarithm identities by [KN11].

##  Part III Scattering diagram method

###  Chapter 7 Algebraic formulation of dilogarithm and mutations

Motivated by the cluster scattering diagram formalism in the next chapter, we
define an infinite group for each skew-symmetric matrix. We focus on certain
distinguished elements in the group called the _dilogarithm elements_. They
are regarded as an algebraic version of the dilogarithm and mutations, and
they satisfy the _pentagon relation_ corresponding to the pentagon periodicity
of the $Y$-pattern of type $A_{2}$. We present several interesting relations
generated by the pentagon relation, including infinite ones,

####  7.1 Lie algebra associated with skew-symmetric form

Let $\Omega=(\omega_{ij})_{i,j=1}^{n}$ be any skew-symmetric rational matrix.
Let us fix a lattice $N=\mathbb{Z}^{n}$ of rank $n$. Let
$\\{\cdot,\cdot\\}_{\Omega}$ be the skew-symmetric rational bilinear form on
$N$ defined by (5.21). Let

$\displaystyle N^{+}=\mathbb{Z}^{n}_{\geq 0}\setminus\\{\mathbf{0}\\}$ (7.1)

be the set of the positive vectors of $N$. The _degree_ $\deg(\mathbf{n})$ of
$\mathbf{n}=(n_{i})\in N^{+}$ is defined by

$\displaystyle\deg(\mathbf{n}):=\sum_{i=1}^{n}n_{i}.$ (7.2)

For any integer $\ell>0$, we write

$\displaystyle(N^{+})^{\leq\ell}:=$ $\displaystyle\\{\mathbf{n}\in
N^{+}\mid\deg(\mathbf{n})\leq\ell\\},$ (7.3) $\displaystyle(N^{+})^{>\ell}:=$
$\displaystyle\\{\mathbf{n}\in N^{+}\mid\deg(\mathbf{n})>\ell\\}.$ (7.4)

We say that $\mathbf{n}\in N^{+}$ is _primitive_ if there is no pair
$j\in\mathbb{Z}_{>1}$ and $\mathbf{n}^{\prime}\in N^{+}$ such that
$\mathbf{n}=j\mathbf{n}^{\prime}$. The set of all primitive vectors are
denoted by $N^{+}_{\mathrm{pr}}$.

Following Gross-Siebert [GS11], Kontsevich-Soibelman [KS08, KS14], and Gross-
Hacking-Keel-Kontsevich (GHKK) [GHKK18], we introduce an $N^{+}$-graded Lie
algebra over $\mathbb{Q}$,

$\displaystyle\mathfrak{g}=\mathfrak{g}_{\Omega}=\bigoplus_{\mathbf{n}\in
N^{+}}\mathbb{Q}X_{\mathbf{n}}$ (7.5)

with generators $X_{\mathbf{n}}$ ($\mathbf{n}\in N^{+}$) and relations

$\displaystyle[X_{\mathbf{n}},X_{\mathbf{n}^{\prime}}]=\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}X_{\mathbf{n}+\mathbf{n}^{\prime}}.$
(7.6)

Clearly, the above Lie bracket is skew-symmetric. The Jacobi identity is also
clear from the following cyclic expression:

$\displaystyle\begin{split}[X_{\mathbf{n}_{1}},[X_{\mathbf{n}_{2}},X_{\mathbf{n}_{3}}]]&=\\{\mathbf{n}_{2},\mathbf{n}_{3}\\}_{\Omega}[X_{\mathbf{n}_{1}},X_{\mathbf{n}_{2}+\mathbf{n}_{3}}]\\\
&=(\\{\mathbf{n}_{1},\mathbf{n}_{2}\\}_{\Omega}\\{\mathbf{n}_{2},\mathbf{n}_{3}\\}_{\Omega}-\\{\mathbf{n}_{2},\mathbf{n}_{3}\\}_{\Omega}\\{\mathbf{n}_{3},\mathbf{n}_{1}\\}_{\Omega})X_{\mathbf{n}_{1}+\mathbf{n}_{2}+\mathbf{n}_{3}}.\end{split}$
(7.7)

We consider the completion of $\mathfrak{g}$ by $\deg$. Namely, for any
integer $\ell>0$, consider a Lie algebra ideal and its quotient

$\displaystyle\mathfrak{g}^{>\ell}$
$\displaystyle=\bigoplus_{\mathbf{n}\in(N^{+})^{>\ell}}\mathbb{Q}X_{\mathbf{n}},$
(7.8) $\displaystyle\mathfrak{g}^{\leq\ell}$
$\displaystyle=\mathfrak{g}/\mathfrak{g}^{>\ell}.$ (7.9)

Note that $\mathfrak{g}^{\leq\ell}$ is nilpotent. Then, we take the inverse
limit

$\displaystyle\widehat{\mathfrak{g}}=\widehat{\mathfrak{g}}_{\Omega}:=\lim_{\longleftarrow}\mathfrak{g}^{\leq\ell}.$
(7.10)

Thus, an element of $\widehat{\mathfrak{g}}$ is given by a formal infinite sum

$\displaystyle\sum_{\mathbf{n}\in
N^{+}}c_{\mathbf{n}}X_{\mathbf{n}}\quad(c_{\mathbf{n}}\in\mathbb{Q}).$ (7.11)

For each $\mathbf{n}\in N^{+}_{\mathrm{pr}}$, let
$\widehat{\mathfrak{g}}_{\mathbf{n}}^{\parallel}$ denote the commutative Lie
subalgebra of $\widehat{\mathfrak{g}}$ spanned by elements
$\sum_{j=1}^{\infty}c_{j\mathbf{n}}X_{j\mathbf{n}}$.

####  7.2 Exponential group

For the Lie algebra $\widehat{g}$ in (7.10), we define the associated
exponential group

$\displaystyle G=G_{\Omega}:=\exp(\widehat{\mathfrak{g}}),$ (7.12)

where $\exp$ is a formal exponential map. Namely, as a set, $G$ is identified
with $\widehat{\mathfrak{g}}$ with $\exp(X)\leftrightarrow X$, and its product
is defined by the _Baker-Campbell-Hausdorff (BCH) formula_ (e.g., [Bou89,
Ch.2, §6], [BF12]),

$\displaystyle\begin{split}&\ \exp(X)\exp(Y)\\\ =&\
\exp\biggl{(}X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+\cdots\biggr{)},\end{split}$
(7.13)

where we do not need the explicit form of the higher-order brackets. The
infinite sum in the RHS converges in $\widehat{\mathfrak{g}}$ due to the
nilpotency of $\mathfrak{g}^{\leq\ell}$ for any $\ell$. Originally, the BCH
formula is a product formula for the exponentials of noncommutative variables
$X$ and $Y$, where $[X,Y]:=XY-YX$. So, the associativity of the product is
guaranteed. See [BF12, Thm. 5.42] for a detailed treatment.

Alternatively, for the Lie algebra ${g}^{\leq\ell}$ in (7.9), we define the
associated exponential group

$\displaystyle G^{\leq\ell}=\exp(\mathfrak{g}^{\leq\ell})$ (7.14)

in the same way. Then, $G$ is identified with the inverse limit

$\displaystyle G=\lim_{\longleftarrow}G^{\leq\ell}.$ (7.15)

For the canonical projection

$\displaystyle\pi_{\ell}:G\longrightarrow G^{\leq\ell},$ (7.16)

we set $G^{>\ell}:=\mathrm{Ker}\,\pi_{\ell}$.

For each $\mathbf{n}\in N^{+}_{\mathrm{pr}}$, let $G_{\mathbf{n}}^{\parallel}$
be the exponential group of $\widehat{\mathfrak{g}}_{\mathbf{n}}^{\parallel}$.
It is an abelian subgroup of $G$. We call it the _parallel subgroup_ of
$\mathbf{n}$.

An _infinite_ product in $G$ makes sense if it is reduced to a finite product
by applying $\pi_{\ell}$ for any $\ell$. For any element $g=\exp(X)$, its
rational power $g^{c}$ ($c\in\mathbb{Q}$) is ambiguously defined by
$g^{c}:=\exp(cX)$.

####  7.3 $y$-representation

We introduce a representation of the group $G=G_{\Omega}$ in the previous
section. Let $\mathbf{y}=(y_{1},\dots,y_{n})$ be an $n$-tuple of variables and
let $\mathbb{Q}[[\mathbf{y}]]$ be the formal power series ring of $\mathbf{y}$
over $\mathbb{Q}$. Namely, any element of $\mathbb{Q}[[\mathbf{y}]]$ is
written as a formal infinite sum

$\displaystyle\sum_{\mathbf{n}\in\mathbb{Z}^{n}_{\geq
0}}c_{\mathbf{n}}y^{\mathbf{n}}\quad(c_{\mathbf{n}}\in\mathbb{Q}),\quad
y^{\mathbf{e}_{i}}=y_{i}.$ (7.17)

The algebra $\mathbb{Q}[[\mathbf{y}]]$ is naturally equipped with the Poisson
bracket defined by

$\displaystyle\\{y^{\mathbf{n}},y^{\mathbf{n}^{\prime}}\\}=\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}y^{\mathbf{n}+\mathbf{n}^{\prime}},$
(7.18)

where $\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}$ is the same skew-
symmetric rational bilinear form in (7.6). The Leibniz rule is confirmed by

$\displaystyle\begin{split}\\{y^{\mathbf{n}},y^{\mathbf{n}_{1}}\\}y^{\mathbf{n}_{2}}+y^{\mathbf{n}_{1}}\\{y^{\mathbf{n}},y^{\mathbf{n}_{2}}\\}&=\\{\mathbf{n},\mathbf{n}_{1}\\}_{\Omega}y^{\mathbf{n}+\mathbf{n}_{1}}y^{\mathbf{n}_{2}}+y^{\mathbf{n}_{1}}\\{\mathbf{n},\mathbf{n}_{2}\\}_{\Omega}y^{\mathbf{n}+\mathbf{n}_{2}}\\\
&=\\{\mathbf{n},\mathbf{n}_{1}+\mathbf{n}_{2}\\}_{\Omega}y^{\mathbf{n}+\mathbf{n}_{1}+\mathbf{n}_{2}},\end{split}$
(7.19)

while the Jacobi identity is verified by the following cyclic expression:

$\displaystyle\\{y^{\mathbf{n}_{1}},\\{y^{\mathbf{n}_{2}},y^{\mathbf{n}_{3}}\\}\\}$
$\displaystyle=(\\{\mathbf{n}_{1},\mathbf{n}_{2}\\}_{\Omega}\\{\mathbf{n}_{2},\mathbf{n}_{3}\\}_{\Omega}-\\{\mathbf{n}_{2},\mathbf{n}_{3}\\}_{\Omega}\\{\mathbf{n}_{3},\mathbf{n}_{1}\\}_{\Omega})y^{\mathbf{n}_{1}+\mathbf{n}_{2}+\mathbf{n}_{3}}.$
(7.20)

Compare (7.20) with (7.7). As a special case of (7.18), we have

$\displaystyle\\{y_{i},y_{j}\\}=\omega_{ij}y_{i}y_{j}.$ (7.21)

Thus, the Poisson bracket on $\mathbb{Q}[[\mathbf{y}]]$ here is essentially
the same as the quadratic Poisson bracket (5.37) on $\mathcal{C}(\mathbf{y})$
under the identification of the matrix $\Omega$ in both situations.

We define an endomorphism $\tilde{X}_{\mathbf{n}}$ ($\mathbf{n}\in N^{+}$) of
$\mathbb{Q}[[\mathbf{y}]]$ by

$\displaystyle\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}^{\prime}}):=\\{y^{\mathbf{n}},y^{\mathbf{n}^{\prime}}\\}=\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}y^{\mathbf{n}+\mathbf{n}^{\prime}}.$
(7.22)

This is a _derivation_ ; namely, we have

$\displaystyle\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}_{1}}y^{\mathbf{n}_{2}})=\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}_{1}})y^{\mathbf{n}_{2}}+y^{\mathbf{n}_{1}}\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}_{2}})$
(7.23)

due to (7.19). Moreover, it preserves the Poisson bracket; namely, we have

$\displaystyle\tilde{X}_{\mathbf{n}}(\\{y^{\mathbf{n}_{1}},y^{\mathbf{n}_{2}})\\})=\\{\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}_{1}}),y^{\mathbf{n}_{2}}\\}+\\{y^{\mathbf{n}_{1}},\tilde{X}_{\mathbf{n}}(y^{\mathbf{n}_{2}})\\}$
(7.24)

due to the Jacobi identity.

Let $\mathrm{Der}(\mathbb{Q}[[\mathbf{y}]])$ be the _derivation Lie algebra_
of $\mathbb{Q}[[\mathbf{y}]]$, i.e., the Lie algebra consisting of all
derivations of $\mathbb{Q}[[\mathbf{y}]]$ [Jac79, §I.2].

######  Proposition 7.1 (Cf. [GHKK18, §1.1]).

The linear map

$\displaystyle\begin{matrix}\rho_{y}:&\widehat{g}&\longrightarrow&\mathrm{Der}(\mathbb{Q}[[\mathbf{y}]])\\\
&\displaystyle\sum_{\mathbf{n}\in
N_{+}}c_{\mathbf{n}}X_{\mathbf{n}}&\mapsto&\displaystyle\sum_{\mathbf{n}\in
N_{+}}c_{\mathbf{n}}\tilde{X}_{\mathbf{n}}\end{matrix}$ (7.25)

yields a Lie algebra homomorphism. Moreover, if $\Omega$ is nonsingular, it is
injective.

###### Proof.

Thanks to the Jacobi identity of the Poisson bracket, we have

$\displaystyle[\tilde{X}_{\mathbf{n}_{1}},\tilde{X}_{\mathbf{n}_{2}}](y^{\mathbf{n}})$
$\displaystyle=\\{y^{\mathbf{n}_{1}},\\{y^{\mathbf{n}_{2}},y^{\mathbf{n}}\\}\\}-\\{y^{\mathbf{n}_{2}},\\{y^{\mathbf{n}_{1}},y^{\mathbf{n}}\\}\\}$
$\displaystyle=\\{\\{y^{\mathbf{n}_{1}},y^{\mathbf{n}_{2}}\\},y^{\mathbf{n}}\\}$
$\displaystyle=\\{\mathbf{n}_{1},\mathbf{n}_{2}\\}_{\Omega}\\{y^{\mathbf{n}_{1}+\mathbf{n}_{2}},y^{\mathbf{n}}\\}$
$\displaystyle=\\{\mathbf{n}_{1},\mathbf{n}_{2}\\}_{\Omega}\tilde{X}_{\mathbf{n}_{1}+\mathbf{n}_{2}}(y^{\mathbf{n}}).$

Thus, it is a Lie algebra homomorphism. If $\Omega$ is nonsingular,
$\tilde{X}_{\mathbf{n}}$ ($\mathbf{n}\in N^{+}$) are linearly independent by
(7.22). Therefore, $\rho_{y}$ is injective. ∎

For any $X\in\widehat{\mathfrak{g}}$, let $\tilde{X}:={\rho}_{y}(X)$. We
define

$\displaystyle\mathrm{Exp}(\tilde{X}):=\sum_{k=0}^{\infty}\frac{1}{k!}\tilde{X}^{k}\in\mathrm{GL}(\mathbb{Q}[[\mathbf{y}]]).$
(7.26)

Since $\tilde{X}$ is a derivation, $\mathrm{Exp}(\tilde{X})$ is an algebra
automorphism of $\mathbb{Q}[[\mathbf{y}]]$ [Jac79, §I.2]. Then, we have the
following group homomorphism, where we abuse the symbol $\rho_{y}$, for
simplicity.

######  Proposition 7.2 (Cf. [GHKK18, §1.1]).

We have a group homomorphism

$\displaystyle\begin{matrix}\rho_{y}:&G&\rightarrow&\mathrm{Aut}(\mathbb{Q}[[\mathbf{y}]])\\\
&\exp(X)&\mapsto&\mathrm{Exp}(\tilde{X}).\end{matrix}$ (7.27)

Moreover, if $\Omega$ is nonsingular, it is injective.

###### Proof.

The map $\rho_{y}$ is a group homomorphism, because the exponential (7.26)
also satisfies the BCH formula. The injectivity of $\rho_{y}$ follows from the
one in Proposition 7.1. ∎

As we see in the next section, the representation $\rho_{y}$ is closely
related to the mutations of $y$-variables. So, we call it the _$y$
-representation_ of $G$.

####  7.4 Dilogarithm elements and mutations

Now we introduce the main player of Part III.

######  Definition 7.3 (Dilogarithm element).

For each $\mathbf{n}\in N^{+}$, we define

$\displaystyle\Psi[\mathbf{n}]:=\exp\Biggl{(}\,\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j^{2}}X_{j\mathbf{n}}\Biggr{)}\in
G_{\mathbf{n}_{0}}^{\parallel},$ (7.28)

where $\mathbf{n}_{0}\in N^{+}_{\mathrm{pr}}$ is the one such that
$\mathbf{n}=h\mathbf{n}_{0}$ for some $h\in\mathbb{Z}_{>0}$. We call
$\Psi[\mathbf{n}]$ the _dilogarithm element_ for $\mathbf{n}$.

The above definition is closely related to the following formula.

######  Proposition 7.4 (Cf. [GHKK18, Lemma 1.3]).

Under the $y$-representation $\rho_{y}$, $\Psi[\mathbf{n}]$ acts on
$\mathbb{Q}[[\mathbf{y}]]$ as

$\displaystyle\Psi[\mathbf{n}](y^{\mathbf{n}^{\prime}})=y^{\mathbf{n}^{\prime}}(1+y^{\mathbf{n}})^{\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}}\quad(\mathbf{n}^{\prime}\in\mathbb{Z}_{\geq
0}^{n}).$ (7.29)

###### Proof.

By (7.22), we have

$\displaystyle\Psi[\mathbf{n}](y^{\mathbf{n}^{\prime}})$
$\displaystyle=y^{\mathbf{n}^{\prime}}\exp\biggl{(}\,\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j^{2}}y^{j\mathbf{n}}\\{j\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}\biggr{)}$
$\displaystyle=y^{\mathbf{n}^{\prime}}\exp\biggl{(}\,\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j}y^{j\mathbf{n}}\biggr{)}^{\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}}$
$\displaystyle=y^{\mathbf{n}^{\prime}}(1+y^{\mathbf{n}})^{\\{\mathbf{n},\mathbf{n}^{\prime}\\}_{\Omega}}.$

∎

We may compare the above calculation with the following one (see also (1.3)):

$\displaystyle\begin{split}x\frac{d}{dx}\biggl{(}-\mathrm{Li}_{2}(-x)\biggr{)}&=x\frac{d}{dx}\biggl{(}\sum_{j=1}^{\infty}{\frac{(-1)^{j+1}}{j^{2}}}x^{j}\biggr{)}\\\
&=\sum_{j=1}^{\infty}{\frac{(-1)^{j+1}}{j}}x^{j}=\log(1+x).\end{split}$ (7.30)

Thus, the dilogarithm elements are regarded as a certain algebraic formulation
of (the exponential of) the Euler dilogarithm.

