Two variations on $(A_3\times A_1\times A_1)^{(1)}$ type discrete Painlev\'e equations

TL;DR

This paper investigates two symmetry variations of discrete Painlevé equations derived from specific Coxeter group normalizers, clarifying their structure and role in integrability using Brink-Howlett theory.

Contribution

It introduces two new symmetry structures for discrete Painlevé equations from Coxeter group normalizers, expanding understanding of their algebraic properties.

Findings

Constructed two normalizer-based symmetry groups from a $(A_3 imes A_1 imes A_1)^{(1)}$ subroot system.

Connected non-translational elements to discrete Painlevé equations.

Applied Brink-Howlett theory to clarify the nature of these symmetry elements.

Abstract

By considering the normalizers of reflection subgroups of types $A_1^{(1)}$ and $A_3^{(1)}$ in $\widetilde{W}\left(D_5^{(1)}\right)$, two normalizers: $\widetilde{W}\left(A_3\times A_1\right)^{(1)}\ltimes {W}(A_1^{(1)})$ and $\widetilde{W}\left(A_1\times A_1\right)^{(1)}\ltimes {W}(A_3^{(1)})$ can be constructed from a $(A_{3}\times A_1\times A_1)^{(1)}$ type subroot system. These two symmetries arose in the studies of discrete \Pa equations \cite{KNY:2002, Takenawa:03, OS:18}, where certain non-translational elements of infinite order were shown to give rise to discrete \Pa equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups \cite{BH}. This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.

Equations

$$\Delta =\{{\alpha }_i\mid 1\le i\le n\}\text{mbox}and{\Delta }^{(1)}=\{{\alpha }_i\mid 0\le i\le n\},$$

$$\Phi =W.\Delta \text{mbox}and{\Phi }^{(1)}=W^{(1)}.{\Delta }^{(1)}.$$

$$\delta ={\alpha }_0+\tilde{\alpha }={\alpha }_0+\sum _{i=1}^n{c}_i{\alpha }_i,$$

$$\delta ={\alpha }_0+\tilde{\alpha }={\alpha }_0+{\alpha }_1+2{\alpha }_2+2{\alpha }_3+{\alpha }_4+{\alpha }_5.$$

$$⟨{\alpha }_i,h_j⟩={\delta }_{ij},⟨{\alpha }_i,h_{\delta }⟩=⟨\delta ,h_j⟩=0\text{mbox}for(1\le i,j\le n),\text{mbox}and⟨\delta ,h_{\delta }⟩=1.$$

$$⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=A_{ij}$$

$${\alpha }_i^{\vee }=\sum _{j=1}^n{A}_{ij}{h}_j,$$

$${\alpha }_0^{\vee }=-\sum _{i=1}^n{c}_i{\alpha }_i^{\vee },$$

$$s_i{\alpha }_j={\alpha }_j-⟨{\alpha }_j,{\alpha }_i^{\vee }⟩{\alpha }_i={\alpha }_j-A_{ji}{\alpha }_i.$$

$$⟨w^{-1}f,h⟩=⟨f,wh⟩,\text{mbox}forf\in V^{(1)},h\in V^{(1)\ast },w\in W^{(1)}.$$

$$s_i{\alpha }_j^{\vee }={\alpha }_j^{\vee }-A_{ij}{\alpha }_i^{\vee },$$

$$s_{w(\alpha )}=w{s}_{\alpha }{w}^{-1},\text{mbox}for\alpha \in {\Delta }^{(1)},w\in W^{(1)}.$$

$$⟨{\alpha }_i,h_j⟩={\delta }_{ij},⟨{\alpha }_i,h_{\delta }⟩=⟨\delta ,h_j⟩=0\text{mbox}for(1\le i,j\le 5),\text{mbox}and⟨\delta ,h_{\delta }⟩=1.$$

$$⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=A_{ij}$$

$$(A_{ij}{)}_{i,j=0}^5=\left(\begin{array}{cccccc}2& 0& -1& 0& 0& 0\\ 0& 2& -1& 0& 0& 0\\ -1& -1& 2& -1& 0& 0\\ 0& 0& -1& 2& -1& -1\\ 0& 0& 0& -1& 2& 0\\ 0& 0& 0& -1& 0& 2\end{array}\right).$$

$${\alpha }_i^{\vee }=\sum _{j=1}^5{A}_{ij}{h}_j\text{mbox}and{\alpha }_0^{\vee }=-({\alpha }_1^{\vee }+2{\alpha }_2^{\vee }+2{\alpha }_3^{\vee }+{\alpha }_4^{\vee }+{\alpha }_5^{\vee })=-h_2.$$

$$H=\{h\in V^{(1)\ast }|⟨\delta ,h⟩=1\},\text{mbox}forh\in H.$$

$$t_{{\mu }^{\vee }}h=h+{\mu }^{\vee },$$

$$t_{{\mu }^{\vee }}{\alpha }_i={\alpha }_i-⟨{\alpha }_i,{\mu }^{\vee }⟩\delta ={\alpha }_i-{\mu }_i\delta .$$

$$0=⟨\delta ,{\mu }^{\vee }⟩$$

$$w{t}_{{\mu }^{\vee }}{w}^{-1}=t_{w{\mu }^{\vee }},w\in {\tilde{W}}^{(1)}.$$

$$t_{{\mu }^{\vee }}^h=\left(\begin{array}{cc}{I}_n& {\mathbf{\mu }}^{\vee }\\ 0& 1\end{array}\right),$$

$$t_{{\mu }^{\vee }}^{\delta }=(t_{{\mu }^{\vee }}^h{)}^{-1}=\left(\begin{array}{cc}{I}_n& -{\mathbf{\mu }}^{\vee }\\ 0& 1\end{array}\right).$$

$${\sigma }_1{s}_{\{0,1,2,3,4,5\}}$$

$${\sigma }_2{s}_{\{0,1,2,3,4,5\}}$$

$${\sigma }_1{\sigma }_2{s}_{\{0,1,2,3,4,5\}}$$

$${\sigma }_1=(10),{\sigma }_2=(05)(14)(23),{\sigma }_1{\sigma }_2=(5140)(23).$$

$$\delta ={\alpha }_0+{\alpha }_1.$$

$$⟨{\alpha }_1,h_1⟩=1,\text{mbox}and⟨\delta ,h_1⟩=0.$$

$${\alpha }_1^{\vee }=2h_1.$$

$$t_{h_1}=\pi s_1,$$

Full text

1. Introduction

Formulation of Painlevé equations and their generalisations as birational representations of Weyl groups provides us with an elegant and efficient way to characterise and study these highly transcendental, nonlinear equations. In particular, it is well-known that discrete evolutions of Painlevé equations are given by translational elements of extended affine Weyl groups. Examples include the work of Okamoto on the symmetries of the six classical Painlevé equations [10], Sakai’s classification of second-order discrete Painlevé equations [12], and generalisations of Painlevé equations such as Kajiwara, Noumi and Yamada’s $\widetilde{W}\left(A_{n}\times A_{m}\right)^{(1)}$ system (KNY) [7], Sasano [14] and Masuda’s [8] $D_{n}$ type systems, and more recently Okubo and Suzuki’s $(A_{2n+1}\times A_{1}\times A_{1})^{(1)}$ system [11].

Theory of Weyl groups, or in general Coxeter groups [4] is a classical area of algebra rich with remarkable properties, for which significant results and breakthroughs are still being made, long-standing conjectures proved [3]. Characterisation of integrable systems in terms of such groups naturally leads us to the question: how much of the intrinsic properties of the group remain in the realisations of integrable equations and what are their implications in this particular context? For example, varieties of subgroup relations exist between Coxeter groups of different type. Do different integrable systems relate in a way that is inherent with these subgroup structures?

An example of this kind was given by an interesting link discovered by Takenawa [17] between a KNY system for the case $n=1,m=3$, that is a $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ type discrete Painlevé equation and Sakai’s $\widetilde{W}\left(D_{5}^{(1)}\right)$ $q$-Painlevé equation. It was thought that KNY’s $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ system should be a second-order $q$-differece Painlevé equation, however it does not coincide in symmetry with any equations in Sakai’s list. In 2003 Takenawa established via algebraic geometrical means, the fact that KNY’s $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ equation can be embedded as a sub-system of Sakai’s $\widetilde{W}\left(D_{5}^{(1)}\right)$ equation. In particular, it was found that in this embedding the element $\phi$ which give rise to $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ type Painlevé equation is not a translation in $\widetilde{W}\left(D_{5}^{(1)}\right)$, whereas the element $\phi^{2}$ is translational. Similar examples have been found in different contexts in the integrable system literature [6, 15, 1, 5]. A common feature of these examples is that the element $\phi$ which gives rise to a discrete Painlevé equation is not translational, whereas some powers of $\phi$ is. We refer to such an element as quasi-translational. In the present work, we show that quasi-translational elements are in fact certain elements of infinite order occur in normalizers of Weyl groups. Moreover, we clarify the quasi-translational nature of such elements in terms of normalizer theory.

Our approach is illustrated through a simple example. We look at the normalizer theory for an underlying $(A_{3}\times A_{1}\times A_{1})^{(1)}$ type subsystem of a $D_{5}^{(1)}$ root system. We show that Takenawa’s embedding of KNY’s $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ equation and the $n=1$ case of Okubo and Suzuki’s $(A_{3}\times A_{1}\times A_{1})^{(1)}$ system coincide with the two variations of normalizers arising from an $(A_{3}\times A_{1}\times A_{1})^{(1)}$ subsystem of Sakai’s $\widetilde{W}\left(D_{5}^{(1)}\right)$ $q$-Painlevé equation.