Next, let us compare the formula (7.29) with the automorphism
$\mathfrak{q}(s)$ in (5.23). Under the identification of the matrix $\Omega$
in both situations, we have

$\displaystyle\mathfrak{q}{(s)}=\rho_{y}(\Psi[\mathbf{c}^{+}_{k_{s}}(s)]^{-\varepsilon_{s}\delta_{k_{s}}}),$
(7.31)

where we regard $\mathfrak{q}(s)$ as acting on $\mathbb{Q}[[\mathbf{y}]]$.
More generally, the time-one flow (5.56) of the Hamiltonian
$\mathscr{H}_{\mathbf{n},a}=a\mathrm{Li}_{2}(-y^{\mathbf{n}})$ is identified
with the action of $\Psi[\mathbf{n}]^{-a}$.

In summary, mutations are described by the action of dilogarithm elements.
This is also regarded as an algebraic reformulation of the Hamiltonian picture
of mutations, where the time-one flow of the Hamiltonian is replaced with the
action of dilogarithm elements. The identification (7.31) serves as a key to
relate a cluster pattern/$Y$-pattern and a forthcoming _cluster scattering
diagram_.

There are several advantages of this algebraic formulation as follows:

  1. 1.

A composition of mutations is expressed as a product of dilogarithm elements
in $G$.

  2. 2.

In particular, a period of mutations is expressed as a relation among
dilogarithm elements in the group $G$.

  3. 3.

The infinite product of dilogarithm elements makes sense in $G$. This enables
us to go beyond mutations in a cluster pattern, where only finite compositions
of mutations are allowed.

  4. 4.

The group $G$ controls mutations locally and globally.

  5. 5.

The quantum case will be treated in a parallel way.

####  7.5 DI for dilogarithm elements

The periodicity condition (3.4) implies a relation of dilogarithm elements in
the group $G$.

######  Theorem 7.5 (DI for dilogarithm elements).

Suppose that the sequence of mutations (3.2) is _$\nu$ -periodic_. Then, the
following relation holds in $G$.

$\displaystyle\Psi[\mathbf{c}^{+}_{k_{P-1}}({P-1})]^{\varepsilon_{P-1}\delta_{k_{P-1}}}\cdots\Psi[\mathbf{c}^{+}_{k_{0}}(0)]^{\varepsilon_{0}\delta_{k_{0}}}=\mathrm{id}.$
(7.32)

###### Proof.

Case 1. Assume $\Upsilon(0)=\Upsilon$. Then, we are in the situation of (5.8).
By (6.75), we have

$\displaystyle\mathfrak{q}(P-1;0)=\mathrm{id}.$ (7.33)

Thus, thanks to the identification (7.31),

$\displaystyle
g=\Psi[\mathbf{c}^{+}_{k_{0}}(0)]^{-\varepsilon_{0}\delta_{k_{0}}}\cdots\Psi[\mathbf{c}^{+}_{k_{P-1}}({P-1})]^{-\varepsilon_{P-1}\delta_{k_{P-1}}}$
(7.34)

acts trivially on $\mathbb{Z}[[\mathbf{y}]]$. Assume that $B$ is nonsingular.
Then, by Proposition 7.2, $g$ is the identity element in $G$. Taking its
inverse, we obtain (7.32). If $B$ is singular, let $\overline{B}$ be the
principal extension in (6.68). Accordingly, the matrix $\Omega$ is extended as

$\displaystyle\overline{\Omega}=\begin{pmatrix}\Omega&-\Delta^{-1}\\\
\Delta^{-1}&O\\\ \end{pmatrix}$ (7.35)

By Proposition 6.9 and (6.70), we obtain the relation (7.32) in the group
$G_{\overline{\Omega}}$. Since $G=G_{\Omega}$ is a subgroup of
$G_{\overline{\Omega}}$, we obtain the relation (7.32) in $G$.

Case 2. Assume $\Upsilon(0)\neq\Upsilon$. By (2.15), we extend the sequence
(3.2) as

$\displaystyle\Upsilon\ {\buildrel{k^{\prime}_{0}}\over{\rightarrow}}\
\Upsilon^{\prime}(1)\ {\buildrel{k^{\prime}_{1}}\over{\rightarrow}}\ \cdots\
{\buildrel{k^{\prime}_{r}}\over{\rightarrow}}\ \Upsilon(0)\
{\buildrel{k_{0}}\over{\rightarrow}}\ \cdots\
{\buildrel{k_{P-1}}\over{\rightarrow}}\ \Upsilon(P)=\nu\Upsilon(0)\
{\buildrel{\nu(k^{\prime}_{r})}\over{\rightarrow}}\cdots\
{\buildrel{\nu(k^{\prime}_{0})}\over{\rightarrow}}\ \nu\Upsilon.$ (7.36)

Then, the extended sequence belongs to Case 1. Note that the $k_{t}$th
$c$-vector of $\Upsilon^{\prime}(t)$ coincides with the $\nu(k_{t})$th
$c$-vector of $\nu\Upsilon^{\prime}(t)$. Moreover, by (2.52), it is the
inverse of the $\nu(k_{t})$th $c$-vector of $\nu\Upsilon^{\prime}(t+1)$. Thus,
the corresponding relation (7.32) has the form

$\displaystyle
h^{-1}(\Psi[\mathbf{c}^{+}_{k_{P-1}}({P-1})]^{\varepsilon_{P-1}\delta_{k_{P-1}}}\cdots\Psi[\mathbf{c}^{+}_{k_{0}}(0)]^{\varepsilon_{0}\delta_{k_{0}}})h,=\mathrm{id}$
(7.37)

where $h\in G$ is the contribution of the mutations from $\Upsilon$ to
$\Upsilon(0)$. Therefore, we have (7.32). ∎

We regard the relation (7.32) as an algebraic version of the dilogarithm
identity (3.13).

######  Example 7.6.

Let us consider the pentagon periodicity of type $A_{2}$. We choose a skew-
symmetric decomposition of the initial exchange matrix $B$ in (2.39) as

$\displaystyle\Delta=I,\quad\Omega=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$
(7.38)

The $c$-vectors are found in (3.18). Then, the relation (7.32) is written as

$\displaystyle\Psi[\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{1}+\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{1}]^{-1}\Psi[\mathbf{e}_{2}]\Psi[\mathbf{e}_{1}]=\mathrm{id}.$
(7.39)

Equivalently, we have

$\displaystyle\Psi[\mathbf{e}_{2}]\Psi[\mathbf{e}_{1}]=\Psi[\mathbf{e}_{1}]\Psi[\mathbf{e}_{1}+\mathbf{e}_{2}]\Psi[\mathbf{e}_{2}].$
(7.40)

We call it the _pentagon relation_.

####  7.6 Pentagon relation

The pentagon relation in Example 7.6 holds in a more general situation.

######  Proposition 7.7.

Let $\mathbf{n}_{1},\mathbf{n}_{2}\in N^{+}$. The following relations hold in
$G$.

(a) (Commutative relation). If
$\\{\mathbf{n}_{2},\mathbf{n}_{1}\\}_{\Omega}=0$, then, for any
$c_{1},c_{2}\in\mathbb{Q}$, we have

$\displaystyle\Psi[\mathbf{n}_{2}]^{c_{2}}\Psi[\mathbf{n}_{1}]^{c_{1}}=\Psi[\mathbf{n}_{1}]^{c_{1}}\Psi[\mathbf{n}_{2}]^{c_{2}}.$
(7.41)

(b) (Pentagon relation [GHKK18, Ex. 1.14], [Nak23, Prop. III.1.14]). If
$\\{\mathbf{n}_{2},\mathbf{n}_{1}\\}_{\Omega}=c\neq 0$, we have

$\displaystyle\Psi[\mathbf{n}_{2}]^{1/c}\Psi[\mathbf{n}_{1}]^{1/c}=\Psi[\mathbf{n}_{1}]^{1/c}\Psi[\mathbf{n}_{1}+\mathbf{n}_{2}]^{1/c}\Psi[\mathbf{n}_{2}]^{1/c}.$
(7.42)

###### Proof.

(a). This is clear from (7.6). (b). Let us consider the rank 2 sublattice
$N^{\prime}$ of $N$ generated by $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$. By the
assumption $\\{\mathbf{n}_{2},\mathbf{n}_{1}\\}_{\Omega}\neq 0$, the form
$\\{\cdot,\cdot\\}_{\Omega}$ restricted on $N^{\prime}$ is nondegenerate. Let
$G^{\prime}$ be the subgroup of $G$ corresponding to $N^{\prime}$. Since the
relation (7.42) involves only elements in $G^{\prime}$, one can prove it by
the $y$-representation of $G^{\prime}$, which is faithful. The LHS is given by

$\displaystyle\quad\
\Psi[\mathbf{n}_{2}]^{1/c}\Psi[\mathbf{n}_{1}]^{1/c}(y^{\mathbf{n}})$
$\displaystyle=\Psi[\mathbf{n}_{2}]^{1/c}(y^{\mathbf{n}}(1+y^{\mathbf{n}_{1}})^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c})$
$\displaystyle=y^{\mathbf{n}}(1+y^{\mathbf{n}_{2}})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c}(1+y^{\mathbf{n}_{1}}(1+y^{\mathbf{n}_{2}}))^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c}.$

The RHS is given by

$\displaystyle\quad\
\Psi[\mathbf{n}_{1}]^{1/c}\Psi[\mathbf{n}_{1}+\mathbf{n}_{2}]^{1/c}\Psi[\mathbf{n}_{2}]^{1/c}(y^{\mathbf{n}})$
$\displaystyle=\Psi[\mathbf{n}_{1}]^{1/c}\Psi[\mathbf{n}_{1}+\mathbf{n}_{2}]^{1/c}(y^{\mathbf{n}}(1+y^{\mathbf{n}_{2}})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c})$
$\displaystyle=\Psi[\mathbf{n}_{1}]^{1/c}(y^{\mathbf{n}}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{\\{\mathbf{n}_{1}+\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c}$
$\displaystyle\qquad\times(1+y^{\mathbf{n}_{2}}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{-1})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c})$
$\displaystyle=\Psi[\mathbf{n}_{1}]^{1/c}(y^{\mathbf{n}}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}}+y^{\mathbf{n}_{2}})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c})$
$\displaystyle=y^{\mathbf{n}}(1+y^{\mathbf{n}_{1}})^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}}(1+y^{\mathbf{n}_{1}})^{-1})^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c}$
$\displaystyle\qquad\times(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}}(1+y^{\mathbf{n}_{1}})^{-1}+y^{\mathbf{n}_{2}}(1+y^{\mathbf{n}_{1}})^{-1})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c}$
$\displaystyle=y^{\mathbf{n}}(1+y^{\mathbf{n}_{1}}+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{\\{\mathbf{n}_{1},\mathbf{n}\\}_{\Omega}/c}(1+y^{\mathbf{n}_{2}})^{\\{\mathbf{n}_{2},\mathbf{n}\\}_{\Omega}/c}.$

Thus, two expressions agree. ∎

######  Example 7.8.

For the matrix $\Omega$ in (7.38), we have
$\\{\mathbf{e}_{2},\mathbf{e}_{1}\\}_{\Omega}=\omega_{21}=1$. Then, we recover
(7.40) by the specialization of the relation (7.42).

####  7.7 Ordering problem

Let us demonstrate that the pentagon relation (7.42) generates several
interesting relations in $G$, including infinite ones.

Concentrate on the rank 2 case, and let $\Omega$ be the one in (7.38). We
consider a possibly infinite product of powers of dilogarithm elements
$\Psi[\mathbf{n}]^{c}$ ($\mathbf{n}\in N^{+}$, $c\in\mathbb{Q}$). Such a
product is well-defined in $G$ if, for each degree $\ell$, there are only
finitely many terms in the product such that $\deg\mathbf{n}\leq\ell$. We say
that such a (possibly infinite) product is _ordered_ (resp. _anti-ordered_)
if, for any pair $\Psi[\mathbf{n}^{\prime}]^{c^{\prime}}$ and
$\Psi[\mathbf{n}]^{c}$ in the product, where the former appears left to the
latter, $\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}\leq 0$ (resp.
$\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}\geq 0$) holds. For example, in
the relation (7.40), the LHS is anti-ordered, while the RHS is ordered.

Then, the following problem arises.

######  Problem 7.9 (Ordering Problem).

Is it possible to order a given anti-ordered product to an ordered one only by
applying the commutative relation (7.41) and the pentagon relation (7.42)
repeatedly (possibly infinitely many times)?

The answer crucially depends on the exponents in the product. For example,
consider the anti-ordered product
$\Psi[\mathbf{e}_{2}]^{1/2}\Psi[\mathbf{e}_{1}]$. Then, it is easy to see that
it cannot be ordered by the pentagon relation (7.40) due to the fractional
power of $\Psi[\mathbf{e}_{2}]$.

The following fact holds.

######  Proposition 7.10 ([Nak23, Prop. III.5.4]).

For any positive integers $\delta_{1}$ and $\delta_{2}$, the anti-ordered
product $\Psi[\mathbf{e}_{2}]^{\delta_{2}}\Psi[\mathbf{e}_{1}]^{\delta_{1}}$
can be transformed into a possibly infinite ordered product only by applying
the commutative relation (7.41) and the pentagon relation (7.42) possibly
infinitely many times.

This is a special case of a more general result in the forthcoming Proposition
8.24 (Ordering Lemma). Later, the integers $\delta_{1}$ and $\delta_{1}$ will
be identified with the diagonal elements of the matrix
$\Delta=\mathrm{diag}(\delta_{1},\delta_{2})$ in the decomposition (5.1) of
$B$. Here we present some examples, which reveal a surprisingly rich structure
of the ordered product obtained in Proposition 7.10. Below, for
$\mathbf{n}=(n_{1},n_{2})$, we write $\Psi[\mathbf{n}]^{c}$ as
$[n_{1},n_{2}]^{c}$ (or column-wise). By (7.38), we have
$\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}=n_{1}n^{\prime}_{2}-n_{2}n^{\prime}_{1}$.
Thus, the condition $\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}\leq 0$ is
equivalent to $n^{\prime}_{1}/n^{\prime}_{2}\geq n_{1}/n_{2}$, where
$n_{1}/n_{2}=\infty$ if $n_{2}=0$.

######  Example 7.11 (Finite type: $\delta_{1}\delta_{2}\leq 3$).

(a) Type $A_{2}$. Let $(\delta_{1},\delta_{2})=(1,1)$. This is already covered
in (7.40). In the current notation, it is written as

$\displaystyle\begin{split}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 0\end{bmatrix}&=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}.\end{split}$ (7.43)

Observe that the vectors appearing in the RHS are identified with the positive
roots $\alpha_{1}$, $\alpha_{1}+\alpha_{2}$, $\alpha_{2}$ of the root system
of type $A_{2}$. See Figure 7.1.

(b) Type $B_{2}$. Here we consider the case $(\delta_{1},\delta_{2})=(1,2)$.
By applying the pentagon relation (7.42) with $c=1$ repeatedly for adjacent
pairs $\mathbf{n}^{\prime}$, $\mathbf{n}$ with
$\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}=1$, we have

$\displaystyle\begin{split}\begin{bmatrix}0\\\
1\end{bmatrix}^{2}\begin{bmatrix}1\\\ 0\end{bmatrix}&=\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}\begin{bmatrix}0\\\ 1\end{bmatrix}=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}\\\ &=\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}^{2}\begin{bmatrix}1\\\ 2\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}^{2}.\end{split}$ (7.44)

This relation coincides with the equality (7.32) for the periodicity of the
free $Y$-pattern of type $B_{2}$ in (3.23). The vectors appearing in the RHS
are identified with the positive roots $\alpha_{1}$, $\alpha_{1}+\alpha_{2}$,
$\alpha_{1}+2\alpha_{2}$, $\alpha_{2}$ of the root system of type $B_{2}$.
Moreover, the exponent is 2 if and only if the corresponding root is a short
one.

(c) Type $G_{2}$. Let $(\delta_{1},\delta_{2})=(1,3)$. In the same way as
above, we have

$\displaystyle\begin{split}\begin{bmatrix}0\\\
1\end{bmatrix}^{3}\begin{bmatrix}1\\\ 0\end{bmatrix}&=\begin{bmatrix}0\\\
1\end{bmatrix}\biggl{(}\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}^{2}\begin{bmatrix}1\\\ 2\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}^{2}\biggr{)}\\\ &=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
2\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\begin{bmatrix}1\\\
2\end{bmatrix}^{2}\begin{bmatrix}1\\\ 3\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}^{3}\\\ &=\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}^{3}\begin{bmatrix}2\\\ 3\end{bmatrix}\begin{bmatrix}1\\\
2\end{bmatrix}^{3}\begin{bmatrix}1\\\ 3\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}^{3}.\end{split}$ (7.45)

This relation coincides with the equality (7.32) for the periodicity of the
free $Y$-pattern of type $G_{2}$ in (3.28). The vectors appearing in the RHS
are identified with the positive roots $\alpha_{1}$, $\alpha_{1}+\alpha_{2}$,
$2\alpha_{1}+3\alpha_{2}$, $\alpha_{1}+2\alpha_{2}$, $\alpha_{1}+3\alpha_{2}$,
$\alpha_{2}$ of the root system of type $G_{2}$. Moreover, the exponent is 3
if and only if the corresponding root is a short one.

$\alpha_{1}$$\alpha_{1}+\alpha_{2}$$\alpha_{2}$$\textstyle{A_{2}}$

$\alpha_{1}$$\alpha_{1}+2\alpha_{2}$$\alpha_{1}+\alpha_{2}$$\alpha_{2}$$\textstyle{B_{2}}$

$\alpha_{1}$$\alpha_{1}+3\alpha_{2}$$2\alpha_{1}+3\alpha_{2}$$\alpha_{1}+2\alpha_{2}$$\alpha_{1}+\alpha_{2}$$\alpha_{2}$$\textstyle{G_{2}}$

Figure 7.1: Rank 2 root systems of finite type.

######  Example 7.12 (Affine type: $\delta_{1}\delta_{2}=4$).

(a) Type $A_{1}^{(1)}$. Let $(\delta_{1},\delta_{2})=(2,2)$. By using the
pentagon relation (7.42) with $c=1$, we obtain the equality

$\displaystyle\begin{bmatrix}0\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
0\end{bmatrix}^{2}$ $\displaystyle=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
2\end{bmatrix}\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}^{2}\begin{bmatrix}1\\\ 2\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}^{2}.$ (7.46)

The expression in the RHS is not yet ordered. To make it ordered, we need to
interchange $[1,2]$ and $[1,0]$ in the middle, where
$\\{\mathbf{n}^{\prime},\mathbf{n}\\}_{\Omega}=2$. As the lowest
approximation, we consider modulo $G^{>3}$. Then, $[1,2]$ and $[1,0]$ commute,
and we have, modulo $G^{>3}$,

$\displaystyle\begin{bmatrix}0\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
0\end{bmatrix}^{2}$ $\displaystyle\equiv\begin{bmatrix}1\\\
0\end{bmatrix}^{2}\begin{bmatrix}2\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
1\end{bmatrix}^{4}\begin{bmatrix}1\\\ 2\end{bmatrix}^{2}\begin{bmatrix}0\\\
1\end{bmatrix}^{2}.$ (7.47)

To proceed to a higher degree, we now apply the pentagon relation (7.42) with
$c=2$ to the pair $[1,2]^{1/2}$ and $[1,0]^{1/2}$ in (7.46), which are
obtained by factorizing the elements $[1,2]$ and $[1,0]$. To clarify the
structure among fractional powers, we use the temporal notation
$[n_{1},n_{2}]_{2}:=[n_{1},n_{2}]^{1/2}$. Then, we have

$\displaystyle\begin{bmatrix}1\\\ 2\end{bmatrix}\begin{bmatrix}1\\\
0\end{bmatrix}=\begin{bmatrix}1\\\ 2\end{bmatrix}_{2}^{2}\begin{bmatrix}1\\\
0\end{bmatrix}_{2}^{2}$ $\displaystyle=\begin{bmatrix}1\\\
0\end{bmatrix}_{2}\begin{bmatrix}2\\\
2\end{bmatrix}_{2}^{2}\begin{bmatrix}3\\\
4\end{bmatrix}_{2}\begin{bmatrix}1\\\ 0\end{bmatrix}_{2}\begin{bmatrix}2\\\
2\end{bmatrix}_{2}^{2}\begin{bmatrix}3\\\
4\end{bmatrix}_{2}\begin{bmatrix}1\\\ 2\end{bmatrix}_{2}^{2}.$ (7.48)

This is parallel with (7.46). As the next approximation, we consider modulo
$G^{>7}$. Then, $[3,4]$ and $[1,0]$ commute, and we have, modulo $G^{>7}$,

$\displaystyle\begin{bmatrix}1\\\ 2\end{bmatrix}\begin{bmatrix}1\\\
0\end{bmatrix}$ $\displaystyle\equiv\begin{bmatrix}1\\\
0\end{bmatrix}_{2}^{2}\begin{bmatrix}3\\\
2\end{bmatrix}_{2}^{2}\begin{bmatrix}2\\\
2\end{bmatrix}_{2}^{4}\begin{bmatrix}3\\\
4\end{bmatrix}_{2}^{2}\begin{bmatrix}1\\\
2\end{bmatrix}_{2}^{2}=\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}3\\\
2\end{bmatrix}\begin{bmatrix}2\\\ 2\end{bmatrix}^{2}\begin{bmatrix}3\\\
4\end{bmatrix}\begin{bmatrix}1\\\ 2\end{bmatrix}.$ (7.49)

Then, we plug it into (7.46) and apply the pentagon relation with $c=1$ again,
and we have, modulo $G^{>7}$,

$\displaystyle\begin{bmatrix}0\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
0\end{bmatrix}^{2}$ $\displaystyle\equiv\begin{bmatrix}1\\\
0\end{bmatrix}^{2}\begin{bmatrix}2\\\ 1\end{bmatrix}^{2}\begin{bmatrix}3\\\
2\end{bmatrix}^{2}\begin{bmatrix}4\\\ 3\end{bmatrix}^{2}\begin{bmatrix}1\\\
1\end{bmatrix}^{4}\begin{bmatrix}2\\\ 2\end{bmatrix}^{2}\begin{bmatrix}3\\\
4\end{bmatrix}^{2}\begin{bmatrix}2\\\ 3\end{bmatrix}^{2}\begin{bmatrix}1\\\
2\end{bmatrix}^{2}\begin{bmatrix}0\\\ 1\end{bmatrix}^{2}.$ (7.50)

By repeating this procedure modulo $G^{>2^{\ell}-1}$, the relations converge
to

$\displaystyle\begin{bmatrix}0\\\ 1\end{bmatrix}^{2}\begin{bmatrix}1\\\
0\end{bmatrix}^{2}$ $\displaystyle=\begin{bmatrix}1\\\
0\end{bmatrix}^{2}\begin{bmatrix}2\\\ 1\end{bmatrix}^{2}\begin{bmatrix}3\\\
2\end{bmatrix}^{2}\cdots\prod_{j=0}^{\infty}\begin{bmatrix}2^{j}\\\
2^{j}\end{bmatrix}^{2^{2-j}}\cdots\begin{bmatrix}2\\\
3\end{bmatrix}^{2}\begin{bmatrix}1\\\ 2\end{bmatrix}^{2}\begin{bmatrix}0\\\
1\end{bmatrix}^{2},$ (7.51)

where an infinite product emerges in the RHS. Also, fractional powers appear
due to the factorization in the procedure. A complete proof of the relation
(7.51) by the pentagon relation was given by Matsushita [Mat21].

(b) Type $A_{2}^{(2)}$. Let $(\delta_{1},\delta_{2})=(1,4)$. The situation is
in parallel with $A_{1}^{(1)}$ though a little more complicated. The resulting
relation is as follows:

$\displaystyle\begin{split}&\begin{bmatrix}0\\\
1\end{bmatrix}^{4}\begin{bmatrix}1\\\ 0\end{bmatrix}=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}^{4}\begin{bmatrix}3\\\
4\end{bmatrix}\begin{bmatrix}2\\\ 3\end{bmatrix}^{4}\begin{bmatrix}5\\\
8\end{bmatrix}\begin{bmatrix}3\\\ 5\end{bmatrix}^{4}\cdots\\\ &\hskip
60.0pt\times\begin{bmatrix}1\\\
2\end{bmatrix}^{6}\prod_{j=1}^{\infty}\begin{bmatrix}2^{j}\\\
2^{j+1}\end{bmatrix}^{2^{2-j}}\cdots\begin{bmatrix}5\\\
12\end{bmatrix}\begin{bmatrix}2\\\ 5\end{bmatrix}^{4}\begin{bmatrix}3\\\
8\end{bmatrix}\begin{bmatrix}1\\\ 3\end{bmatrix}^{4}\begin{bmatrix}1\\\
4\end{bmatrix}\begin{bmatrix}0\\\ 1\end{bmatrix}^{4}.\end{split}$ (7.52)

Again, a complete proof of the equality (7.52) by the pentagon relation was
given by [Mat21].

######  Example 7.13 (Non-affine infinite type: $\delta_{1}\delta_{2}\geq 5$).

The explicit expression of the ordered product is not yet known, but it is
quite intriguing. There is a program to compute the ordered product written
for SageMath [Sag] whose source code is available in [Nak23, III. Appendix A].
The program can calculate the ordered product up to a given degree; however,
in practice, once some exponent exceeds 1,000, it is difficult to continue
calculation due to the limitation of time. As an example, for
$(\delta_{1},\delta_{2})=(1,5)$, the product has the following form modulo
$G^{>16}$:

$\displaystyle\begin{split}\begin{bmatrix}0\\\
1\end{bmatrix}^{5}\begin{bmatrix}1\\\ 0\end{bmatrix}&\equiv\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}^{5}\begin{bmatrix}4\\\
5\end{bmatrix}\begin{bmatrix}3\\\
4\end{bmatrix}^{5}\biggl{\\{}\begin{bmatrix}5\\\
7\end{bmatrix}^{10}\begin{bmatrix}2\\\ 3\end{bmatrix}^{10}\begin{bmatrix}4\\\
6\end{bmatrix}^{10}\begin{bmatrix}6\\\ 9\end{bmatrix}^{15}\begin{bmatrix}5\\\
8\end{bmatrix}^{85}\begin{bmatrix}3\\\ 5\end{bmatrix}^{27}\\\
&\quad\times\begin{bmatrix}6\\\ 10\end{bmatrix}^{302}\begin{bmatrix}4\\\
7\end{bmatrix}^{85}\begin{bmatrix}5\\\ 9\end{bmatrix}^{295}\begin{bmatrix}1\\\
2\end{bmatrix}^{10}\begin{bmatrix}2\\\ 4\end{bmatrix}^{10}\begin{bmatrix}3\\\
6\end{bmatrix}^{15}\begin{bmatrix}4\\\ 8\end{bmatrix}^{45}\begin{bmatrix}5\\\
10\end{bmatrix}^{130}\\\ &\quad\times\begin{bmatrix}5\\\
11\end{bmatrix}^{1095}\begin{bmatrix}4\\\
9\end{bmatrix}^{295}\begin{bmatrix}3\\\ 7\end{bmatrix}^{85}\begin{bmatrix}2\\\
5\end{bmatrix}^{27}\begin{bmatrix}4\\\
10\end{bmatrix}^{302}\begin{bmatrix}3\\\
8\end{bmatrix}^{85}\begin{bmatrix}4\\\
11\end{bmatrix}^{295}\begin{bmatrix}1\\\ 3\end{bmatrix}^{10}\\\
&\quad\times\begin{bmatrix}2\\\ 6\end{bmatrix}^{10}\begin{bmatrix}3\\\
9\end{bmatrix}^{15}\begin{bmatrix}4\\\ 12\end{bmatrix}^{45}\begin{bmatrix}3\\\
10\end{bmatrix}^{27}\begin{bmatrix}2\\\
7\end{bmatrix}^{10}\biggr{\\}}\begin{bmatrix}3\\\
11\end{bmatrix}^{5}\begin{bmatrix}1\\\ 4\end{bmatrix}^{5}\begin{bmatrix}1\\\
5\end{bmatrix}\begin{bmatrix}0\\\ 1\end{bmatrix}^{5}.\end{split}$ (7.53)

The behavior of the terms outside the bracket are completely determined by
combining the results by GHKK [GHKK18] and Reading [Rea14]. Namely, they are

$\displaystyle[\mathbf{c}_{0}][\mathbf{c}_{1}]^{5}[\mathbf{c}_{2}][\mathbf{c}_{3}]^{5}\cdots\\{\cdots\\}\cdots[\mathbf{c}^{\prime}_{3}][\mathbf{c}^{\prime}_{2}]^{5}[\mathbf{c}^{\prime}_{1}][\mathbf{c}^{\prime}_{0}]^{5},$
(7.54)

where

$\displaystyle\mathbf{c}_{0}$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix},\ \mathbf{c}_{1}=\begin{pmatrix}1\\\ 1\end{pmatrix},\
\mathbf{c}_{2}=\begin{pmatrix}4\\\ 5\end{pmatrix},\
\mathbf{c}_{3}=\begin{pmatrix}3\\\ 4\end{pmatrix},\
\mathbf{c}_{4}=\begin{pmatrix}11\\\ 15\end{pmatrix},\,\cdots,$ (7.55)
$\displaystyle\mathbf{c}^{\prime}_{0}$ $\displaystyle=\begin{pmatrix}0\\\
1\end{pmatrix},\ \mathbf{c}^{\prime}_{1}=\begin{pmatrix}1\\\ 5\end{pmatrix},\
\mathbf{c}^{\prime}_{2}=\begin{pmatrix}1\\\ 4\end{pmatrix},\
\mathbf{c}^{\prime}_{3}=\begin{pmatrix}4\\\ 15\end{pmatrix},\
\mathbf{c}^{\prime}_{4}=\begin{pmatrix}3\\\ 11\end{pmatrix},\,\cdots.$ (7.56)

These vectors are uniquely determined from the recursion relations

$\displaystyle\mathbf{c}_{i}=\begin{cases}5\mathbf{c}_{i-1}-\mathbf{c}_{i-2}&\text{$i$:
even},\\\ \mathbf{c}_{i-1}-\mathbf{c}_{i-2}&\text{$i$:
odd},\end{cases}\quad\mathbf{c}^{\prime}_{i}=\begin{cases}\mathbf{c}^{\prime}_{i-1}-\mathbf{c}^{\prime}_{i-2}&\text{$i$:
even},\\\ 5\mathbf{c}^{\prime}_{i-1}-\mathbf{c}^{\prime}_{i-2}&\text{$i$:
odd}.\end{cases}$ (7.57)

The ratios of the components of these vectors have the limits

$\displaystyle\mathbf{c}_{i}$ $\displaystyle=\begin{pmatrix}n_{1;i}\\\
n_{2;i}\end{pmatrix},\quad\lim_{i\rightarrow\infty}\frac{n_{1;i}}{n_{2;i}}=\frac{2}{5-\sqrt{5}}=0.723\dots,$
(7.58) $\displaystyle\mathbf{c}^{\prime}_{i}$
$\displaystyle=\begin{pmatrix}n^{\prime}_{1;i}\\\
n^{\prime}_{2;i}\end{pmatrix},\quad\lim_{i\rightarrow\infty}\frac{n^{\prime}_{1;i}}{n^{\prime}_{2;i}}=\frac{2}{5+\sqrt{5}}=0.276\dots.$
(7.59)

In contrast, inside the bracket in (7.53), every $\mathbf{n}\in N^{+}$ with
$\deg(\mathbf{n})\leq 16$ such that

$\displaystyle\frac{2}{5+\sqrt{5}}<\frac{n_{1}}{n_{2}}<\frac{2}{5-\sqrt{5}}$
(7.60)

appears. It is recently shown by Gräfnitz and Luo [GL23] that this is true for
any $\mathbf{n}$ satisfying (7.60). However, the explicit description of the
exponents is known only partially (e.g., [Aka23]).

The meaning of the relations in the above examples will be clarified in the
next chapter in view of cluster scattering diagrams.

Notes

The class of Lie algebras and their exponential groups in this chapter were
introduced in [GS11, KS08, KS14] in more generality in the study of mirror
symmetry and wall-crossing structures for the Donaldson-Thomas invariants. The
group $G$ here appeared in [GHKK18] in the disguised form of the
$x$-representation. The dilogarithm elements and the pentagon relation
appeared in [KS08, KS14] in a slightly different setup, and also in [GHKK18]
in the disguised form of the $x$-representation. Here we follow the
presentation in the companion monograph [Nak23]. Theorem 7.5 is new here, but
the rank 2 examples of DIs appeared in [Nak23]. The ordering problem is
closely related to the wall-crossing formula in [KS08].

###  Chapter 8 Quick course on cluster scattering diagrams

Following GHKK [GHKK18], we introduce an algebraic and geometrical object
called a _cluster scattering diagram (CSD)_. For any cluster pattern, the fan
constructed from its $G$-matrices are embedded in the corresponding CSD.
Therefore, it is regarded as an extension of a cluster pattern. It turns out
that the dilogarithm elements and the pentagon relation play fundamental roles
in CSDs. Our presentation is minimal without proofs. The proofs and details
are found in the companion monograph [Nak23].

####  8.1 Scattering diagrams

Continuing from the previous chapter, we fix a given skew-symmetric rational
matrix $\Omega$. Let $G=G_{\Omega}$ be the group constructed in Section 7.2.

Following [GHKK18], we introduce several notions related to scattering
diagrams. For the lattice $N=\mathbb{Z}^{n}$, let

$\displaystyle M:=\mathrm{Hom}(N,\mathbb{Z}),\quad
M_{\mathbb{R}}:=M\otimes\mathbb{R}.$ (8.1)

Let $\langle\mathbf{n},m\rangle$ be the canonical paring $N\times
M\rightarrow\mathbb{Z}$ or its extension $N\times
M_{\mathbb{R}}\rightarrow\mathbb{R}$, depending on the context. For each
$\mathbf{n}\in N$, $\mathbf{n}\neq\mathbf{0}$, let

$\displaystyle\mathbf{n}^{\perp}:=\\{z\in
M_{\mathbb{R}}\mid\langle\mathbf{n},z\rangle=0\\}$ (8.2)

be the hyperplane in $M_{\mathbb{R}}$ which is orthogonal to $\mathbf{n}$. For
any $m_{1},\,\dots,\,m_{k}\in M$, the positive span

$\displaystyle\sigma(m_{1},\dots,m_{k}):=\biggl{\\{}\sum_{i=1}^{k}a_{i}m_{i}\,\bigg{|}\,a_{i}\in\mathbb{R}_{\geq
0}\biggr{\\}}\subset M_{\mathbb{R}}$ (8.3)

is called a _convex rational cone_. We simple call it a _cone_. We also set
$\sigma(\emptyset):=\\{0\\}$, which is also a cone. A cone $\sigma$ is
_strongly convex_ if $\sigma\cap(-\sigma)=\\{0\\}$.

######  Definition 8.1 (Wall/Scattering diagram).

Let $G=G_{\Omega}$ be the group as above.

(a). A _wall_ $\mathbf{w}=(\mathfrak{d},g)_{\mathbf{n}}$ is a triplet such
that $\mathbf{n}\in N^{+}_{\mathrm{pr}}$, a cone
$\mathfrak{d}\subset\mathbf{n}^{\perp}$ of dimension $n-1$, and $g\in
G_{\mathbf{n}}^{\parallel}$. We call $\mathbf{n}$, $\mathfrak{d}$, $g$ the
_normal vector_ , the _support_ , and the _wall element_ of $\mathbf{w}$,
respectively.

(b). A _scattering diagram_ $\mathfrak{D}$ is a collection of walls
$\\{\mathbf{w}_{\lambda}=(\mathfrak{d}_{\lambda},g_{\lambda})_{\mathbf{n}_{\lambda}}\\}_{\lambda\in\Lambda}$
with a (possibly infinite) index set $\Lambda$ satisfying the following
_finiteness condition_ :

  * •

For any integer $\ell\geq 1$, there are only finitely many walls such that
$\pi_{\ell}(g_{\lambda})\neq\mathrm{id}$, where $\pi_{\ell}:G\rightarrow
G^{\leq\ell}$ is the canonical projection (7.16).

The group $G$ is called the _structure group_ of $\mathfrak{D}$.

(c). For each integer $\ell\geq 1$, the subcollection $\mathfrak{D}_{\ell}$
consisting of the walls with $\pi_{\ell}(g_{\lambda})\neq\mathrm{id}$ is
called the _reduction_ of $\mathfrak{D}$ at degree $\ell$. By the finiteness
condition, there are only finitely may walls of $\mathfrak{D}_{\ell}$.

We give more related notions.

######  Definition 8.2 (Support/Singular locus).

For a scattering diagram
$\mathfrak{D}=\\{\mathbf{w}_{\lambda}=(\mathfrak{d}_{\lambda},g_{{\lambda}})_{\mathbf{n}_{\lambda}}\\}_{\lambda\in\Lambda}$,
the _support_ and the _singular locus_ of $\mathfrak{D}$ are defined by

$\displaystyle\mathrm{Supp}(\mathfrak{D})$
$\displaystyle:=\bigcup_{\lambda\in\Lambda}\mathfrak{d}_{\lambda},$ (8.4)
$\displaystyle\mathrm{Sing}(\mathfrak{D})$
$\displaystyle:=\bigcup_{\lambda\in\Lambda}\partial\mathfrak{d}_{\lambda}\cup\bigcup_{\scriptstyle\lambda,\,\lambda^{\prime}\in\Lambda\atop\scriptstyle\dim\mathfrak{d}_{\lambda}\cap\mathfrak{d}_{\lambda^{\prime}}\,=\,n-2}\mathfrak{d}_{\lambda}\cap\mathfrak{d}_{\lambda^{\prime}}.$
(8.5)

######  Definition 8.3 (Admissible curve).

A curve $\gamma:[0,1]\rightarrow M_{\mathbb{R}}$ is _admissible_ for a
scattering diagram $\mathfrak{D}$ if it satisfies the following conditions:

  * (1).

The endpoints of $\gamma$ are in
$M_{\mathbb{R}}\setminus\mathrm{Supp}(\mathfrak{D})$.

  * (2).

It is a smooth curve and intersects $\mathrm{Supp}(\mathfrak{D})$
transversally.

  * (3).

$\gamma$ does not intersect $\mathrm{Sing}(\mathfrak{D})$.