The paper is organised as follows. In Section 2 we give a brief summary of some well-known facts and properties of Coxeter groups relevant to our discussion. The main results are given in Section 3, where we show that two subgroups of $\widetilde{W}\left(D_{5}^{(1)}\right)$ with an underlying $(A_{3}\times A_{1}\times A_{1})^{(1)}$ subroot system : $\widetilde{W}\left(A_{3}\times A_{1}\right)^{(1)}\ltimes{W}(A_{1}^{(1)})$ and $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{3}^{(1)})$ arise as the normalizers of Weyl subgroups of type $A_{1}^{(1)}$ and $A_{3}^{(1)}$, respectively. We define quasi-translational elements as certain elements of the normalizer, their properties are described in detail in Sections 3.2 and 3.3. In Section 4, the normalizer theory developed in Section 3 is placed in the context of discrete Painlevé equations. In particular, we discuss the quasi-translational nature of the elements which give rise to discrete Painlevé equations given by Takenawa [17], and Okubo and Suzuki [11] in terms of the normalizer theory. Concluding remarks and some future directions are given in Section 5.

2. Properties of Coxeter groups

Let $W=\langle s_{i}\mid s_{i}^{2}=1,(s_{i}s_{j})^{m_{ij}}=1,\;1\leq i,j\leq n\rangle$ be a finite reflection group or Coxeter group whose defining relations of the generators are encoded in a corresponding Dynkin diagram. When parameter $m_{ij}\in\{2,3,4,6\}$, known as the crystallographic condition, W is called a finite Weyl group. Its affine extension $W^{(1)}=\langle s_{i}\mid s_{i}^{2}=1,(s_{i}s_{j})^{m_{ij}}=1,\;0\leq i,j\leq n\rangle$, is a group of infinite order with a corresponding extended Dynkin diagram. Dynkin diagram is a diagram consisting of vertices and bonds: each vertex of the diagram represents a generator $s_{i}$ is labeled by $i$, for $0\leq i\leq n$. The parameter $m_{ij}$ takes value of: $2,3,4$ or $6$ when two vertices labeled $i$ and $j$ are respectively: disconneted, joined by a single, a double, or a triple bond. Diagrams which have only single bonds are called simply-laced, they are of types $A_{n}$, $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. The non-simply laced types are $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$. For each $s_{i}$ in the generating set of $W^{(1)}$, known as a simple reflection, we have a corresponding simple root $\alpha_{i}$. Vertices of Dynkin diagram can be equivalently labelled by either simple reflections or simple roots. We have

[TABLE]

which are the simple systems of finite and affine type, respectively. The finite and affine root systems are then given respectively by

[TABLE]

Any root of $\Phi^{(1)}$ can be expressed in the form $\alpha+k\delta$ for $k\in\mathbb{Z}$ and $\alpha\in\Phi$, where

[TABLE]

is called the null root, and $\tilde{\alpha}$ the highest root. The value of $c_{i}$ for all Weyl groups can be found in any reference books on Coxeter groups, [4] for example.

Example 1**.**

A Weyl group of $D^{(1)}_{5}$ type. Let $W(D_{5})=\langle s_{i}\mid 1\leq i\leq 5\rangle$ and $W(D^{(1)}_{5})=\langle s_{i}\mid 0\leq i\leq 5\rangle$ be the finite and affine Weyl group of type $D_{5}$, respectively. Their defining relations are encoded in the Dynkin diagram of Figure 1.

Simple systems of finite and affine type are given respectively by $\Delta=\{\alpha_{i}\mid 1\leq i\leq 5\}$ and $\Delta^{(1)}=\{\alpha_{i}\mid 0\leq i\leq 5\}$; $\Phi=W({D}_{5}).\Delta$ and $\Phi^{(1)}=W({D}^{(1)}_{5}).\Delta^{(1)}$ the finite and affine root systems. The null root in this case is given by

[TABLE]

From now on we adapt to the notation: $\alpha_{i}+...+\alpha_{j}=\alpha_{i...j}$ to express a sum of simple roots, and $s_{i}...s_{j}=s_{i...j}$ for product of simple reflections. For example we now write $\delta=\alpha_{01223345}$.

Finite and affine Weyl groups $W$ and $W^{(1)}$ can be realised as groups of reflections in real vector spaces $V$ and $V^{(1)}$ with bases $\Delta$ and $\Delta^{(1)}$, respectively. From definition of $\delta$, we see that the set $\{\alpha_{1},\ldots,\alpha_{n},\delta\}$ is also a basis of $V^{(1)}$. For our purpose, it is instructive to introduce the dual spaces of $V$ and $V^{(1)}$, given by the bilinear pairing $\langle{}\,,{}\rangle:$ $V^{(1)}\times V^{(1)\ast}\to\mathbb{R}$,

[TABLE]

Dual spaces $V^{\ast}$ and $V^{(1)\ast}$ have bases $\{h_{1},\ldots,h_{n}\}$ and $\{h_{1},\ldots,h_{n},h_{\delta}\}$, respectively. Vectors $h_{i}\;(1\leq i\leq n)$ are called the fundamental weights, and $P=\mathbb{Z}\{h_{1},\ldots,h_{n}\}$ is the weight lattice.