######  Definition 8.4 (Path-ordered product).

Let $\mathfrak{D}$ be any scattering diagram. For any admissible curve
$\gamma$ for $\mathfrak{D}$, we define an element
$\mathfrak{p}_{\gamma,\mathfrak{D}}\in G$ as follows: For each integer
$\ell>0$, let $\mathfrak{D}_{\ell}$ be the reduction of $\mathfrak{D}$ at
degree $\ell$. Suppose that $\gamma=\gamma(t)$ crosses walls
$\mathbf{w}_{i}=(\mathfrak{d}_{i},g_{i})_{\mathbf{n}_{i}}$ ($i=1$, …, $s$) of
$\mathfrak{D}_{\ell}$ in this order at $t=t_{i}$ with

$\displaystyle 0<t_{1}\leq t_{2}\leq\cdots\leq t_{s}<1.$ (8.6)

Since $\gamma(t_{i})\notin\mathrm{Sing}(\mathfrak{D})$, when $\gamma$ crosses
multiple walls at a time, these walls have a common normal vector. We define
the _intersection sign_ $\epsilon_{i}$ ($i=1$, …, $s$) by

$\displaystyle\epsilon_{i}=\begin{cases}1&\langle
n_{i},\gamma^{\prime}(t_{i})\rangle<0,\\\ -1&\langle
n_{i},\gamma^{\prime}(t_{i})\rangle>0,\\\ \end{cases}$ (8.7)

where $\gamma^{\prime}(t_{i})$ is the velocity vector of $\gamma(t)$ at
$t_{i}$. Then, we define

$\displaystyle\mathfrak{p}_{\gamma,\mathfrak{D}_{\ell}}:=$ $\displaystyle\
g_{s}^{\epsilon_{s}}\cdots
g_{1}^{\epsilon_{1}},\quad\mathfrak{p}_{\gamma,\mathfrak{D}}:=\lim_{\ell\rightarrow\infty}\mathfrak{p}_{\gamma,\mathfrak{D}_{\ell}}\in
G.$ (8.8)

We call $\mathfrak{p}_{\gamma,\mathfrak{D}}$ the _path-ordered product_ (of
wall elements in $\mathfrak{D}$) along $\gamma$.

Note that $\mathfrak{p}_{\gamma,\mathfrak{D}}$ only depends on the homotopy
class of $\gamma$ in $M_{\mathbb{R}}\setminus\mathrm{Sing}(\mathfrak{D})$.

######  Definition 8.5 (Equivalence).

Two scattering diagrams $\mathfrak{D}$ and $\mathfrak{D}^{\prime}$ with a
common structure group $G$ are _equivalent_ if, for any curve $\gamma$ that is
admissible for both $\mathfrak{D}$ and $\mathfrak{D}^{\prime}$, the equality
$\mathfrak{p}_{\gamma,\mathfrak{D}}=\mathfrak{p}_{\gamma,\mathfrak{D}^{\prime}}$
holds.

For a given scattering diagram $\mathfrak{D}$, one can obtain infinitely many
scattering diagrams that are equivalent to $\mathfrak{D}$ by joining and
splitting the supports and wall elements and also by adding and removing the
walls with the trivial wall element $g=\mathrm{id}$ (trivial walls) possibly
infinitely many times.

Finally, we introduce a crucial notion for scattering diagrams.

######  Definition 8.6 (Consistency).

A scattering diagram $\mathfrak{D}$ is _consistent_ if for any admissible
curve $\gamma$ for $\mathfrak{D}$, the associated path-ordered product
$\mathfrak{p}_{\gamma,\mathfrak{D}}$ depends only on the endpoints of
$\gamma$. In other words, $\mathfrak{D}$ is consistent if

$\displaystyle\mathfrak{p}_{\gamma,\mathfrak{D}}=\mathrm{id}$ (8.9)

for any admissible loop (i.e., closed admissible curve) $\gamma$ for
$\mathfrak{D}$.

Let

$\displaystyle\begin{split}\mathcal{C}^{+}&:=\\{z\in
M_{\mathbb{R}}\mid\langle\mathbf{e}_{i},z\rangle\geq 0\ \text{for any
$i$}\\},\\\ \mathcal{C}^{-}&:=\\{z\in
M_{\mathbb{R}}\mid\langle\mathbf{e}_{i},z\rangle\leq 0\ \text{for any
$i$}\\}.\end{split}$ (8.10)

It is easy to see that the support of a scattering diagram $\mathfrak{D}$
intersects $\mathcal{C}^{\pm}$ only in the boundary of $\mathcal{C}^{\pm}$.
For a given consistent scattering diagram $\mathfrak{D}$, let $\gamma_{+-}$ be
any admissible path with the initial point in $\mathrm{Int}(\mathcal{C}^{+})$
and the final point in $\mathrm{Int}(\mathcal{C}^{-})$. Then, thanks to the
consistency of $\mathfrak{D}$, the unique element
$g(\mathfrak{D}):=\mathfrak{p}_{\gamma_{+-},\mathfrak{D}}\in G$ is assigned to
$\mathfrak{D}$.

The following fact reveals the intrinsic nature of consistent scattering
diagrams.

######  Theorem 8.7 ([KS14, Thm. 2.1.6], [GHKK18, Thm. 1.17]).

The assignment $\mathfrak{D}\mapsto g(\mathfrak{D})$ gives a one-to-one
correspondence between the consistent scattering diagrams up to equivalence
and the elements in $G$.

######  Definition 8.8.

We say that a scattering diagram $\mathfrak{D}$ is with _minimal support_ if
its support $\mathrm{Supp}(\mathfrak{D})$ is the minimal set among the
supports of all equivalent scattering diagrams to $\mathfrak{D}$.

The proof of Theorem 8.7 also ensures the following fact.

######  Proposition 8.9 (e.g., [Nak23, Prop. III.2.25]).

For any consistent scattering diagram $\mathfrak{D}$, there exists a (not
unique) scattering diagram with minimal support that is equivalent to
$\mathfrak{D}$.

####  8.2 Cluster scattering diagrams

Next, we will define a unique consistent scattering diagram $\mathfrak{D}(B)$
(up to equivalence and isomorphism) associated with a skew-symmetrizable
integer matrix $B$ called a _cluster scattering diagram_.

First, we choose an arbitrary skew-symmetric decomposition (5.1) of $B$. We
then construct the group $G=G_{\Omega}$ for the matrix $\Omega$ as in Section
7.1.

Let us introduce an abelian group homomorphism

$\displaystyle\begin{matrix}p^{*}\colon&N&\longrightarrow&M_{\mathbb{R}}\\\
&\mathbf{n}&\mapsto&\\{\cdot,\mathbf{n}\\}_{\Omega}.\end{matrix}$ (8.11)

In particular,

$\displaystyle\langle\mathbf{n},p^{*}(\mathbf{n})\rangle=\\{\mathbf{n},\mathbf{n}\\}_{\Omega}=0.$
(8.12)

Thus,

$\displaystyle p^{*}(\mathbf{n})\in\mathbf{n}^{\perp}$ (8.13)

holds for any $\mathbf{n}\neq\mathbf{0}$.

######  Definition 8.10.

A wall $\mathbf{w}=(\mathfrak{d},g)_{\mathbf{n}}$ is said to be _incoming_
(resp. _outgoing_) if $p^{*}(\mathbf{n})\in\mathfrak{d}$ (resp. otherwise).
See figure 8.1.

$\bullet$$0$$n^{\perp}$$\bullet$$p^{*}(n)$incoming wall

$\small\bullet$$\bullet$outgoing wall

Figure 8.1: Incoming and outgoing walls.

For example, a wall $\mathbf{w}=(\mathbf{n}^{\perp},g)_{\mathbf{n}}$ is
incoming for any $\mathbf{n}\in N^{+}_{\mathrm{pr}}$ by (8.13).

So far, we only use the data $\Omega$ in the decomposition (5.1). The
following definition involves $\Delta$ and $\Omega$ together.

######  Definition 8.11.

For a skew-symmetrizable matrix $B$ with a skew-symmetric decomposition (5.1),
a _cluster scattering diagram_ (CSD) for $B$ is a consistent scattering
diagram $\mathfrak{D}$ satisfying the following properties:

  * •

The structure group of $\mathfrak{D}$ is $G_{\Omega}$.

  * •

The incoming walls
$\mathbf{w}_{i}:=(\mathbf{e}_{i}^{\perp},\Psi[\mathbf{e}_{i}]^{\delta_{i}})_{\mathbf{e}_{i}}$
($i=1,\dots,n$) belong to $\mathfrak{D}$, where $\delta_{i}$ is the $i$th
diagonal entry of $\Delta$.

  * •

All other walls of $\mathfrak{D}$ are outgoing.

Now we present the first fundamental result on CSDs.

######  Theorem 8.12 ([GHKK18, Thm. 1.21]).

For any skew-symmetrizable matrix $B$ with a skew-symmetric decomposition
(5.1), there exists a unique CSD $\mathfrak{D}(B)$ up to equivalence.

The decomposition (5.1) is not unique at all. For example, one many rescale
$\Delta^{\prime}=\lambda^{-1}\Delta$ and $\Omega^{\prime}=\lambda\Omega$ such
that the entries of $\Delta^{\prime}$ are integers. However, $G_{\Omega}$ and
$G_{\Omega^{\prime}}$ are isomorphic by the correspondence $g\mapsto
g^{\prime}=g^{\lambda}$. So, the CSDs for the two decompositions are naturally
identified under this isomorphism. Moreover, this is generalized to any
decomposition of $B$ [Nak23, Prop. III.1.23]. Therefore, a CSD essentially
depends only on $B$.

The matrix $\Delta$ also determines the data

$\displaystyle
N^{\circ}:=\bigoplus_{i=1}^{n}\mathbb{Z}\delta_{i}\mathbf{e}_{i},\quad
M^{\circ}:=\mathrm{Hom}(N^{\circ},\mathbb{Z}),$ (8.14)

so that $N^{\circ}\subset N$ and $M\subset M^{\circ}\subset M_{\mathbb{R}}$.
For the standard basis $\mathbf{e}_{1}$, …, $\mathbf{e}_{n}$ of $N$, let
$e^{*}_{1}$, …, $e^{*}_{n}$ be the dual basis of $M$. Then, we have a basis of
$\delta_{1}\mathbf{e}_{1}$, …, $\delta_{n}\mathbf{e}_{n}$ of $N^{\circ}$ and
its dual basis $f_{1}=e^{*}_{1}/\delta_{1}$, …, $f_{n}=e^{*}_{n}/\delta_{n}$
of $M^{\circ}$. We identify $M_{\mathbb{R}}\simeq\mathbb{R}^{n}$ with
$f_{i}\mapsto\mathbf{e}_{i}$, not with $e^{*}_{i}\mapsto\mathbf{e}_{i}$. Under
the identification, the canonical pairing $\langle\mathbf{n},z\rangle\colon
N\times M_{\mathbb{R}}\rightarrow\mathbb{R}$ is given by

$\displaystyle\langle\mathbf{n},\mathbf{z}\rangle=\mathbf{n}^{T}\Delta^{-1}\mathbf{z}.$
(8.15)

We also have

$\displaystyle
p^{*}(\mathbf{e}_{j})(\delta_{i}\mathbf{e}_{i})=\\{\delta_{i}\mathbf{e}_{i},\mathbf{e}_{j}\\}_{\Omega}=\delta_{i}\omega_{ij}=b_{ij}.$
(8.16)

This means that the representation matrix of the map $p^{*}$ in (8.11) with
respect the basis $\mathbf{e}_{1}$, …, $\mathbf{e}_{n}\in N$ and $f_{1}$, …,
$f_{n}\in M^{\circ}\subset M_{\mathbb{R}}$ is given by $B$. Thus,
$p^{*}(\mathbf{n})\in M_{\mathbb{R}}$ is represented by the vector
$B\mathbf{n}\in\mathbb{R}^{n}$.

####  8.3 Rank 2 CSDs: finite type

The proof of Theorem 8.12 in [GHKK18] is based on certain abstract
construction. For the rank 2 case, however, it is possible to describe CSDs
more explicitly, thanks to the results in Section 7.7. Here we concentrate on
the finite type to illustrate the idea of a CSD.

$\textstyle{\gamma_{1}}$$\textstyle{\gamma_{2}}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\begin{bmatrix}1\\\
0\end{bmatrix}^{\delta_{1}}$$\begin{bmatrix}0\\\ 1\end{bmatrix}^{\delta_{2}}$
Figure 8.2: CSD of type $A_{1}\times A_{1}$.

Let us first consider the simplest case.

######  Example 8.13 (Type $A_{1}\times A_{1}$).

When $B=O$, we have a decomposition (5.1) with $\Omega=O$ and arbitrary
$\Delta$. By the commutativity (7.41), we have

$\displaystyle\Psi[\mathbf{e}_{2}]^{\delta_{2}}\Psi[\mathbf{e}_{1}]^{\delta_{1}}=\Psi[\mathbf{e}_{1}]^{\delta_{1}}\Psi[\mathbf{e}_{2}]^{\delta_{2}}.$
(8.17)

This relation is naturally interpreted as the (unique) consistency condition
for a scattering diagram of rank 2 in Figure 8.2. Namely, it consists of two
incoming walls

$\displaystyle(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}]^{\delta_{1}})_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}]^{\delta_{2}})_{\mathbf{e}_{2}}.$
(8.18)

The LHS of the relation (8.17) is the path-ordered product
$\mathfrak{p}_{\gamma_{1},\mathfrak{D}}$ along $\gamma_{1}$, while the RHS is
the one $\mathfrak{p}_{\gamma_{2},\mathfrak{D}}$ along $\gamma_{2}$. Thus, the
relation (8.17) is exactly the consistency relation
$\mathfrak{p}_{\gamma_{1},\mathfrak{D}}=\mathfrak{p}_{\gamma_{1},\mathfrak{D}}$.
Thus, this is a CSD for $B$.

Next, let us assume

$\displaystyle B=\begin{pmatrix}0&-\delta_{1}\\\ \delta_{2}&0\end{pmatrix}$
(8.19)

for some positive integers $\delta_{1}$ and $\delta_{2}$. We consider a skew-
symmetric decomposition of $B$ with

$\displaystyle\Delta=\begin{pmatrix}\delta_{1}&0\\\
0&\delta_{2}\end{pmatrix},\quad\Omega=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}.$ (8.20)

The advantage of this decomposition is that one can work with the common
structure group $G_{\Omega}$ for any matrix $B$ in (8.19). Note that the
matrix $\Omega$ is the same one in (7.38). So, we can use the results in
Section 7.7. Also, by (8.19), $B\mathbf{n}$ ($\mathbf{n}\in
N^{+}_{\mathrm{pr}}$) is in the closure of the _second_ quadrant excluding the
origin $\mathbf{0}$ in $\mathbb{R}^{2}$. Thus, a wall
$(\mathfrak{d},g)_{\mathbf{n}}$ is incoming if and only if the support
$\mathfrak{d}$ intersects with the closure of the second quadrant excluding
the origin $\mathbf{0}$.

(a) $A_{2}$$\begin{bmatrix}1\\\ 0\end{bmatrix}$$\begin{bmatrix}0\\\
1\end{bmatrix}$$\begin{bmatrix}1\\\ 1\end{bmatrix}$

(b) $B_{2}$$\begin{bmatrix}1\\\ 0\end{bmatrix}$$\begin{bmatrix}0\\\
1\end{bmatrix}^{2}$$\begin{bmatrix}1\\\
1\end{bmatrix}^{2}$$\begin{bmatrix}1\\\ 2\end{bmatrix}$

(c) $G_{2}$$\begin{bmatrix}1\\\ 0\end{bmatrix}$$\begin{bmatrix}0\\\
1\end{bmatrix}^{3}$$\begin{bmatrix}1\\\
1\end{bmatrix}^{3}$$\begin{bmatrix}2\\\ 3\end{bmatrix}$$\begin{bmatrix}1\\\
2\end{bmatrix}^{3}$$\begin{bmatrix}1\\\ 3\end{bmatrix}$

Figure 8.3: Rank 2 CSDs of finite type.

######  Example 8.14 (Finite type: $\delta_{1}\delta_{2}\leq 3$).

We use the results in Example 7.11.

(a) Type $A_{2}$. Let $(\delta_{1},\delta_{2})=(1,1)$. The relation (7.43) is
interpreted as the consistency relation for the scattering diagram in Figure
8.3 (a). It consists of three walls

$\displaystyle(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}])_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}])_{\mathbf{e}_{2}},\quad(\mathbb{R}_{\geq
0}(1,-1),\Psi[(1,1)])_{(1,1)},$ (8.21)

where

$\displaystyle\mathbb{R}_{\geq 0}(1,-1)\subset(1,1)^{\perp}$ (8.22)

by (8.15). This ray is contained in the closure of the fourth quadrant. Thus,
it is outgoing. Therefore, this is a CSD for $B$.

(b) Type $B_{2}$. Let $(\delta_{1},\delta_{2})=(1,2)$. The relation (7.44) is
interpreted as the consistency relation for the scattering diagram in Figure
8.3 (b), which consists of four walls

$\displaystyle\begin{split}&(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}])_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}]^{2})_{\mathbf{e}_{2}},\\\
&(\mathbb{R}_{\geq 0}(1,-2),\Psi[(1,1)]^{2})_{(1,1)},\quad(\mathbb{R}_{\geq
0}(1,-1),\Psi[(1,2)])_{(1,2)}.\end{split}$ (8.23)

This is a CSD for $B$.

(c) Type $G_{2}$. Let $(\delta_{1},\delta_{2})=(1,3)$. The relation (7.45) is
interpreted as the consistency relation for the scattering diagram in Figure
8.3 (c), which consists of six walls

$\displaystyle\begin{split}&(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}])_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}]^{3})_{\mathbf{e}_{2}},\\\
&(\mathbb{R}_{\geq 0}(1,-3),\Psi[(1,1)]^{3})_{(1,1)},\quad(\mathbb{R}_{\geq
0}(1,-2),\Psi[(2,3)])_{(2,3)},\\\ &(\mathbb{R}_{\geq
0}(2,-3),\Psi[(1,2)]^{3})_{(1,2)},\quad(\mathbb{R}_{\geq
0}(1,-1),\Psi[(1,3)])_{(1,3)}.\end{split}$ (8.24)

This is a CSD for $B$.

We see that the above CSDs are constructed only with dilogarithm elements and
the pentagon relation.

####  8.4 $G$-fan and CSD

Let us explain the relation between a CSD $\mathfrak{D}(B)$ and a cluster
pattern/$Y$-pattern with the initial exchange matrix $B$.

Recall that the $G$-pattern
$\mathbf{G}=\mathbf{G}^{t_{0}}=\\{G_{t}\\}_{t\in\mathbb{T}_{n}}$ for the
initial exchange matrix $B$ in Definition 2.17 encodes all essential
information of the corresponding cluster pattern/$Y$-pattern by Theorem 2.31.
One can geometrically represent each $G$-matrix $G_{t}$ by a _$G$ -cone_ in
$\mathbb{R}^{n}$ defined by

$\displaystyle\sigma(G_{t}):=\sigma(\mathbf{g}_{1;t},\dots,\mathbf{g}_{n;t}).$
(8.25)

Due to the unimodularity (2.77), the dimension of $\sigma(G_{t})$ is $n$. Note
that the cone $\sigma(G_{t})$ is invariant under the action of a permutation
$\nu$ on a $G$-matrix $G_{t}$ in (2.80). Each cone generated by a subset of
$\mathbf{g}_{1;t}$, …, $\mathbf{g}_{n;t}$ is called a _face_ of
$\sigma(G_{t})$, including the one $\sigma(\emptyset)=\\{\mathbf{0}\\}$. In
particular, let

$\displaystyle\sigma_{i}(G_{t}):=\sigma(\mathbf{g}_{1;t},\dots,\check{\mathbf{g}}_{i;t}\dots,\mathbf{g}_{n;t}),$
(8.26)

which is a face of dimension $n-1$, Then, by (2.54), if $t$,
$t^{\prime}\in\mathbb{T}_{n}$ are $k$-adjacent, the $G$-cones $\sigma(G_{t})$
and $\sigma(G_{t^{\prime}})$ intersect each other in their common face
$\sigma_{k}(G_{t})=\sigma_{k}(G_{t^{\prime}})$.

######  Definition 8.15.

A set of cones $\mathcal{F}$ is a _fan_ if the intersection
$\sigma\cap\sigma^{\prime}$ of any pair $\sigma$,
$\sigma^{\prime}\in\mathcal{F}$ belongs to $\mathcal{F}$.

The following property is a consequence of the sign-coherence in Theorem 2.24.

######  Proposition 8.16.

[Rea14, Thm. 8.7] The set $\mathcal{G}(B)$ consisting of all $G$-cones and
their faces is a fan in $\mathbb{R}^{n}$.

We call the above fan the _$G$ -fan_ for a skew-symmetrizable matrix $B$ or
the corresponding cluster pattern/$Y$-pattern. The _support_ of the $G$-fan
$\mathcal{G}(B)$ is defined by

$\displaystyle|\mathcal{G}(B)|:=\bigcup_{t\in\mathbb{T}^{n}}\sigma(G_{t}).$
(8.27)

We say that the $G$-fan $\mathcal{G}(B)$ is _complete_ if
$|\mathcal{G}(B)|=\mathbb{R}^{n}$.

The following fact is known.

######  Proposition 8.17 (e.g., [Nak23, Prop. II.2.18]).

The $G$-fan $\mathcal{G}(B)$ is complete if and only if the corresponding
cluster pattern/$Y$-pattern is of finite type.

Let $\mathcal{G}_{n-1}(B)$ be the set of all cones in $\mathcal{G}(B)$ of
dimension $n-1$. Let $|\mathcal{G}_{n-1}(B)|$ be the union of the supports of
all cones in $\mathcal{G}_{n-1}(B)$.

Here is the second fundamental result on CSDs.

######  Theorem 8.18 ([GHKK18, Construction 1.30], [Rea20, Cor. 4.4]).

Let $\mathfrak{D}(B)$ be a CSD for $B$ with minimal support. Let
$\mathfrak{S}(B)\subset\mathbb{R}^{n}$ be the image of the
$\mathrm{Supp}(\mathfrak{D}(B))\subset M_{\mathbb{R}}$ under the
identification $M_{\mathbb{R}}\simeq\mathbb{R}^{n}$ with
$f_{i}\mapsto\mathbf{e}_{i}$. Then, we have the inclusion

$\displaystyle|\mathcal{G}_{n-1}(B)|\subset\mathfrak{S}(B).$ (8.28)

Also, the interior $\mathrm{Int}(\sigma(G_{t}))$ of each $G$-cone dones not
intersect $\mathfrak{S}(B)$. Moreover, for a cluster pattern/$Y$-pattern of
finite type, we have the equality

$\displaystyle|\mathcal{G}_{n-1}(B)|=\mathfrak{S}(B).$ (8.29)

Therefore, all essential information of a cluster pattern/$Y$-pattern is
contained in the corresponding CSD.

######  Example 8.19.

Let us check the equality (8.29) in Examples 8.13 and 8.14. For type
$A_{1}\times A_{1}$, there are four $G$-matrices

$\displaystyle\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\quad\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\quad\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix},\quad\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (8.30)

Comparing it with Figure 8.2, we see that the equality (8.29) holds. For type
$A_{2}$, as we see in Example 2.21, there are five $G$-matrices (up to the
action of the permutation $\tau_{12}$)

$\displaystyle\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\quad\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\quad\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix},\quad\begin{pmatrix}1&0\\\
-1&-1\end{pmatrix},\quad\begin{pmatrix}1&1\\\ -1&0\end{pmatrix}.$ (8.31)

Comparing it with Figure 8.3 (a), we see that the equality (8.29) holds. Other
cases are similar and left as exercises for the reader.

####  8.5 Rank 2 CSDs: infinite type

Let us present examples of rank 2 CSDs of _infinite type_ using the results in
Section 7.7. It turns out that the infinite type is more interesting because
CSDs contain some additional information outside the $G$-fan.

######  Example 8.20 (Affine type: $\delta_{1}\delta_{2}=4$).

(a). Type $A_{1}^{(1)}$. Let $(\delta_{1},\delta_{2})=(2,2)$. The relation
(7.51) is translated into the following consistent scattering diagram:

$\displaystyle(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}]^{2})_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}]^{2})_{\mathbf{e}_{2}},$
(8.32) $\displaystyle(\mathbb{R}_{\geq
0}(p,-p-1),\Psi[(p+1,p)]^{2})_{(p+1,p)}\quad(p\in\mathbb{Z}_{>0}),$ (8.33)
$\displaystyle(\mathbb{R}_{\geq
0}(p+1,-p),\Psi[(p,p+1)]^{2})_{(p,p+1)}\quad(p\in\mathbb{Z}_{>0}),$ (8.34)
$\displaystyle(\mathbb{R}_{\geq
0}(1,-1),\prod_{j=0}^{\infty}\Psi[2^{j}(1,1)]^{2^{2-j}})_{(1,1)}.$ (8.35)

Equivalently, one can split the wall in (8.35) into infinitely many walls

$\displaystyle(\mathbb{R}_{\geq
0}(1,-1),\Psi[2^{j}(1,1)]^{2^{2-j}})_{(1,1)}\quad(j=0,\,1,\,\dots).$ (8.36)

See Figure 8.4 (a). All walls except for the ones in (8.32) are outgoing.
Thus, it is a CSD for $B$. The equality (8.29) does not hold anymore. Indeed,
the complement of $|\mathcal{G}(B)|$ in $\mathbb{R}^{2}$ is given by the ray
$\mathbb{R}_{>0}(1,-1)$, which is depicted as a thick ray in the figure. As we
see in (8.36), there is some rich structure of walls in the ray.

(b). Type $A_{2}^{(2)}$. Let $(\delta_{1},\delta_{2})=(1,4)$. The relation
(7.52) is translated into the following consistent scattering diagram:

$\displaystyle(\mathbf{e}_{1}^{\perp},\Psi[\mathbf{e}_{1}])_{\mathbf{e}_{1}},\quad(\mathbf{e}_{2}^{\perp},\Psi[\mathbf{e}_{2}]^{4})_{\mathbf{e}_{2}},$
(8.37) $\displaystyle(\mathbb{R}_{\geq
0}(p,-2p-1),\Psi[(2p+1,4p)])_{(2p+1,4p)}\quad(p\in\mathbb{Z}_{>0}),$ (8.38)
$\displaystyle(\mathbb{R}_{\geq
0}(2p-1,-4p),\Psi[(p,2p-1)]^{4})_{(p,2p-1)}\quad(p\in\mathbb{Z}_{>0}),$ (8.39)
$\displaystyle(\mathbb{R}_{\geq
0}(p,-2p+1),\Psi[(2p-1,4p)])_{(2p-1,4p)}\quad(p\in\mathbb{Z}_{>0}),$ (8.40)
$\displaystyle(\mathbb{R}_{\geq
0}(2p+1,-4p),\Psi[(p,2p+1)]^{4})_{(p,2p+1)}\quad(p\in\mathbb{Z}_{>0}),$ (8.41)
$\displaystyle\begin{split}&\quad\ (\mathbb{R}_{\geq
0}(1,-2),\Psi[(1,2)]^{6}\prod_{j=1}^{\infty}\Psi[2^{j}(1,2)]^{2^{2-j}})_{(1,2)}.\end{split}$
(8.42)

See Figure 8.4 (b). Again, it is a CSD for $B$. The complement of
$|\mathcal{G}(B)|$ in $\mathbb{R}^{2}$ is given by the ray
$\mathbb{R}_{>0}(1,-2)$.

(a) $A_{1}^{(1)}$$(b,c)=(5,1)$

(b) $A_{2}^{(2)}$$(a,b)=(5,1)$

(c) Non-affine infinite type$(a,b)=(5,1)$

Figure 8.4: Rank 2 CSDs of infinite type. The shaded area in (c) is called the
Badlands.

The remaining case is most intriguing, as already seen in Example 7.13, though
the entire structure of a CSD is not known yet.

######  Example 8.21 (Non-affine infinite type: $\delta_{1}\delta_{2}>4$).

We generalize and translate the results and facts in Example 7.13 into the CSD
language. Let $\sigma(\mathbf{v}_{+},\mathbf{v}_{-})$ be the irrational cone
spanned by two vectors

$\displaystyle\mathbf{v}_{\pm}=\begin{pmatrix}\delta_{1}\delta_{2}\pm\sqrt{\delta_{1}\delta_{2}(\delta_{1}\delta_{2}-4)}\\\
-2\delta_{2}\\\ \end{pmatrix}.$ (8.43)

The complement of $|\mathcal{G}(B)|$ in $\mathbb{R}^{2}$ is
$\sigma(\mathbf{v}_{+},\mathbf{v}_{-})\setminus\\{0\\}$, which is informally
called the _Badlands_. For example, in the case
$(\delta_{1},\delta_{2})=(1,5)$, the cone
$\sigma(\mathbf{v},\mathbf{v}^{\prime})$ is depicted as a shaded region in
Figure 8.4 (c). As in Example 7.13, every rational ray in the cone
$\sigma(\mathbf{v},\mathbf{v}^{\prime})$ appears as a nontrivial wall of
$\mathfrak{D}(B)$ [GL23]. However, their wall elements are not known yet
except for the limited case.

####  8.6 Positive realization of CSDs and pentagon relation

To conclude this quick course, let us explain that any CSD of any rank can be
constructed only by the dilogarithm elements and the pentagon relation.

Recall that, for a given skew-symmetric decomposition (5.1) of $B$, a
sublattice $N^{\circ}$ of $N$ is defined by (8.14).

######  Definition 8.22.

For each $\mathbf{n}\in N^{+}$, the _normalization factor_
$\delta(\mathbf{n})$ of $\mathbf{n}$ is the smallest positive rational number
such that $\delta(\mathbf{n})\mathbf{n}\in N^{\circ}$.

For example, for $\mathbf{n}=\mathbf{e}_{i}$, we have
$\delta(\mathbf{e}_{i})=\delta_{i}$. Also, for any $h\in\mathbb{Z}_{>0}$ and
$\mathbf{n}\in N^{+}$, we have $\delta(h\mathbf{n})=\delta(\mathbf{n})/h$.

Now we present the third fundamental result on CSDs.

######  Theorem 8.23 ([GHKK18, Theorem 1.13]).

Let $\mathfrak{D}(B)$ be a CSD for $B$. Then, there is a consistent scattering
diagram $\mathfrak{D}$ that is equivalent to $\mathfrak{D}(B)$ such that the
wall element of any wall of $\mathfrak{D}$ has the following form

$\displaystyle g=\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})}\quad(\mathbf{n}\in
N_{\mathrm{pr}}^{+};\,s,\,h\in\mathbb{Z}_{>0}).$ (8.44)

We call a CSD with the property (8.44) a _positive realization_ due to the
positivity of the exponents $s$. Such a CSD is always with minimal support due
to the positivity of $s$.

Below we present the construction of the positive realization of a CSD by the
pentagon relation. Let us temporarily concentrate on the rank 2 case. The
following is a more general answer to Problem 7.9 than Proposition 7.10.

######  Proposition 8.24 (Ordering Lemma [Nak23, Prop. III.5.4]).

For the rank 2 case, let

$\displaystyle
C^{\mathrm{in}}=\Psi[h^{\prime}_{j}\mathbf{n}^{\prime}_{j}]^{s^{\prime}_{j}\delta(h^{\prime}_{j}\mathbf{n}^{\prime}_{j})}\cdots\Psi[h^{\prime}_{1}\mathbf{n}^{\prime}_{1}]^{s^{\prime}_{1}\delta(h^{\prime}_{1}\mathbf{n}^{\prime}_{1})}\quad(\mathbf{n}^{\prime}_{i}\in
N_{\mathrm{pr}}^{+};\,\ s^{\prime}_{i},\,h^{\prime}_{i}\in\mathbb{Z}_{>0})$
(8.45)

be any finite anti-ordered product. Then, by applying the relations in
Proposition 7.7 possibly infinitely many times, $C^{\mathrm{in}}$ is
transformed into a (possibly infinite) ordered product $C^{\mathrm{out}}$ of
factors of the same form

$\displaystyle\Psi[h_{i}\mathbf{n}_{i}]^{s_{i}\delta(h_{i}\mathbf{n}_{i})}\quad(\mathbf{n}_{i}\in
N_{\mathrm{pr}}^{+};\,\ s_{i},\,h_{i}\in\mathbb{Z}_{>0}).$ (8.46)

Note the positivity of exponents $s_{i}$. The proof of Proposition 8.24 is
based on an explicit algorithm (Ordering Algorithm) [Nak23, Algorithm
III.5.7], which generalizes the procedure in the examples in Section 7.7. The
positivity is guaranteed because the pentagon relation involves only positive
exponents throughout.

Now we go back to the general rank case. We need the following notion from
[GHKK18].

######  Definition 8.25 (Joint).

Let $\mathfrak{D}$ be a scattering diagram.

(a). For any pair of walls
$\mathbf{w}_{i}=(\mathfrak{d}_{i},g_{i})_{\mathbf{n}_{i}}$ ($i=1,2$), the
intersection of their supports
$\mathfrak{j}=\mathfrak{d}_{1}\cap\mathfrak{d}_{2}$ is called a _joint_ of
$\mathfrak{D}$ if $\mathfrak{j}$ is a cone of dimension $n-2$.

(b) For each joint $\mathfrak{j}$, let

$\displaystyle N_{\mathfrak{j}}:=\\{\mathbf{n}\in
N\mid\langle\mathbf{n},z\rangle=0\ (z\in\mathfrak{j})\\},$ (8.47)

which is a rank 2 sublattice of $N$. Then, a joint $\mathfrak{j}$ is
_parallel_ (resp.  _perpendicular_) if $\\{\mathbf{n}_{1},\mathbf{n}_{2}\\}=0$
(resp. otherwise).

By the topological reason, the consistency condition (8.9) is reduced to the
one for loops around joints.

The construction of a positive realization in Theorem 8.23 is given as
follows. (Here we present the modified version in [Nak23] with the help of
Proposition 8.24.)

######  Construction 8.26 ([GHKK18, Construction C.1, Appendix C.3], [Nak23,
Construction III.5.14]).

The construction proceeds inductively on the degree $\ell$.

  1. 1.

We start with the scattering diagram $\mathfrak{D}_{1}$ consisting of only
incoming walls
$\mathbf{w}_{i}:=(\mathbf{e}_{i}^{\perp},\Psi[\mathbf{e}_{i}]^{\delta_{i}})_{\mathbf{e}_{i}}$
($i=1$, …, $n$).

  2. 2.

For each perpendicular joint
$\mathfrak{j}:=\mathbf{e}_{i}^{\perp}\cap\mathbf{e}_{j}^{\perp}$ ($i\neq j$)
of $\mathfrak{D}_{1}$ with $c=\\{\mathbf{e}_{j},\mathbf{e}_{i}\\}>0$, we apply
Proposition 8.24 for
$\Psi[\mathbf{e}_{i}]^{\delta_{i}}\Psi[\mathbf{e}_{j}]^{\delta_{j}}$. In the
ordered product, we have the degree 2 element
$\Psi[\mathbf{e}_{i}+\mathbf{e}_{j}]^{c\delta_{i}\delta_{j}}$. Accordingly, we
add a new outgoing wall

$\displaystyle(\sigma(\mathfrak{j},-p^{*}(\mathbf{e}_{i}+\mathbf{e}_{j})),\Psi[\mathbf{e}_{i}+\mathbf{e}_{j}]^{c\delta_{i}\delta_{j}})_{\mathbf{e}_{i}+\mathbf{e}_{j}},$
(8.48)

where $\sigma(\mathfrak{j},-p^{*}(\mathbf{e}_{i}+\mathbf{e}_{j}))$ is the cone
spanned by $\mathfrak{j}$ and $-p^{*}(\mathbf{e}_{i}+\mathbf{e}_{j})$.

  3. 3.

Suppose that the procedure is completed up to degree $\ell$ and we have the
scattering diagram $\mathfrak{D}_{\ell}$. By splitting walls if necessary, we
may assume that any pair of distinct joints of $\mathfrak{D}_{\ell}$ intersect
each other only in their boundaries. For each perpendicular joint
$\mathfrak{j}$, we set $N^{+}_{\mathfrak{j}}=N_{\mathfrak{j}}\cap N^{+}$ and
$N^{+}_{\mathfrak{j},\mathrm{pr}}=N_{\mathfrak{j}}\cap N^{+}_{\mathrm{pr}}$.
Then, there is a unique $\tilde{\mathbf{e}}_{1},\tilde{\mathbf{e}}_{2}\in
N^{+}_{\mathfrak{j},\mathrm{pr}}$ such that

$\displaystyle N^{+}_{\mathfrak{j}}\subset\mathbb{Q}_{\geq
0}\tilde{\mathbf{e}}_{1}+\mathbb{Q}_{\geq
0}\tilde{\mathbf{e}}_{2},\quad\\{\tilde{\mathbf{e}}_{2},\tilde{\mathbf{e}}_{1}\\}>0.$
(8.49)

There are walls of $\mathfrak{D}_{\ell}$ containing $\mathfrak{j}$ and
intersecting the “second quadrant” with respect to $\tilde{\mathbf{e}}_{1}$
and $\tilde{\mathbf{e}}_{2}$. We take the anti-ordered product of wall
elements of those walls and apply Proposition 8.24. Then, for each element
$\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})}$ with $\deg(h\mathbf{n})=\ell+1$ in
the ordered product, we add a new outgoing wall

$\displaystyle(\sigma(\mathfrak{j},-p^{*}(\mathbf{n})),\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})})_{\mathbf{n}}$
(8.50)

in the “fourth quadrant”.

  4. 4.

The desired positive realization for $\mathfrak{D}(B)$ in Theorem 8.23 is
obtained as the limit of $\mathfrak{D}_{\ell}$.

The consistency around each perpendicular joint is satisfied by the
construction. On the other hand, the consistency around each parallel joint is
not obvious; however, it is guaranteed thanks to Theorem 8.12.

By construction, the consistency relation around each perpendicular joint is
reduced to a trivial one (i.e., $g=g$) by applying the pentagon relation in
the reverse way to the ordereing. On the other hand, the one around each
parallel joint is reduced to a trivial one by applying the commutative
relation (7.41). Thus, as a by-product, the above construction also reveals
the fourth fundamental fact on the structure of a CSD.

######  Theorem 8.27 ([Nak23, Thm. III.5.17] ).

Let $\mathfrak{D}(B)$ be a CSD for $B$. For any admissible loop $\gamma$ for
$\mathfrak{D}(B)$, the consistency relation
$\mathfrak{p}_{\gamma,\mathfrak{D}(B)}=\mathrm{id}$ is reduced to a trivial
one by applying the relations in Proposition 7.7 possibly infinitely many
times.

Notes

Scattering diagrams were introduced by [KS06] and [GS11] in the study of
homological mirror symmetry, and developed further in [KS08, GPS10, CPS,
Gro11, KS14]. The connection to cluster algebras was established by [GHKK18],
where CSDs were introduced. With the scattering diagram method, several
important conjectures on cluster algebras were proved, including Theorems 2.24
and 2.29. Here, we follow the presentation of CSDs in the companion monograph
[Nak23], where the roles of the dilogarithm elements and the pentagon relation
are highlighted. Proofs of all results in this chapter are conveniently found
in [Nak23] as specified below. (We omit the ones already cited in the text.)