Simple coroots ${\alpha}^{\vee}_{i}\in V^{(1)\ast}$ are defined by

[TABLE]

for all $0\leq i,j\leq n$, and $(A_{ij})_{i,j=0}^{n}$ is the generalised Cartan matrix. Let $Q=\mathbb{Z}\{{\alpha}^{\vee}_{1},\ldots,{\alpha}^{\vee}_{n}\}$ be the coroot lattice. It can be easily checked that as a consequence of Equations (5) and (6) we have

[TABLE]

and in particular for simply-laced Weyl groups we have

[TABLE]

where $c_{i}$ are given in Equation (3). Let $s_{\alpha}$ denotes reflection along the hyperplane orthogonal to a root $\alpha\in\Phi^{(1)}$. The generator $s_{i}=s_{\alpha_{i}}$ of $W^{(1)}$ then is reflection along the hyperplane orthogonal to the simple root $\alpha_{i}\in\Delta^{(1)}$, given by the formula,

[TABLE]

$W^{(1)}$ acts on $V^{(1)\ast}$ via the contragredient action:

[TABLE]

That is $s_{i}$ acts on the dual space $V^{(1)\ast}$ by ,

[TABLE]

for all $0\leq i,j\leq n$. Moreover we have the following property for elements of reflection

[TABLE]

The group $W^{(1)}$ acts transitively on the root system, that is for all $\beta_{1},\beta_{2}\in\Phi^{(1)}$, there is a $w\in W^{(1)}$ such that $w\beta_{1}=\beta_{2}$, which means that reflection associated to any root of the root system can be presented as a conjugation to a simple reflection.

Example 2**.**

Weyl group of $D_{5}^{(1)}$ type realised as group of reflections. Real vector spaces $V$ and $V^{(1)}$ for which the Weyl group of $D^{(1)}_{5}$ type can be realised as groups of reflections have bases $\Delta=\{\alpha_{i}\mid 1\leq i\leq 5\}$ and $\Delta^{(1)}=\{\alpha_{i}\mid 0\leq i\leq 5\}$, respectively. The corresponding dual spaces $V^{\ast}$ and $V^{(1)\ast}$ with bases $\{\,h_{1},h_{2},h_{3},h_{4},h_{5}\}$ and $\{\,h_{1},h_{2},h_{3},h_{4},h_{5},h_{\delta}\}$ are defined by the bilinear pairing $\langle{}\,,{}\rangle:$ $V^{(1)}\times V^{(1)\ast}\to\mathbb{R}$,

[TABLE]

Vectors $h_{i}\;(1\leq i\leq 5)$ are the fundamental weights, and $P=\mathbb{Z}\{h_{1},h_{2},h_{3},h_{4},h_{5}\}$ is the $D_{5}$ weight lattice. The set of simple coroots in $V^{(1)\ast}$ is ${\Delta^{(1)}}^{\vee}=\{{\alpha}^{\vee}_{0},{\alpha}^{\vee}_{1},{\alpha}^{\vee}_{2},{\alpha}^{\vee}_{3},{\alpha}^{\vee}_{4},{\alpha}^{\vee}_{5}\}$ where we have

[TABLE]

for all $0\leq i,j\leq 5$, and $(A_{ij})_{i,j=0}^{5}$ is the generalised Cartan matrix of type $D_{5}$ given by

[TABLE]

Moreover we have

[TABLE]

The group $W({D}^{(1)}_{5})$ is realised as a group of reflections in $V^{(1)}$ and $V^{(1)\ast}$ by formulae (9) and (11), respectively.

2.1. Translations in the weight lattice.

One can form an extended affine Weyl group $\widetilde{W}^{(1)}=W^{(1)}\rtimes A$, where $A$ is a group of certain Dynkin diagram automorphisms acting on $W^{(1)}$ via conjugation. It is a remarkable property of the Weyl group that with appropriate extension $\widetilde{W}^{(1)}$ decomposes into a semidirect product of the finite Weyl group $W$ and translations on weight lattice $P$: $\widetilde{W}^{(1)}=W\ltimes P$. Translational elements of $\widetilde{W}^{(1)}$ are best understood by looking at their actions on a hyperplane $H$ in the dual space $V^{(1)\ast}$ defined by,

[TABLE]

Let ${\mu}^{\vee}\in V^{\ast}$ be a point on the weight lattice $P$, that is ${\mu}^{\vee}=\sum_{i=1}^{n}\mu_{i}h_{i}$, for $\mu_{i}\in\mathbb{Z}$. We define a translational element $t_{{\mu}^{\vee}}\in\widetilde{W}^{(1)}$ such that it acts on $h\in H$ by

[TABLE]

whereas its action on simple affine roots $\alpha_{i}\in\Delta^{(1)}$ is given by

[TABLE]

That is, $t_{{\mu}^{\vee}}$ shifts $\alpha_{i}$ by $-\mu_{i}$ multiples of $\delta$ for $1\leq i\leq n$. By property of the null root given in Equation (5), coefficients $\mu_{i}$ ($0\leq i\leq n$) satisfy the constraint

[TABLE]

where $c_{0}=1$ . Moreover, we have the following property for translational elements,

[TABLE]