  * •

Theorem 8.7: See [Nak23, Thm. III.2.20].

  * •

Theorem 8.12: See [Nak23, Thm. III.3.12].

  * •

Proposition 8.16: See [Nak23, Thm. II.2.17].

  * •

Theorem 8.18: See [Nak23, Thm. II.3.3 (a)].

  * •

Theorem 8.23: See [Nak23, Thm. III.5.2].

###  Chapter 9 Dilogarithm identities in cluster scattering diagrams

We derive the DI associated with a loop in a CSD by combining the results on
CSDs and the classical dynamical method in Part II. This generalizes the DI
associated with a period of a $Y$-pattern to the ones with infinite sum.
Thanks to the structure theorem for CSDs, such a DI is always reduced to the
trivial one by applying the pentagon identity possibly infinitely many times.
Thus, it is _infinitely reducible_. Moreover, this result provides yet another
proof of the DIs in Theorem 3.5.

####  9.1 $y$-variables in CSD

To formulate DIs in a CSD, we first need to extend the notion of $y$-variables
in a $Y$-pattern to the corresponding CSD. From now on, we assume that a CSD
$\mathfrak{D}(B)$ is a positive realization in Theorem 8.23. In particular,
$\mathfrak{D}(B)$ is with minimal support.

We continue to identify $M_{\mathbb{R}}\simeq\mathbb{R}^{n}$ with
$f_{i}=e^{*}_{i}/\delta_{i}\mapsto\mathbf{e}_{i}$ as in Section 8.4. Note
that, under the identification, the lattice $M^{\circ}\subset M_{\mathbb{R}}$
corresponds to $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$. Through this
identification, we regard a $c$-vector $\mathbf{c}_{i;t}$ and a $g$-vector
$\mathbf{g}_{i;t}$ as elements of $N$ and $M^{\circ}$, respectively. Then, the
canonical pairing of them is given by

$\displaystyle\langle\mathbf{c}_{i;t},\mathbf{g}_{j;t}\rangle=(\mathbf{c}_{i;t})^{T}\Delta^{-1}\mathbf{g}_{j;t}$
(9.1)

by (8.15). Let $\mathbf{c}_{i;t}^{+}=\varepsilon_{i;t}\mathbf{c}_{i;t}$ be the
$c^{+}$-vector in (5.22). Also, let $\sigma_{i}(G_{t})$ be the cone of
dimension $n-1$ in (8.26).

######  Lemma 9.1 ([Nak23, §II.5.2]).

The following facts hold.

(a). $\mathbf{c}_{i;t}^{+}$ is normal to $\sigma_{i}(G_{t})$ with respect to
the canonical pairing (9.1).

(b). $\mathbf{c}_{i;t}^{+}$ is primitive.

(c). $\delta(\mathbf{c}_{i;t}^{+})=\delta_{i}$.

###### Proof.

(a). By the duality (2.76), the canonical pairing (9.1) is written as

$\displaystyle\langle\delta_{i}\mathbf{c}_{i;t},\mathbf{g}_{j;t}\rangle=\delta_{ij}.$
(9.2)

(b). Since the $C$-matrix $C_{t}$ is unimodular by (2.77), $\mathbf{c}_{1;t}$,
…, $\mathbf{c}_{n;t}$ are a basis of $N$. Thus, $\mathbf{c}_{i;t}$ is
primitive.

(c). Since the $G$-matrix $G_{t}$ is unimodular by (2.77), $\mathbf{g}_{1;t}$,
…, $\mathbf{g}_{n;t}$ are a basis of $M^{\circ}$. Then, by the duality (9.2)
again, we see that $\delta_{i}\mathbf{c}_{i;t}^{+}\in N^{\circ}$. Moreover, it
also tells that $\delta_{i}$ is the minimal one such that
$\delta_{i}\mathbf{c}_{i;t}^{+}\in N^{\circ}$. Therefore,
$\delta(\mathbf{c}_{i;t}^{+})=\delta_{i}$. ∎

Let $\sigma_{i}(G_{t})$ be the cone in (8.26). Recall that it is contained in
$\mathrm{Supp}(\mathfrak{D}(B))$ by Theorem 8.18. This means that, if
$z\in\sigma_{i}(G_{t})$, then there is a wall
$\mathbf{w}=(\mathfrak{d},g)_{\mathbf{n}}$ of $\mathfrak{D}(B)$ such that
$z\in\mathfrak{d}$.

######  Lemma 9.2 ([Nak23, Prop. III.6.24]).

Let $z\in M_{\mathbb{R}}$ be a point in $\sigma_{i}(G_{t})$ with
$z\not\in\mathrm{Sing}(\mathfrak{D}(B))$. Suppose that
$\mathbf{w}=(\mathfrak{d},g)_{\mathbf{n}}$ is a wall of $\mathfrak{D}(B)$ such
that $z\in\mathfrak{d}$. Then, we have

$\displaystyle\mathbf{n}=\mathbf{c}_{i;t}^{+},\quad
g=\Psi[\mathbf{c}_{i;t}^{+}]^{\delta_{i}}.$ (9.3)

Moreover, there is no other wall
$\mathbf{w}^{\prime}=(\mathfrak{d}^{\prime},g^{\prime})_{\mathbf{n}}$ of
$\mathfrak{D}(B)$ such that $z\in\mathfrak{d}^{\prime}$.

We omit the proof. See [Nak23, §III.6.6] for details.

In Definition 8.3 of an admissible curve $\gamma$, we weaken Condition (1) so
that only the ending point $\gamma(1)$ is prohibited from being in
$\mathrm{Supp}(\mathfrak{D})$. Namely, the starting point $\gamma(0)$ is now
allowed to be in $\mathrm{Supp}(\mathfrak{D})$ (but not allowed to be in
$\mathrm{Sing(\mathfrak{D})}$ by Condition (3)). Such a curve
$\gamma:[0,1]\rightarrow M_{\mathbb{R}}$ is said to be _weakly admissible_ for
a scattering diagram $\mathfrak{D}$. The definition of the path-ordered
product $\mathfrak{p}_{\gamma,\mathfrak{D}}$ in Definition 8.4 is applied to a
weakly admissible curve $\gamma$. In particular, the contribution of the wall
at the starting point $\gamma(0)$ is _not_ counted in
$\mathfrak{p}_{\gamma,\mathfrak{D}}$.

Let us extend the notion of $y$-variables and $c^{+}$-vectors in (5.22) for a
CSD.

######  Definition 9.3 ($y$-variable/$c^{+}$-vector for CSD).

Let $\mathbf{w}=(\mathfrak{d},g)_{\mathbf{n}}$ be any wall of
$\mathfrak{D}(B)$ with $g=\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})}$ as in
(8.44). Let $z\in\mathfrak{d}$ and $z\not\in\mathrm{Sing}(\mathfrak{D}(B))$.
Let $\mathcal{C}^{+}$ be the positive orthant in (8.10). Let $\gamma_{z}$ be
any weakly admissible curve in $\mathfrak{D}(B)$ from $z$ to any point in
$\mathrm{Int}(\mathcal{C}^{+})$. Then, we define a _$y$ -variable
$y_{z}[h\mathbf{n}]$ at $z$ with the $c^{+}$-vector_ $h\mathbf{n}$ by

$\displaystyle
y_{z}[h\mathbf{n}]:=\mathfrak{p}_{\gamma_{z},\mathfrak{D}(B)}(y^{h\mathbf{n}})\in\mathbb{Q}[[\mathbf{y}]],$
(9.4)

where the path-ordered product $\mathfrak{p}_{\gamma_{z},\mathfrak{D}(B)}\in
G$ acts on $y^{h\mathbf{n}}$ under the $y$-representation $\rho_{y}$ in
Proposition 7.2. (Even though $y_{z}[h\mathbf{n}]$ only depends on $z$ as we
see below, to represent its $c^{+}$-vector $h\mathbf{n}$ explicitly is useful
in our application.)

######  Remark 9.4.

Let $\gamma_{z}$ and $\gamma^{\prime}_{z}$ be weakly admissible curves. Then,
the equality
$\mathfrak{p}_{\gamma_{z},\mathfrak{D}(B)}=\mathfrak{p}_{\gamma^{\prime}_{z},\mathfrak{D}(B)}$
does _not_ hold in general because the contribution of the wall element
$\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})}$ at $z$ is not counted.
Nevertheless, the RHS of (9.4) does not depend on the choice of $\gamma_{z}$
because $\Psi[h\mathbf{n}]^{s\delta(h\mathbf{n})}$ acts on $y^{h\mathbf{n}}$
trivially.

The following fact justifies the above definition.

######  Proposition 9.5 ([Nak24, Remark 3.2 (a)]).

In (9.4), if the point $z$ lies in $\sigma_{i}(G_{t})$, then we have

$\displaystyle y_{z}[\mathbf{c}^{+}_{i;t}]=y_{i;t}^{\varepsilon_{i;t}},$ (9.5)

where the RHS is regarded as an element of $\mathbb{Q}[[\mathbf{y}]]$ by the
mutations.

###### Proof.

For clarity, the proof is separated into three steps.

(i). Let $t_{0}$, $t_{1}$, …, $t_{P}=t$ be a sequence in $\mathbb{T}_{n}$ such
that $t_{0}$ is the initial vertex of $\mathbb{T}_{n}$ and $t_{s}$ and
$t_{s+1}$ are adjacent. For each $s=0$, …, $P-1$, let
$k_{s}\in\\{1,\,\dots,\,n\\}$ be the one such that $t_{s}$ and $t_{s+1}$ are
$k_{s}$-adjacent. We take a weakly admissible curve $\gamma_{z}$ in (9.4) so
that, starting form $z\in\sigma_{i}(G_{t_{P}})$, it passes through the
$G$-cones $\sigma(G_{t_{P}})$, $\sigma(G_{t_{P-1}})$, …,
$\sigma(G_{t_{0}})=\mathcal{C}^{+}$ in this order. (To be precise, if
$k_{P-1}=i$, the curve $\gamma_{z}$ may stay in $\sigma(G_{t_{P}})$ only at
the starting point $\gamma_{z}(0)$ depending on the choice of $\gamma_{z}$;
otherwise, it stays in $\sigma(G_{t_{P}})$ for a while. See Figure 9.1.) Let
us take into account Lemma 9.2. Also, if $k_{P-1}=i$, then
$\mathbf{c}^{+}_{k_{P-1};t_{P-1}}=\mathbf{c}^{+}_{i;t}$. Thus,
$\Psi[\mathbf{c}^{+}_{k_{P-1};t_{P-1}}]$ acts on $y^{\mathbf{c}^{+}_{i;t}}$
trivially. Then, in any situation, we have

$\displaystyle
y_{z}[\mathbf{c}^{+}_{i;t}]=\mathfrak{p}_{\gamma_{z},\mathfrak{D}(B)}(y^{\mathbf{c}^{+}_{i;t}})=\Psi[\mathbf{c}_{k_{0};t_{0}}^{+}]^{\epsilon_{0}\delta_{k_{0}}}\cdots\Psi[\mathbf{c}_{k_{P-1};t_{P-1}}^{+}]^{\epsilon_{P-1}\delta_{k_{P-1}}}(y^{\mathbf{c}^{+}_{i;t}}),$
(9.6)

where $\epsilon_{s}$ is the intersection sign (8.7) of $\gamma_{z}$ at the
wall $\sigma_{k_{s}}(G_{s})=\sigma_{k_{s}}(G_{s+1})$.

$\sigma(G_{t(P-1)})$$\sigma(G_{t(P)})$$\mathcal{C}^{+}$$\gamma$$z$(a)
$k_{P-1}=i$

$\sigma(G_{t(P-1)})$$\sigma(G_{t(P)})$$\mathcal{C}^{+}$$\gamma$$z$(b)
$k_{P-1}\neq i$

Figure 9.1: Admissible curve $\gamma_{z}$. The shaded region represents
$\sigma(G_{t(P)})$.

(ii). Consider the the mutation sequence (5.8) corresponding to the above
sequence $k_{0}$, …, $k_{P-1}$. Let us find the relation between the
intersection sign $\epsilon_{s}$ and the tropical sign
$\varepsilon_{k_{s};t_{s}}$. The duality (9.2) implies that, when the curve
$\gamma_{z}$ crosses the wall $\sigma_{k_{s}}(G_{s})=\sigma_{k_{s}}(G_{s+1})$
from $\sigma(G_{s+1})$ to $\sigma(G_{s})$, its derivative
$\gamma^{\prime}_{z}$ satisfies

$\displaystyle\langle\mathbf{c}_{k_{s};t_{s}},\gamma^{\prime}_{z}\rangle>0.$
(9.7)

Since the vector $\mathbf{n}_{i}$ in (8.7) is given by
$\mathbf{c}_{k_{s};t_{s}}^{+}:=\varepsilon_{k_{s};t_{s}}\mathbf{c}_{k_{s};t_{s}}$,
the sign of $\langle\mathbf{n}_{i},\gamma^{\prime}_{z}\rangle$ coincides with
the tropical sign $\varepsilon_{k_{s};t_{s}}$. Thus, by (8.7), we conclude
that

$\displaystyle\epsilon_{s}=-\varepsilon_{k_{s};t_{s}}.$ (9.8)

Putting it into (9.6), we have

$\displaystyle
y_{z}[\mathbf{c}^{+}_{i;t}]=\Psi[\mathbf{c}_{k_{0};t_{0}}^{+}]^{-\varepsilon_{k_{0};t_{0}}\delta_{k_{0}}}\cdots\Psi[\mathbf{c}_{k_{P-1};t_{P-1}}^{+}]^{-\varepsilon_{k_{P-1};t_{P-1}}\delta_{k_{P-1}}}(y^{\mathbf{c}^{+}_{i;t}}).$
(9.9)

(iii). Let us identify $y_{i;t_{s}}$ with $y_{i}(s)$ in the sequence of
mutations in (5.8) and switch to the notation therein. In the formalism of
(5.10), the equality (9.9) is written as

$\displaystyle y_{z}[\mathbf{c}^{+}_{i}(P)]$
$\displaystyle{=}(\mathfrak{q}(0)\circ\cdots\circ\mathfrak{q}(P-1))(y^{\mathbf{c}^{+}_{i}(P)})$
$\displaystyle{=}\mathfrak{q}(P-1;0)(y^{\mathbf{c}^{+}_{i}(P)})$
$\displaystyle{=}(\mathfrak{q}(P-1;0)\circ\tau(P-1;0))(y_{i}(P)^{\varepsilon_{i}(P)})\quad$
$\displaystyle{=}\mu(P-1;0)(y_{i}(P)^{\varepsilon_{i}(P)})).$

This is the desired result. ∎

The above proof also tells that a sequence of mutations of $y$-variables in a
$Y$-pattern is expressed by a path-ordered product in the corresponding CSD.
By the same token, the periodicity condition of a $Y$-pattern is expressed by
the consistency relation along a loop in the CSD.

######  Example 9.6.

Suppose that the sequence of mutations of $Y$-seeds in (3.2) is
$\nu$-periodic; namely, $\Upsilon(P)=\nu\Upsilon(0)$. Here we do not assume
$\Upsilon(0)$ is the initial $Y$-seed $\Upsilon$. By Theorem 2.31, we have
$G(P)=\nu G(0)$; hence, $\sigma(G(P))=\sigma(G(0))$. Correspondingly, we
consider an admissible loop $\gamma$ with the base point
$w=\gamma(0)=\gamma(1)$ in $\mathfrak{D}(B)$ such that

  * •

it is entirely contained in the support of the $G$-fan $\mathcal{G}(B)$,

  * •

the base point $w$ is in the internal of $\sigma(G(0))=\sigma(G(P))$,

  * •

it crosses the $G$-cones $\sigma(G(P-1))$, …, $\sigma(G(0))=\sigma(G(P))$ in
this order.

See Figure 9.2. We have the consistency relation along the loop $\gamma$

$\displaystyle\mathfrak{p}_{\gamma,\mathfrak{D}(B)}=\mathrm{id}.$ (9.10)

This relation coincides with the relation (7.32).

$w$$\gamma$ Figure 9.2: Admissible loop corresponding to a period of a
$Y$-pattern. The shaded region represents $\sigma(G(0))=\sigma(G(P))$.

####  9.2 DI associated with a loop in a CSD

Now we extend our leitmotif, Theorem 3.5 for an arbitrary loop in a CSD. Since
such a DI involves an infinite sum in general, it is natural to regard it as
an identity for formal power series in the initial variables $\mathbf{y}$.
However, as an additional issue, the modified Rogers dilogarithm
$\tilde{L}(x)$ in (1.26) is not analytic at $x=0$; instead, it has the
following expansion around $x=0$ with $\log$ factor,

$\displaystyle\tilde{L}(x)$
$\displaystyle=\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j^{2}}x^{j}-\frac{1}{2}\log
x\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j}x^{j}.$ (9.11)

Accordingly, for variables $\mathbf{y}=(y_{1},\dots,y_{n})$, we consider the
following formal sum

$\displaystyle f(\mathbf{y})+\sum_{i=1}^{n}(\log
y_{i})g_{i}(\mathbf{y})\quad(f(\mathbf{y}),g_{i}(\mathbf{y})\in\mathbb{Q}[[\mathbf{y}]]),$
(9.12)

and call it a _formal Puiseux series in $\mathbf{y}$ with $\log$ factor_. For
each integer $\ell\geq 1$, let $\mathcal{F}^{>\ell}$ be the set of all formal
power series $f(\mathbf{y})\in\mathbb{Q}[[\mathbf{y}]]$ such that all
coefficients of $f(\mathbf{y})$ vanish up to the total degree $\ell$. Let

$\displaystyle\mathcal{F}^{>\ell}_{\log}:=\mathcal{F}^{>\ell}+\sum_{i=1}^{n}(\log
y_{i})\mathcal{F}^{>\ell}.$ (9.13)

Let $\gamma$ be any admissible loop in $\mathfrak{D}(B)$ with the base point
$w$. Let us fix $\ell\geq 1$. Let $\mathfrak{D}_{\ell}(B)$ the reduction of
$\mathfrak{D}(B)$ at degree $\ell$. Suppose that $\gamma$ intersects walls of
$\mathfrak{D}_{\ell}(B)$ at points $z_{0}$, …, $z_{P-1}$ in this order. There
might be multiple walls with a common normal vector intersected by $\gamma$ at
a time. We distinguish them by allowing the multiplicity
$z_{a}=z_{a+1}=\cdots=z_{a+k}$. Then, the walls crossed at $z_{0}$, …,
$z_{P-1}$ are parametrized as $\mathbf{w}_{0}$, …, $\mathbf{w}_{P-1}$. By our
assumption, each wall $\mathbf{w}_{a}$ has the form

$\displaystyle\mathbf{w}_{a}=(\mathfrak{d}_{a},\Psi[h_{a}\mathbf{n}_{a}]^{s_{a}\delta(h_{a}\mathbf{n}_{a})})_{\mathbf{n}_{a}}.$
(9.14)

By the consistency of $\mathfrak{D}(B)$, we have
$\mathfrak{p}_{\gamma,\mathfrak{D}(B)}=\mathrm{id}$. Thus, we have

$\displaystyle\begin{split}\mathfrak{p}_{\gamma,\mathfrak{D}_{\ell}(B)}&=\Psi[h_{P-1}\mathbf{n}_{P-1}]^{\epsilon_{P}s_{P-1}\delta(h_{P-1}\mathbf{n}_{P-1})}\cdots\Psi[h_{0}\mathbf{n}_{0}]^{\epsilon_{0}s_{0}\delta(h_{0}\mathbf{n}_{0})}\equiv\mathrm{id}\\\
&\hskip 200.0pt\mod G^{>\ell},\end{split}$ (9.15)

where $\epsilon_{a}$ is the intersection sign. Let us turn it into an identity
for the modified Rogers dilogarithm $\tilde{L}$ that is parallel with the one
in Theorem 3.5.