Let the column vector $(a_{1},\ldots,a_{n},a_{\delta})^{T}$ where $a_{1},\ldots,a_{n},a_{\delta}\in\mathbb{R}$ be the coordinate vector of the dual space $V^{(1)\ast}$ in basis $\{h_{1},\ldots,h_{n},h_{\delta}\}$. Then by definition given in Equation (17), all vectors in $H\subset V^{(1)\ast}$ are of the form $(a_{1},\ldots,a_{n},1)^{T}$. Translational element $t_{{\mu}^{\vee}}$ as an $(n+1)\times(n+1)$ matrix of linear transformation on $V^{(1)\ast}$ in basis $\{h_{1},\ldots,h_{n},h_{\delta}\}$ acting from the left given by Equation (18) is then:

[TABLE]

where $I_{n}$ is the $n\times n$ Identity matrix, and $\bm{{\mu}^{\vee}}=(\mu_{1},...,\mu_{n})^{T}$ denotes the $n$-column coordinate vector of ${\mu}^{\vee}$. By contragredient action element $t_{{\mu}^{\vee}}$ as an $(n+1)\times(n+1)$ matrix of linear transformation on $V^{(1)}$ in basis $(\alpha_{1},\ldots,\alpha_{n},\delta)$ acting from the right is given by Equation (19):

[TABLE]

Example 3**.**

Translations in $\widetilde{W}\left(D_{5}^{(1)}\right).$

Let $\sigma_{1},\sigma_{2}$ be two diagram automorphisms of $D_{5}^{(1)}$ type Dynkin diagram (see Figure 1), acting on the simple reflections of $W(D^{(1)}_{5})$ via conjugation:

[TABLE]

If we write the above actions as permutations of the index set $\{0,1,2,3,4,5\}$ we have:

[TABLE]

Then it is easily seen that element $\sigma_{1}\sigma_{2}$ is of order 4. The fact that the index of connection between the weight and coroot lattice for $D_{odd}$ type Weyl groups is 4 tells us that we need to extend $W(D^{(1)}_{5})$ by $A=\langle\sigma_{1}\sigma_{2}\rangle$, a cyclic group of order four, in order for the extended affine Weyl group to have the decomposition of finite Weyl group and weight lattice $P$, that is \widetilde{W}\bigl{(}D^{(1)}_{5}\bigr{)}=W(D^{(1)}_{5})\rtimes A=W(D_{5})\ltimes P.

In the following, we give explicit examples of elements of translation in \widetilde{W}\bigl{(}A^{(1)}_{1}\bigr{)} and \widetilde{W}\bigl{(}A^{(1)}_{3}\bigr{)} which will be relevant for our later discussion of quasi-translations in the context of discrete Painlevé equations.

Example 4**.**

Translations in \widetilde{W}\bigl{(}A^{(1)}_{1}\bigr{)}.

For a simple system $\Delta^{(1)}=\{\alpha_{1},\alpha_{0}\}\subset V^{(1)}$ of type $A^{(1)}_{1}$ , we have the extended affine Weyl group \widetilde{W}\bigl{(}A^{(1)}_{1}\bigr{)}=\langle s_{1},s_{0},\pi\rangle, where $\pi$ is the Dynkin diagram automorphism that exchanges the two roots $\alpha_{1}$ and $\alpha_{0}$. Defining relations of the generators are encoded by the Dynkin diagram in Figure 2.

The null root in this case is given by

[TABLE]

The fundamental weight $h_{1}\in V^{(1)\ast}$ is defined by

[TABLE]

This together with the definition of the null root in Equation (24) implies that $\langle\alpha_{0},h_{1}\rangle=-1$. Coroot ${\alpha}^{\vee}_{1}$ is related to $h_{1}$ by the Cartan matrix of type $A_{1}$,

[TABLE]

The weight lattice is $P=\mathbb{Z}\{h_{1}\}$. Translation on the weight lattice by $h_{1}$ is given by the element

[TABLE]

which as matrices of linear transforms on vector spaces $V^{(1)\ast}$ and $V^{(1)}$ in bases $\{h_{1},h_{\delta}\}$ and $\{\alpha_{1},\delta\}$ are respectively:

[TABLE]

For example, we can read off the action of element $t_{h_{1}}$ on root $\alpha_{1}$ by looking at the first row of the matrix $t_{h_{1}}^{\delta}$ , that is we have

[TABLE]

The action of $t_{h_{1}}$ on $\alpha_{0}$ can then be inferred by definition (24) of $\delta$ and the condition given in Equation (20), hence we have

[TABLE]

Example 5**.**

Translations in \widetilde{W}\bigl{(}A^{(1)}_{3}\bigr{)}.

For a simple system $\Delta^{(1)}=\{\beta_{0},\beta_{1},\beta_{2},\beta_{3}\}$ of type $A^{(1)}_{3}$, we have \widetilde{W}\bigl{(}A^{(1)}_{3}\bigr{)}=\langle s_{\beta_{0}},s_{\beta_{1}},s_{\beta_{2}},s_{\beta_{3}},p_{1}p_{2}\rangle, whose defining relations are encoded in the Dynkin diagram of Figure 3.