######  Theorem 9.7 ([Nak24]).

In the situation (9.15), the following identity holds as a formal Puiseux
series in $\mathbf{y}$ with $\log$ factor:

$\displaystyle\sum_{a=0}^{P-1}\epsilon_{a}s_{a}\delta(h_{a}\mathbf{n}_{a})\tilde{L}(y_{z_{a}}[h_{a}\mathbf{n}_{a}])\equiv
0\mod\mathcal{F}^{>\ell}_{\log}.$ (9.16)

###### Proof.

We give a proof based on the classical mechanical method in Chapter 6
(especially, Proposition 6.13). Let us consider an admissible curve
$\gamma_{w}$ from the base point $w$ of $\gamma$ to a point in
$\mathrm{Int}({\mathcal{C}}^{+})$. Then, we concatenate $\gamma_{w}^{-1}$,
$\gamma$, and $\gamma_{w}$ in this order to have an admissible loop
$\gamma^{\prime}$ with the base point in $\mathrm{Int}({\mathcal{C}}^{+})$.
Then, the contribution to the LHS of (9.16) from $\gamma_{w}^{-1}$ and the one
from $\gamma_{w}$ cancel because the intersection signs therein are opposite.
Therefore, we assume that the base point $w$ of $\gamma$ is in
$\mathrm{Int}({\mathcal{C}}^{+})$ without loss of generality. We separate the
proof in two cases.

(a). Assume that $B$ is nonsingular. We use a parallel notation in Section
6.7. We consider the Hamiltonian system on the small phase space
$\tilde{M}_{0}$ for the time span $[0,P]$. The Hamiltonian for the time span
$[a,a+1]$ ($a=0$, …, $P-1$) is given by

$\displaystyle\mathscr{H}_{a}=\epsilon_{a}s_{a}\delta(h_{a}\mathbf{n}_{a})\mathrm{Li}_{2}(-y^{h_{a}\mathbf{n}_{a}}).$
(9.17)

Accordingly, for a trajectory
$\alpha=(\overline{\mathbf{u}}(t),\overline{\mathbf{p}}(t))_{t\in[0,P]}$ in
the small phase space $\tilde{M}_{0}$, we consider the quantity

$\displaystyle
S[\alpha]=\sum_{a=0}^{P-1}\mathscr{L}_{a},\quad\mathscr{L}_{a}=\epsilon_{a}s_{a}\delta(h_{a}\mathbf{n}_{a})\tilde{L}(\overline{y}(a)^{h_{a}\mathbf{n}_{a}}),$
(9.18)
$\displaystyle\overline{y}_{i}(a)=\exp\left(\sqrt{2}\delta^{-1}_{i}\overline{p}_{i}(a)\right)=\exp\biggl{(}\sqrt{2}\sum_{j=1}^{n}b_{ji}\overline{u}_{j}(a)\biggr{)},$
(9.19)

which is the counterpart of the “action integral” in (6.87). Recall that the
time-one flow of the Hamiltonian
$\mathscr{H}_{\mathbf{n},a}=a\mathrm{Li}_{2}(-y^{\mathbf{n}})$ coincides with
the action of $\Psi[\mathbf{n}]^{-a}$ on the function space as explained in
Section 7.4. For the $i$th coordinate function ${y}_{i}$ at time 0, let
${y}^{\prime}_{i}$ be the one at time $P$ after the above time development.
Then, it is described as a function of $\mathbf{y}$ as

$\displaystyle y^{\prime}_{i}$
$\displaystyle=\mathfrak{p}_{\gamma^{-1},\mathfrak{D}_{\ell}(B)}(y_{i}),$
(9.20) $\displaystyle\mathfrak{p}_{\gamma^{-1},\mathfrak{D}_{\ell}(B)}$
$\displaystyle=\Psi[h_{0}\mathbf{n}_{0}]^{-\epsilon_{0}s_{0}\delta(h_{0}\mathbf{n}_{0})}\cdots\Psi[h_{P-1}\mathbf{n}_{P-1}]^{-\epsilon_{P-1}s_{P-1}\delta(h_{P-1}\mathbf{n}_{P-1})}.$
(9.21)

Note that the dilogarithm elements act in the opposite order along the time
development on the coordinate functions.

For simplicity, suppose that the equality (9.15) is the exact one _without_
modulo $G^{>\ell}$. So, by (9.20), we have

$\displaystyle y^{\prime}_{i}=y_{i}.$ (9.22)

Then, we repeat the argument in Section 6.7. Namely, by (6.107)–(6.109), we
have $u^{\prime}_{i}=u_{i}$ and $p^{\prime}_{i}=p_{i}$. It follows from
Proposition 6.13 that

$\displaystyle\delta
S[\alpha]=\sum_{a=0}^{P-1}\delta\mathcal{L}_{\mathbf{c},a}(\alpha_{a})=\sum_{i=1}^{n}\overline{p}^{\prime}_{i}\delta\overline{u}^{\prime}_{i}-\sum_{i=1}^{n}\overline{p}_{i}\delta\overline{u}_{i}=0$
(9.23)

under any infinitesimal deformation $\alpha+\delta\alpha$ in $\tilde{M}_{0}$.
This implies that $S[\alpha]$ is constant. Moreover, the constant term is
evaluated as zero in the limit
$\overline{\mathbf{y}}=\overline{\mathbf{y}}(0)\rightarrow\mathbf{0}$ because,
in (9.18), $\overline{y}_{i}(a)\rightarrow 0$ and $h_{a}\mathbf{n}_{a}$ is
positive. By replacing $\overline{y}(a)^{h_{a}\mathbf{n}_{a}}$ with
$y_{z_{a}}[h_{a}\mathbf{n}_{a}]$, $S[\alpha]$ can be viewed as a a formal
Puiseux series in $\mathbf{y}=\mathbf{y}(0)$ with $\log$ factor. Thus, we
obtain the desired formula (9.16).

Now we consider the general case in (9.15). The equality (9.22) is replaced
with

$\displaystyle y^{\prime}_{i}=y_{i}(1+h_{i}(\mathbf{y}))$ (9.24)

for some $h_{i}(\mathbf{y})\in\mathcal{F}^{>\ell}$ due to the action of
elements in $G^{>\ell}$. Then, again by Proposition 6.13, under any
infinitesimal variation $\alpha+\delta\alpha$, we have

$\displaystyle\delta
S[\alpha]=\sum_{i=1}^{n}(\overline{p}^{\prime}_{i}\delta\overline{u}^{\prime}_{i}-\overline{p}_{i}\delta\overline{u}_{i}),$
(9.25)
$\displaystyle\delta^{-1}_{i}\overline{p}_{i}=\sum_{j=1}^{n}b_{ji}\overline{u}_{j}=\frac{1}{\sqrt{2}}\log\overline{y}_{i},\quad\delta^{-1}_{i}\overline{p}^{\prime}_{i}=\sum_{j=1}^{n}b_{ji}\overline{u}^{\prime}_{j}=\frac{1}{\sqrt{2}}\log\overline{y}^{\prime}_{i},$
(9.26)

where we set $\overline{p}_{i}=\overline{p}_{i}(0)$ and
$\overline{p}^{\prime}_{i}=\overline{p}_{i}(P)$, etc. From now on, we view
$\overline{p}_{i}$, $\overline{p}^{\prime}_{i}$, $\overline{u}_{i}$,
$\overline{u}^{\prime}_{i}$ as formal Puiseux series in
$\mathbf{y}=\mathbf{y}(0)$ with $\log$ factor, and we omit bars. By (9.24) and
(9.26), we can write them as

$\displaystyle p^{\prime}_{i}-p_{i}=f_{i}\quad
u^{\prime}_{i}-u_{i}=g_{i}\quad(f_{i},\,g_{i}\in\mathcal{F}^{>\ell}).$ (9.27)

Thus, we have

$\displaystyle\begin{split}\quad\ \delta
S[\alpha]&=\sum_{i=1}^{n}\biggl{(}(p_{i}+f_{i})\biggl{(}\delta
u_{i}+\sum_{j=1}^{n}\frac{\partial g_{i}}{\partial y_{j}}\frac{\partial
y_{j}}{\partial u_{i}}\delta u_{i}\biggr{)}-p_{i}\delta u_{i}\biggr{)}\\\
&=\sum_{i=1}^{n}((\log y_{i})\tilde{f}_{i}+\tilde{g}_{i})\delta
u_{i}\quad(\tilde{f}_{i},\,\tilde{g}_{i}\in\mathcal{F}^{>\ell})\\\
&=\sum_{i=1}^{n}((\log y_{i})\hat{f}_{i}+\hat{g}_{i})\delta
y_{i}\quad(\hat{f}_{i},\,\hat{g}_{i}\in\mathcal{F}^{>\ell-1}).\end{split}$

It follows that

$\displaystyle\frac{\partial S[\alpha]}{\partial
y_{i}}\in\mathcal{F}_{\log}^{>\ell-1}.$ (9.28)

Therefore, we have

$\displaystyle S[\alpha]\in\mathcal{F}_{\log}^{>\ell},$ (9.29)

where the constant term is evaluated as $0$ as before. This is the desired
formula (9.16).

(b). When $B$ is singular, as discussed in Section 6.6, the problem is that
those $\mathbf{y}$ represented in $\eqref{eq:ty2}$ are not algebraically
independent on the small phase space $\tilde{M}_{0}$. So, let us take the
principal extension $\overline{B}$ in (6.68), which is nonsingular. Let
$\overline{\Omega}$ be the one in (7.35). Meanwhile, let $G=G_{\Omega}$ be
unchanged, so that the CSD $\mathfrak{D}(B)$ is also unchanged. We extend
$\mathbf{y}=(y_{1},\dots,y_{n})$ to $\tilde{\mathbf{y}}=(y_{1},\dots,y_{2n})$.
Then, we extend the $y$-representation of $G$ on $\mathbb{Q}[[\mathbf{y}]]$ in
(7.22) to the _principal $y$-representation_ on
$\mathbb{Q}[[\tilde{\mathbf{y}}]]$ by

$\displaystyle\tilde{X}_{\mathbf{n}}(y^{(\mathbf{n}^{\prime},\mathbf{m}^{\prime})}):=\\{(\mathbf{n},\mathbf{0}),(\mathbf{n}^{\prime},\mathbf{m}^{\prime})\\}_{\overline{\Omega}}\,y^{(\mathbf{n}+\mathbf{n}^{\prime},\mathbf{m}^{\prime})}.$
(9.30)

Also, we introduce canonical variables
$\tilde{\mathbf{u}}=(u_{1},\dots,u_{2n})$ and
$\tilde{\mathbf{p}}=(p_{1},\dots,p_{2n})$ and represent $\tilde{\mathbf{y}}$
in the same way as (6.8), where the matrix $B$ is replaced with
$\overline{B}$. Then, $\mathbf{y}=(y_{1},\dots,y_{n})$ are now algebraically
independent on the small phase space for
$(\tilde{\mathbf{u}},\tilde{\mathbf{p}})$. After this, the proof of (a) is
applicable. ∎

Now we take the limit $\ell\rightarrow\infty$. We parametrize the intersection
of $\gamma$ and the walls in $\mathfrak{D}(B)$ as $z_{i}$ ($i\in J$) by a
countable and totally ordered set $J$ so that $\gamma$ crosses $z_{i}$ earlier
than $z_{j}$ only if $i<j$. Then, the consistent relation along $\gamma$ is
presented by an infinite product in the increasing order in $J$ from right to
left as follows:

$\displaystyle\mathfrak{p}_{\gamma,\mathfrak{D}(B)}=\prod_{i\in
J}^{\leftarrow}\Psi[h_{i}\mathbf{n}_{i}]^{\epsilon_{i}s_{i}\delta(h_{i}\mathbf{n}_{i})}=\mathrm{id}.$
(9.31)

######  Theorem 9.8 ([Nak24]).

For any admissible loop $\gamma$ with the consistent relation (9.31), the
following identity holds as a formal Puiseux series in $\mathbf{y}$ with
$\log$ factor:

$\displaystyle\sum_{i\in
J}\epsilon_{i}s_{i}\delta(h_{i}\mathbf{n}_{i})\tilde{L}(y_{z_{i}}[h_{i}\mathbf{n}_{i}])=0.$
(9.32)

###### Proof.

This is obtained by taking the limit $\ell\rightarrow\infty$ of (9.16). ∎

We call the identity (9.32), together with its reduction (9.16) at degree
$\ell$, the _DI associated with a loop $\gamma$ in $\mathfrak{D}(B)$_. When
the loop $\gamma$ is contained in the support of $G$-fan $\mathcal{G}(B)$ as
in Example 9.6, it is regarded as a $\nu$-period of a $Y$-pattern. Then, the
sum in the identity is finite, and the identity coincides with the one in
Theorem 3.5.

######  Example 9.9 (Type $A_{1}^{(1)}$).

For the consistency relation (7.51), the associated DI (3.11) is written as
follows:

$\displaystyle\tilde{L}(y_{2}(1+y_{1})^{2})+\tilde{L}(y_{1})=\tilde{L}(y_{1}(1+y_{2}(1+y_{1})^{2})^{-2})+\Lambda+\tilde{L}(y_{2}),$
(9.33)
$\displaystyle\begin{split}&\Lambda=\tilde{L}(y[(2,1)])+\tilde{L}(y[(3,2)])+\cdots\\\
&\qquad+\sum_{j=0}^{\infty}2^{1-j}\tilde{L}(y[2^{j}(1,1)])+\cdots+\tilde{L}(y[(2,3)])+\tilde{L}(y[(1,2)]).\end{split}$
(9.34)

In the sum $\Lambda$, $y_{z_{i}}[h_{i}\mathbf{n}_{i}]$ in (9.16) is
unambiguously parametrized by $h_{i}\mathbf{n}_{i}$, so that $z_{i}$ is
omitted. Also, the common factor 2 is omitted. Since the LHS is a finite sum,
the RHS converges as a function of $\mathbf{y}\in\mathbb{R}^{2}_{>0}$.
Unfortunately, we do not know the explicit expressions of $y[\mathbf{n}]$ in
the RHS in general.

####  9.3 Alternative proof of DIs and infinite reducibility

Let us show that the reduction of the consistency relations in Theorem 8.27
implies a parallel reduction of the corresponding DIs in Theorem 9.8
(including the ones in Theorem 3.5). This also provides an alternative proof
of Theorem 9.8 and also Theomem 3.5.

First, we reformulate the pentagon identity (1.35) into the form that is
closer to the pentagon relation (7.42). We continue to work on the situation
of the previous section.

######  Lemma 9.10 (Pentagon identity [Nak24]).

Let $\mathbf{n}_{1}$, $\mathbf{n}_{2}\in N^{+}$. If
$\\{\mathbf{n}_{2},\mathbf{n}_{1}\\}_{\Omega}=c\neq 0$, the following identity
as functions of $\mathbf{y}\in\mathbb{Z}_{>0}^{n}$ holds:

$\displaystyle\begin{split}&\quad\
\frac{1}{c}\Psi[\mathbf{n}_{1}]^{-1/c}(\tilde{L}(y^{\mathbf{n}_{2}}))+\frac{1}{c}\tilde{L}(y^{\mathbf{n}_{1}})\\\
&=\frac{1}{c}\Psi[\mathbf{n}_{2}]^{-1/c}\Psi[\mathbf{n}_{1}+\mathbf{n}_{2}]^{-1/c}(\tilde{L}(y^{\mathbf{n}_{1}}))+\frac{1}{c}\Psi[\mathbf{n}_{2}]^{-1/c}(\tilde{L}(y^{\mathbf{n}_{1}+\mathbf{n}_{2}}))+\frac{1}{c}\tilde{L}(y^{\mathbf{n}_{2}}).\end{split}$
(9.35)

###### Proof.

We have

$\displaystyle\Psi[\mathbf{n}_{1}]^{-1/c}(y^{\mathbf{n}_{2}})$
$\displaystyle=y^{\mathbf{n}_{2}}(1+y^{\mathbf{n}_{1}}),$
$\displaystyle\Psi[\mathbf{n}_{2}]^{-1/c}(y^{\mathbf{n}_{1}+\mathbf{n}_{2}})$
$\displaystyle=y^{\mathbf{n}_{1}+\mathbf{n}_{2}}(1+y^{\mathbf{n}_{2}})^{-1},$
$\displaystyle\begin{split}\Psi[\mathbf{n}_{2}]^{-1/c}\Psi[\mathbf{n}_{1}+\mathbf{n}_{2}]^{-1/c}(y^{\mathbf{n}_{1}})&=\Psi[\mathbf{n}_{2}]^{-1/c}(y^{\mathbf{n}_{1}}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{-1})\\\
&=y^{\mathbf{n}_{1}}(1+y^{\mathbf{n}_{2}})^{-1}(1+y^{\mathbf{n}_{1}+\mathbf{n}_{2}}(1+y^{\mathbf{n}_{2}})^{-1})^{-1}\\\
&=y^{\mathbf{n}}(1+y^{\mathbf{n}_{2}}+y^{\mathbf{n}_{1}+\mathbf{n}_{2}})^{-1}.\end{split}$

Then, the pentagon identity (1.35) is written in the form (9.35) by setting
$y_{1}=y^{\mathbf{n}_{1}}$ and $y_{2}=y^{\mathbf{n}_{2}}$. ∎

The overall factor $1/c$ was put so that the identity (9.35) exactly matches
(the log form of) the pentagon relation (7.42). Based on the correspondence
between (9.35) and (7.42), we obtain a parallel result with Theorem 8.27.

######  Theorem 9.11 ([Nak24]).

For any admissible loop $\gamma$ for $\mathfrak{D}(B)$, the associated DI in
(9.32) is constructed from and reduced to a trivial one by applying the
pentagon identity in (1.35) possibly infinitely many times.

$\mathcal{C}^{+}$$z^{\prime}_{1}$$z^{\prime}_{j^{\prime}}$$z_{1}$$z_{j}$$w$$\vdots$$\vdots$$\mathfrak{j}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$

Figure 9.3: Configuration for the consistency relation in (9.36).

###### Proof.

Fix $\ell>0$, and consider the reduction $\mathfrak{D}_{\ell}(B)$ at $\ell$.
We may concentrate on a sufficiently small loop $\gamma$ around a joint
$\mathfrak{j}$ with the base point $w$. If the joint $\mathfrak{j}$ is
parallel, the consistency relation is reduced to a trivial by applying the
commutative relation (7.41). Accordingly, the corresponding DI is also
trivial. So, let us assume that the joint $\mathfrak{j}$ is perpendicular. The
consistency relation around $\mathfrak{j}$ has the form

$\displaystyle\begin{split}&\quad\
\Psi[h^{\prime}_{j^{\prime}}\mathbf{n}^{\prime}_{j^{\prime}}]^{s^{\prime}_{j^{\prime}}\delta(h^{\prime}_{j^{\prime}}\mathbf{n}^{\prime}_{j^{\prime}})}\cdots\Psi[h^{\prime}_{1}\mathbf{n}^{\prime}_{1}]^{s^{\prime}_{1}\delta(h^{\prime}_{1}\mathbf{n}^{\prime}_{1})}\\\
&\equiv\Psi[h_{j}\mathbf{n}_{j}]^{s_{j}\delta(h_{j}\mathbf{n}_{j})}\cdots\Psi[h_{1}\mathbf{n}_{1}]^{s_{1}\delta(h_{1}\mathbf{n}_{1})}\mod
G^{>{\ell}},\end{split}$ (9.36)

where the LHS and RHS are the anti-ordered and ordered products in
Construction 8.26 (3), respectively. See Figure 9.3. The corresponding DI
(9.32) has the form

$\displaystyle\sum_{i=1}^{j^{\prime}}s^{\prime}_{i}\delta(h^{\prime}_{i}\mathbf{n}^{\prime}_{i})\tilde{L}(y_{z^{\prime}_{i}}[h^{\prime}_{i}\mathbf{n}^{\prime}_{i}])\equiv\sum_{i=1}^{j}s_{i}\delta(h_{i}\mathbf{n}_{i})\tilde{L}(y_{z_{i}}[h_{i}\mathbf{n}_{i}])\mod\mathcal{F}^{>\ell}_{\log}.$
(9.37)

In view of Figure 9.3, this is equivalent to

$\displaystyle\begin{split}&\quad\
\sum_{i=1}^{j^{\prime}}s^{\prime}_{i}\delta(h^{\prime}_{i}\mathbf{n}^{\prime}_{i})\Psi[h^{\prime}_{1}\mathbf{n}^{\prime}_{1}]^{-s^{\prime}_{1}\delta(h^{\prime}_{1}\mathbf{n}^{\prime}_{1})}\cdots\Psi[h^{\prime}_{i-1}\mathbf{n}^{\prime}_{i-1}]^{-s^{\prime}_{i-1}\delta(h^{\prime}_{i-1}\mathbf{n}^{\prime}_{i-1})}(\tilde{L}(y^{h^{\prime}_{i}\mathbf{n}^{\prime}_{i}}))\\\
&\equiv\sum_{i=1}^{j}s_{i}\delta(h_{i}\mathbf{n}_{i})\Psi[h_{1}\mathbf{n}_{1}]^{-s_{1}\delta(h_{1}\mathbf{n}_{1})}\cdots\Psi[h_{i-1}\mathbf{n}_{i-1}]^{-s_{i-1}\delta(h_{i-1}\mathbf{n}_{i-1})}(\tilde{L}(y^{h_{i}\mathbf{n}_{i}}))\\\
&\hskip 240.0pt\mod\mathcal{F}^{>\ell}_{\log}.\end{split}$ (9.38)

Along the procedure, say, $\mathcal{P}$ of obtaining (8.46) from (8.45) by
Proposition 8.24 (more precisely, by [Nak23, Algorithm III.5.7]), we do the
following procedure:

  1. 1.

As the initial input, we set $S(y)$ to be the LHS of (9.38).

  2. 2.

If the pentagon relation (7.42) is applied to the adjacent pair
$\Psi[h^{\prime}\mathbf{n}^{\prime}]^{1/c}$ and $\Psi[t\mathbf{n}]^{1/c}$ in
the procedure $\mathcal{P}$, we apply the pentagon identity (9.35) to the
corresponding terms in $S(y)$. Simultaneously, we apply the pentagon relation
(7.42) itself to the corresponding pair
$\Psi[h^{\prime}\mathbf{n}^{\prime}]^{1/c}$ and $\Psi[t\mathbf{n}]^{1/c}$ in
$S(y)$. (This does not change $S(y)$ as a formal Puiseux series in
$\mathbf{y}$ with $\log$ factor.)

  3. 3.

If the commutative relation modulo $G^{>\ell}$ is applied to the adjacent pair
$\Psi[h^{\prime}\mathbf{n}^{\prime}]^{c^{\prime}}$ and $\Psi[t\mathbf{n}]^{c}$
in the procedure $\mathcal{P}$, we also apply it to the corresponding pair
$\Psi[h^{\prime}\mathbf{n}^{\prime}]^{c^{\prime}}$ and $\Psi[t\mathbf{n}]^{c}$
in $S(y)$. (This occurs for the relation (7.41) or the truncation of the
relation (7.42) modulo $G^{>\ell}$. In the former case, this does not change
$S(y)$, while in the latter case the result equals to $S(y)$ modulo
$\mathcal{F}^{>\ell}_{\log}$.)

Then, thanks to the correspondence between (9.35) and (7.42), the final result
is the RHS of (9.38). Thus, the identity (9.38) is obtained from a trivial one
$S(y)=S(y)$. Also, by reversing the procedure, the equality (9.38) is reduced
to a trivial one. ∎

Note that the above proof of Theorem 9.11 is completely formal and relies only
on the results on the CSDs in Section 8.6 and the correspondence between the
pentagon identity (9.35) and the pentagon relation (7.42). Therefore, it
provides an alternative proof of Theorem 9.8, which is independent of the
classical mechanical method in the previous section. In particular, it gives
yet another proof of Theorem 3.5.

######  Example 9.12 (Type $B_{2}$).

Let us demonstrate how the procedure in the proof of Theorem 9.7 works in
practice. Consider the consistency relation (7.44) for type $B_{2}$. Here, we
present again all steps of ordering by the pentagon relation therein, where
the pentagon relation is applied to pairs in the parentheses.

$\displaystyle\begin{split}\begin{bmatrix}0\\\
1\end{bmatrix}\left(\begin{bmatrix}0\\\ 1\end{bmatrix}\begin{bmatrix}1\\\
0\end{bmatrix}\right)&=\left(\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 0\end{bmatrix}\right)\begin{bmatrix}1\\\
1\end{bmatrix}\begin{bmatrix}0\\\ 1\end{bmatrix}\\\ &=\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\left(\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\right)\begin{bmatrix}0\\\
1\end{bmatrix}\\\ &=\begin{bmatrix}1\\\ 0\end{bmatrix}\begin{bmatrix}1\\\
1\end{bmatrix}\begin{bmatrix}1\\\ 1\end{bmatrix}\begin{bmatrix}1\\\
2\end{bmatrix}\begin{bmatrix}0\\\ 1\end{bmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}.\end{split}$

Accordingly, we apply the procedure in the proof of Theorem 9.11 as follows:

$\displaystyle\
\uwave{\Psi[\mathbf{e}_{1}]^{-1}\Psi[\mathbf{e}_{2}]^{-1}}\tilde{L}(y^{\mathbf{e}_{2}})+\uline{\Psi[\mathbf{e}_{1}]^{-1}\tilde{L}(y^{\mathbf{e}_{2}})}+\uline{\tilde{L}(y^{\mathbf{e}_{1}})}$
$\displaystyle=$ $\displaystyle\
\Psi[\mathbf{e}_{2}]^{-1}\Psi[(1,1)]^{-1}\uline{\Psi[\mathbf{e}_{1}]^{-1}\tilde{L}(y^{\mathbf{e}_{2}})}+\Psi[\mathbf{e}_{2}]^{-1}\Psi[(1,1)]^{-1}\uline{\tilde{L}(y^{\mathbf{e}_{1}})}$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\tilde{L}(y^{(1,1)})+\tilde{L}(y^{\mathbf{e}_{2}})$
$\displaystyle=$ $\displaystyle\
\Psi[\mathbf{e}_{2}]^{-1}\uwave{\Psi[(1,1)]^{-1}\Psi[\mathbf{e}_{2}]^{-1}}\Psi[(1,1)]^{-1}\tilde{L}(y^{\mathbf{e}_{1}})$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\uwave{\Psi[(1,1)]^{-1}\Psi[\mathbf{e}_{2}]^{-1}}\tilde{L}(y^{(1,1)})$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\uline{\Psi[(1,1)]^{-1}\tilde{L}(y^{\mathbf{e}_{2}})}+\Psi[\mathbf{e}_{2}]^{-1}\uline{\tilde{L}(y^{(1,1)})}+\tilde{L}(y^{\mathbf{e}_{2}})$
$\displaystyle=$ $\displaystyle\
\Psi[\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{2}]^{-1}\Psi[(1,2)]^{-1}\Psi[(1,1)]^{-1}\Psi[(1,1)]^{-1}\tilde{L}(y^{\mathbf{e}_{1}})$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{2}]^{-1}\Psi[(1,2)]^{-1}\Psi[(1,1)]^{-1}\tilde{L}(y^{(1,1)})$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{2}]^{-1}\Psi[(1,2)]^{-1}\tilde{L}(y^{(1,1)})$
$\displaystyle\quad+\Psi[\mathbf{e}_{2}]^{-1}\Psi[\mathbf{e}_{2}]^{-1}\tilde{L}(y^{(1,2)})+\Psi[\mathbf{e}_{2}]^{-1}\tilde{L}(y^{\mathbf{e}_{2}})+\tilde{L}(y^{\mathbf{e}_{2}}).$

Here, the pentagon identity (9.35) is applied to the pairs of terms with
underline, while the pentagon relation (7.42) is applied to the terms with
wavy underline. Thus, we obtain the associated DI in the form (9.38).

Let us specialize Theorem 9.7 to the case of a period of a $Y$-pattern.

######  Corollary 9.13 ([Nak24]).

For any $\nu$-period of a $Y$-pattern, the following facts hold.

(a). The corresponding consistency relation
$\mathfrak{p}_{\gamma,\mathfrak{D}(B)}=\mathrm{id}$ in the CSD
$\mathfrak{D}(B)$ is reduced to a trivial one by applying the commutative and
pentagon relations in Proposition 7.7 possibly infinitely many times.

(b). The associated DI in Theorem 3.5 is reduced to a trivial one by applying
the pentagon identity (1.35) possibly infinitely many times.

Now we come back to the reduction problem in Section 1.5 for the DI associated
with a loop in a CSD. (All DIs in this text fall into this class.) As shown in
Theorem 9.7, it is reduced to the trivial one by applying the pentagon
identity possibly infinitely many times. We call this property _infinite
reducibility_. Of course, infinitely many operations are necessary for DIs
with an infinite product. However, our method only proves the infinite
reducibility even for DIs with a finite product. This is a somewhat unexpected
answer to the problem.

We give two remarks on the reducibility problem.

  * (1)

Corollary 9.13 (b) does not deny the finite reducibility for a DI with a
finite sum as we see in Example 9.12. Also, a finite reduction may occur
differently from the procedure in Theorem 9.11. So, the finite reducibility
problem for the above DIs with a finite product is still open.

  * (2)

Corollary 9.13 (a) does not imply that any $\nu$-period of a $Y$-pattern is
reduced to the trivial one by the square and pentagon periodicities _in the
$Y$-pattern itself_. Again, Example 9.12 is such a case because there is only
the hexagon periodicity in the $Y$-pattern. Therefore, the reducibility of a
period in a $Y$-pattern and the one in a CSD are _different_ problems. Some
other examples of the periodicity that are not reducible in a $Y$-pattern are
known [FST08, KY20].

Notes

The whole content of this chapter is due to [Nak24].

##  Part IV Quantum dilogarithm identities

###  Chapter 10 Quantum dilogarithm

A natural $q$-deformation of the dilogarithm called the _quantum dilogarithm_
was introduced by Faddeev, Kashaev, Volkov [FV93, FK94] and modified by Fock-
Goncharov [FG09a]. The function satisfies the pentagon identity for
$q$-commutative variables, which is regarded as the counterpart of the
pentagon identity for the dilogarithm.

####  10.1 $q$-exponential

Let $q$ and $x$ be variables. Let $j$ be a nonnegative integer. We introduce
the _$q$ -Pochhammer symbol_

$\displaystyle(x;q)_{\infty}=\prod_{k=0}^{\infty}(1-xq^{k}),\quad(q)_{j}=\prod_{k=1}^{j}(1-q^{k}),\quad(q)_{0}=1,$
(10.1)

and the _$q$ -number_ ($a\in\mathbb{Z}$)

$\displaystyle[a]_{q}=\frac{q^{a}-q^{-a}}{q-q^{-1}},\quad\lim_{q\rightarrow
1}[a]_{q}=a.$ (10.2)

We define the _$q$ -exponential_ $e_{q}(x)$, first as a formal power series in
$x$ with coefficients in $\mathbb{Q}(q)$, by the following three equivalent
expressions:

$\displaystyle e_{q}(x)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{1}{(q)_{j}}x^{j}=\sum_{j=0}^{\infty}\frac{1}{(1-q)\cdots(1-q^{j})}x^{j}$
(10.3)
$\displaystyle=\frac{1}{(x;q)_{\infty}}=\prod_{k=0}^{\infty}(1-q^{k}x)^{-1}$
(10.4)
$\displaystyle=\exp\biggl{(}\sum_{j=1}^{\infty}\frac{1}{j(1-q^{j})}x^{j}\biggr{)}.$
(10.5)

Here, we follow the notation in [Kir95]. Observe the similarity between the
third expression (10.5) and (the exponential of) the Euler dilogarithm
$\mathrm{Li}_{2}(x)$ in (1.1).

The above three expressions are equivalent due to the following
characterization of $e_{q}(x)$.

######  Proposition 10.1.

(a). The properties

$\displaystyle e_{q}(0)$ $\displaystyle=1\quad\text{(initial condition)},$
(10.6) $\displaystyle e_{q}(qx)$
$\displaystyle=(1-x)e_{q}(x)\quad\text{($q$-difference relation)}$ (10.7)

uniquely determine a formal power series $e_{q}(x)$ in $x$ with coefficients
in $\mathbb{Q}(q)$.

(b). All expressions (10.3), (10.4), and (10.5) satisfy the conditions (10.6)
and (10.7).

###### Proof.

(a). We set

$\displaystyle
e_{q}(x)=\sum_{n=0}^{\infty}f_{n}(q)x^{n}\quad(f_{n}(q)\in\mathbb{Q}(q)),$
(10.8)

where $f_{0}(q)=1$ by (10.6). Then, by (10.7), we have the recursion relation

$\displaystyle q^{n}f_{n}(q)=f_{n}(q)-f_{n-1}(q)\quad(n\geq 1).$ (10.9)

This determines $f_{n}(q)$ uniquely as

$\displaystyle
f_{n}(q)=\frac{1}{1-q^{n}}f_{n-1}(q)=\frac{1}{(1-q)\cdots(1-q^{n})}.$ (10.10)

(b). We have already seen in (10.10) that the expression (10.3) satisfies the
conditions in (a). Thus, it is enough to check the relation (10.7) for the
expressions (10.4) and (10.5). For (10.4), we see it as

$\displaystyle\prod_{k=0}^{\infty}(1-q^{k}qx)^{-1}=(1-x)\prod_{k=0}^{\infty}(1-q^{k}x)^{-1}.$
(10.11)

For (10.5), we see it as

$\displaystyle\begin{split}\exp\biggl{(}\sum_{j=1}^{\infty}\frac{1}{j(1-q^{j})}q^{j}x^{j}\biggr{)}&=\exp\biggl{(}-\sum_{j=1}^{\infty}\frac{1}{j}x^{j}\biggr{)}\exp\biggl{(}\sum_{j=1}^{\infty}\frac{1}{j(1-q^{j})}x^{j}\biggr{)}\\\
&=(1-x)\exp\biggl{(}\sum_{j=1}^{\infty}\frac{1}{j(1-q^{j})}x^{j}\biggr{)}.\end{split}$
(10.12)

∎

Next, we specialize $q\in\mathbb{C}^{\times}$ such that $q$ is not a root of
1. Then, every expression for $e_{q}(x)$ in (10.3)–(10.5) makes sense as a
formal power series in $x$ with coefficients in $\mathbb{C}$. Furthermore,
suppose that $|q|<1$. Then, its convergence radius is 1 by (10.3). Thus, we
may also regard $e_{q}(x)$ as a complex function of $x\in\mathbb{C}$ with
$|x|<1$; which is analytically continued to the meromorphic function in
(10.4).

We may consider the complex parameter $q$ ($|q|<1$) as a “quantization
parameter” so that the limit $q\rightarrow 1^{-}$ is the _classical limit_.
Equivalently, we introduce the _Planck constant_ $\hbar\in\mathbb{C}$ with
$\mathrm{Im}\,\hbar>0$ by

$\displaystyle q=\exp(i\hbar),\quad i=\sqrt{-1}.$ (10.13)

Then, the limit $\hbar\rightarrow 0^{+i}$ is the classical limit, where
$0^{+i}$ indicates that $\hbar$ approaches to $0$ with $\mathrm{Im}\,\hbar>0$
in the complex plane.

######  Proposition 10.2 ([FK94, Eq. (1.5)], [Kir95, Cor. 10]).

The following asymptotic expansion holds in the limit $\hbar\rightarrow
0^{+i}$:

$\displaystyle
e_{q}(x)=(1-x)^{-1/2}\exp\biggl{(}\frac{-\mathrm{Li}_{2}(x)}{i\hbar}\biggr{)}(1+\mathcal{O}(\hbar)).$
(10.14)

###### Proof.

Let $q^{1/2}=\exp(i\hbar/2)$. By the expression (10.5), we have

$\displaystyle\begin{split}\log
e_{q}(x)&=\sum_{j=1}^{\infty}\frac{1}{j(1-q^{j})}x^{j}\\\
&=\sum_{j=1}^{\infty}\frac{1}{j}x^{j}-\sum_{j=1}^{\infty}\frac{q^{j/2}}{j(q^{j/2}-q^{-j/2})}x^{j}\\\
&=-\log(1-x)-\frac{1}{q^{1/2}-q^{-1/2}}\sum_{j=1}^{\infty}\frac{q^{j/2}}{j[j]_{q^{1/2}}}x^{j}.\end{split}$

The following expansion formulas hold:

$\displaystyle q^{j/2}$
$\displaystyle=1+\frac{j}{2}i\hbar+\mathcal{O}(\hbar^{2}),$ (10.15)
$\displaystyle[j]_{q^{1/2}}$ $\displaystyle=j+\mathcal{O}(\hbar^{2}),$ (10.16)
$\displaystyle\frac{1}{q^{1/2}-q^{-1/2}}$
$\displaystyle=\frac{1}{i\hbar+\mathcal{O}(\hbar^{3})}=\frac{1}{i\hbar}+\mathcal{O}(\hbar).$
(10.17)

Thus, we have

$\displaystyle\begin{split}\log
e_{q}(x)&=-\log(1-x)-\frac{1}{i\hbar}\mathrm{Li}_{2}(x)+\frac{1}{2}\log(1-x)+\mathcal{O}(\hbar).\end{split}$
(10.18)

This gives the desired result (10.14). ∎

The function $1/e_{q}(x)$ was called the _quantum dilogarithm_ in [FK94].
Here, we reserve the name for its variant introduced in the next section.

####  10.2 Quantum dilogarithm

Following Fock-Goncharov [FG09a], we define the _quantum dilogarithm_
$\Psi_{q}(x)$ by

$\displaystyle\Psi_{q}(x)=e_{q^{2}}(-qx).$ (10.19)

Thus, it is essentially the $q$-exponential in the previous section with a
slightly different parametrization. As we will see, this function fits our
approach to the quantum dilogarithm identities.

In parallel with (10.3)–(10.5), we have the following three equivalent
expressions:

$\displaystyle\Psi_{q}(x)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-q)^{j}}{(q^{2})_{j}}x^{j}=\sum_{j=0}^{\infty}\frac{1}{(q-q^{-1})^{j}}\frac{1}{[1]_{q}\cdots[n]_{q}}x^{j},$
(10.20)