Elements $p_{1}$ and $p_{2}$ are diagram automorphisms of order 2,

[TABLE]

whereas the element $p_{1}p_{2}$ is of order 4,

[TABLE]

corresponding to a clockwise rotation by $\frac{\pi}{2}$ of the type $A^{(1)}_{3}$ Dynkin diagram. The null root in this case is given by

[TABLE]

The fundamental weights $h_{1},h_{2},h_{3}$ of \widetilde{W}\bigl{(}A^{(1)}_{3}\bigr{)} defined by Equation (5), are related to coroots ${\beta}^{\vee}_{1},{\beta}^{\vee}_{2},{\beta}^{\vee}_{3}$ by Cartan matrix of type $A_{3}$,

[TABLE]

Translation of $h_{1}$ is given by the element

[TABLE]

which as matrices of linear transforms in bases $\{h_{1},h_{2},h_{3},h_{\delta}\}\subset V^{(1)\ast}$ and $\{\beta_{1},\beta_{2},\beta_{3},\delta\}\subset V^{(1)}$ are respectively:

[TABLE]

From matrix $t_{h_{1}}^{\delta}$, expression of $\delta$ given in Equation (32) and condition given by Equation (20) we see that action of $t_{h_{1}}$ on the $A^{(1)}_{3}$ type simple system is given by

[TABLE]

The weight lattice $P=\mathbb{Z}\{h_{1},h_{2},h_{3}\}$ of \widetilde{W}\bigl{(}A^{(1)}_{3}\bigr{)} is three dimensional. In order to describe translation in a general direction on the weight lattice we need two more linearly independent directions. Using Equation (21), we can define

[TABLE]

That is we have,

[TABLE]

where subscript in the lower case $t$ notations indicates the direction of translation on $P$ in terms of the three fundamental weight vectors. Notice that we have $t_{1}t_{2}t_{3}t_{4}=1$, so essentially there are only three independent directions.

2.2. Centralizer and normalizer

Two objects relevant for our later discussion are centralizer and normalizer of a subgroup of a Coxeter group. For this purpose, let us equip the real vector space $V^{(1)}$ with a degenerate bilinear form $({}\,,{}):$ $V^{(1)}\times V^{(1)}\to\mathbb{R}$, given by

[TABLE]

where $(\alpha_{i},\alpha_{i})$ gives the squared length of the root $\alpha_{i}$. Note that we have

[TABLE]

For simply-laced types we have $A_{ji}=A_{ij}=-1$, that is all the simple roots are of the same length which we set to be $2$ without any lost of generality.

In the case of non-simply-laced types, we have two different root length given by

[TABLE]

for the case of $m_{ij}=4$ or $6$ respectively, where we have let $\alpha_{i}$ with $(\alpha_{i},\alpha_{i})=2$, to be the long root and $\alpha_{j}$ the short root. The bilinear form is degenerate since by definition of $\delta$ and the generalised Cartan matrix we have $(\delta\,,\alpha_{i})=0$ for all $i\in\{0,1,...,n\}$. In particular, we have $(\delta\,,\delta)=0$.

In terms of the bilinear form $({}\,,{})$, action of the reflection element $s_{\alpha}\in\widetilde{W}^{(1)}$ for $\alpha\in\Phi^{(1)}$, is given by

[TABLE]

for any $v\in V^{(1)}$. We see that $s_{\alpha}$ fixes $v$ whenever $(\alpha\,,v)=0$. Hence, $s_{\alpha}$ acts trivially on $\delta$ for all $\alpha\in\Phi^{(1)}$.

Let $\beta$ be any root of $\Phi^{(1)}$. The centralizer of $\beta$ denoted by $C(\beta)$ is a reflection subgroup of $W^{(1)}$, generated by $\langle s_{\alpha}\mid\alpha\in\Phi^{(1)}\;\mbox{and}\;(\alpha,\beta)=0\rangle$. We can chose $\beta=\alpha_{0}$ without any lost of generality since $W^{(1)}$ acts transitively on $\Phi^{(1)}$ and look at $C(\alpha_{0})=\langle s_{\alpha}\mid\alpha\in\Phi^{(1)}\;\mbox{and}\;(\alpha,\alpha_{0})=0\rangle$.

Let $J\subset\Delta^{(1)}$, the group $W_{J}=\langle s_{i}\mid\alpha_{i}\in J\rangle$ is called standard parabolic subgroup of $\widetilde{W}^{(1)}$, and its conjugates are parabolic subgroups. The normaliser of $W_{J}$ in $\widetilde{W}^{(1)}$ is defined by

[TABLE]

which says that the normalizer is the semidirect product of $W_{J}$ by a complement $N_{J}$. In particular, we have

[TABLE]

That is, element of $N_{J}$ either fixes or permutes the elements of $J$. In other word $N_{J}$ is the setwise stabilizer of $J$.

Remark 1**.**

For an arbitraty group $G$, the normaliser of a subgroup usually is no more than the subgroup itself. However, it was shown by Brink and Howlett [2] that when $G$ is a Coxeter group and the subgroup a parabolic one, one can have interesting and highly non-trivial $N_{J}$. In [2], a systematic description and an algorithm to calculate normalizers of parabolic subgroups for an arbitrary Coxeter group was introduced.

In the present work, we are interested in the normalizer of $W_{J^{\prime}}=\langle s_{i}\mid\alpha_{i}\in J^{\prime}\rangle$, the affine extension of a parabolic subgroup $W_{J}$, that is

[TABLE]

where $J\subset\Delta^{(1)}$ and $\tilde{\alpha}_{J}$ is the highest root of the finite root system generated by $J$. That is we want to investigate

[TABLE]

The group $N_{J^{\prime}}$ is the setwise stabilizer of $J^{\prime}$, consisting of elements that stabilise the set $J$ and those that exchange elements of $J$ with the root $\delta-\widetilde{\alpha}_{J}$. The former are just the elements of $N_{J}$ whose description is given in [2]. What is left to do is to describe the group elements of $\widetilde{W}^{(1)}$ which exchange the roots in $J$ with $\delta-\widetilde{\alpha}_{J}$. So although Brink-Howlett theory of normalizers of parabolic subgroups of Coxeter groups do not apply directly to the problems considered in this paper, a large part of the work is done.

3. Centralizer and normalizers in $\widetilde{W}\left(D_{5}^{(1)}\right)$

3.1. $(A_{3}\times A_{1}\times A_{1})^{(1)}$ subroot system

Let us consider the group $\widetilde{W}\left(D_{5}^{(1)}\right)$ in Example 1. The centralizer $C(\alpha_{0})$ is a reflection subgroup of $\widetilde{W}(D^{(1)}_{5})$, generated by all the roots in the $D_{5}^{(1)}$ root system which are orthogonal to $\alpha_{0}$: $\langle s_{\alpha}\mid\alpha\in\Phi^{(1)}\;\mbox{and}\;(\alpha,\alpha_{0})=0\rangle$. First let us describe such roots in the finite $D_{5}$ root system, that is $\Omega=\{\alpha\in\Phi\mid(\alpha,\alpha_{0})=0\}$. It is easily seen that $\Omega=\Omega_{1}\cup\Omega_{2}$ is a standard parabolic subsystem of $\Phi$ of type $A_{1}\times A_{3}$ generated by two disjoint simple systems of types $A_{1}$ and $A_{3}$: $\{\alpha_{1}\}\cup\{\alpha_{3},\alpha_{4},\alpha_{5}\}$, corresponding to the vertices of the $D_{5}^{(1)}$ Dynkin diagram that are not joint to the one representing $\alpha_{0}$ (see Figure 1). The root system of $C(\alpha_{0})$ given by $\Omega^{(1)}=\{\alpha\in\Phi^{(1)}\mid(\alpha,\alpha_{0})=0\}$ has the form

[TABLE]

The generating set of root system $\Omega_{i}^{(1)}$ contains those of $\Omega_{i}$ and one extra root: $\delta-\tilde{\alpha}_{i}$, where $\tilde{\alpha}_{i}$ denotes the highest root of the finite root system $\Omega_{i}$. This ensures that the affine root systems have the form $\Omega_{i}^{(1)}=\{\alpha+k\delta\;|\;k\in\mathbb{Z}\;\mbox{and}\;\alpha\in\Omega_{i}\}$, for $i=1,2$. Here we have $\tilde{\alpha}_{1}=\alpha_{1}$ and $\tilde{\alpha}_{2}=\alpha_{345}$. Hence $\Omega^{(1)}$ is generated by two disjoint simple systems $\{\alpha_{1},\delta-\alpha_{1}\}\cup\{\alpha_{3},\alpha_{4},\alpha_{5},\delta-\alpha_{345}\}=\{\alpha_{1},\alpha_{0223345}\}\cup\{\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{01223}\}$.

To match the embedding given in [17], let $w=s_{132}$, such that we have $w\alpha_{0}=\alpha_{0123}=\gamma_{0}$, and

[TABLE]

We refer to the root system generated by $\{\beta_{0},\beta_{1},\beta_{2},\beta_{3}\}$, $\{\gamma_{0},\gamma_{1}\}$ and $\{\eta_{0},\eta_{1}\}$ which is of $(A_{3}\times A_{1}\times A_{1})^{(1)}$ type as $\beta-\gamma-\eta$ -system. The corresponding Dynkin diagram is given in Figure 4.

From Equation (49) we know that the $\left(A_{1}\times A_{3}\right)^{(1)}$ type root system of $C(\gamma_{0})=C(w\alpha_{0})$ is given by $w\Omega^{(1)}=w\Omega_{1}^{(1)}\cup w\Omega_{2}^{(1)}$, generated by $\{w\alpha_{1},w\alpha_{0223345}\}\cup\{w\alpha_{3},w\alpha_{4},w\alpha_{5},w\alpha_{01223}\}=\{\eta_{0},\eta_{1}\}\cup\{\beta_{0},\beta_{1},\beta_{2},\beta_{3}\}$. The group $C(\gamma_{0})$ is then given by

[TABLE]

One can easily find the reflection elements that generate the groups $W_{\gamma}$, $W_{\eta}$ and $W_{\beta}$ using the fact $s_{w(\alpha)}=ws_{\alpha}w^{-1}$, for $\alpha\in\Delta^{(1)}$ and $w\in W(D^{(1)}_{5})$. We have:

[TABLE]

3.2. The first variation on the $(A_{3}\times A_{1}\times A_{1})^{(1)}$

subsystem

Now consider the $A_{1}^{(1)}$ type Weyl group $W_{\gamma}=\langle s_{\gamma_{0}},s_{\gamma_{1}}\rangle$. According to Equation (48), normalizer of $W_{\gamma}$ in $\widetilde{W}\left(D_{5}^{(1)}\right)$ is given by $N(W_{\gamma})=N_{\gamma}\ltimes W_{\gamma}$, where $N_{\gamma}$ is the setwise stabiliser of the simple $\gamma$-system $\{{\gamma_{0}},{\gamma_{1}}\}$. Centraliser $C(\gamma_{0})$ fixes point-wisely the simple $\gamma$-system $\{{\gamma_{0}},{\gamma_{1}}\}$. It can be easily checked that diagram automorphism $\sigma_{1}\sigma_{2}$ exchanges ${\gamma_{0}}$ and ${\gamma_{1}}$, hence is in the normalizer. Let the element $g^{\prime}$ be a minimum length representative in the subset of elements of ${W}\left(D_{5}^{(1)}\right)$ that exchange ${\gamma_{0}}$ and ${\gamma_{1}}$. It can be shown that $g^{\prime}=s_{0145}$. So we have $N_{\gamma}=C(\gamma_{0})\rtimes\langle g^{\prime},\sigma_{1}\sigma_{2}\rangle$. Moreover $g^{\prime}$ commutes with $\sigma_{1}\sigma_{2}$ so that the group $\langle g^{\prime},\sigma_{1}\sigma_{2}\rangle$ is of order order 8, acting as diagram automorphisms on the simple systems of $\gamma$, $\eta$ and $\beta$-systems. We list actions of some of its elements in Equation (53). Element that exchanges ${\gamma_{0}}$ and ${\gamma_{1}}$ is indicated by having a $\pi_{\gamma}$ in the corresponding row. Similarly, actions that exchanges the two roots of the $\eta$-system is indicated by a $\pi_{\eta}$ in the corresponding row. Recall that diagram automorphisms on an $A_{3}^{(1)}$ $\beta$-system (see Figure 4) are generated by $p_{1}$ and $p_{2}$,

[TABLE]

We use these to indicate the actions of elements of $\langle g^{\prime},\sigma_{1}\sigma_{2}\rangle$ on the $\beta$-system. Whenever an element acts trivially on a system, the symbol $-$ is used.

[TABLE]

From Equation (53), we can see that the element $g^{\prime}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}$ acts on $\eta$-$\gamma$-systems only, while products of $\sigma_{1}$ and $\sigma_{2}$ acts on $\beta$-$\gamma$-systems only. Hence we have the following decomposition of $N_{\gamma}$ as a direct products of two extended affine Weyl groups,

[TABLE]

Finally we have the normalizer of $W_{\gamma}$ in $\widetilde{W}(D^{(1)}_{5})$:

[TABLE]

Given the $A_{1}^{(1)}$ and $A_{3}^{(1)}$ type extended affine Weyl groups of $\gamma$ and $\beta$-systems in the normalizer above one can follow the recipe in Examples 4 and 5 to construct elements of quasi-translation. First, take the $A_{1}^{(1)}$ type $\eta$-system $\widetilde{W}_{\eta}=\langle s_{\eta_{0}},s_{\eta_{1}},g^{\prime}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}\rangle$. By Equation (27), define element $t_{h_{\eta_{1}}}$ by

[TABLE]

where the subscript indicates its association with the fundamental weight $h_{\eta_{1}}$ of $\eta$-system. Action of $t_{h_{\eta_{1}}}$ on the simple roots of the $D^{(1)}_{5}$ system is given by,

[TABLE]

We can clearly see that $t_{h_{\eta_{1}}}$ is not an element of translation in $\widetilde{W}(D^{(1)}_{5})$. This fact is better understood by looking at its action on the $\gamma-\eta-\beta$-system:

[TABLE]

While $t_{h_{\eta_{1}}}$ behaves like a translation on the $\eta$ and $\beta$-systems, it acts as a permutation of order 2 on the $\gamma$-system. This permutative effect on the $\gamma$-system can be eliminated by iterating it twice, that is

[TABLE]

which in fact is a translation in $\widetilde{W}(D^{(1)}_{5})$ by $2h_{\eta_{1}}=\eta_{1}$. We refer to elements such as $t_{h_{\eta_{1}}}$ quasi-translations. We do not see this permutative action of $t_{h_{\eta_{1}}}$ when considering only the subspace of $V$ orthogonal to $\gamma$-system. That is, we have $V=V_{\gamma}\oplus V_{\gamma}^{\perp}$, where $V_{\gamma}=\mbox{Span}(\{\gamma_{0},\gamma_{1}\})$, and $V_{\gamma}^{\perp}=\mbox{Span}(\{\eta_{0},\eta_{1},\beta_{0},\beta_{1},\beta_{2},\beta_{3}\})$. The element $t_{h_{\eta_{1}}}$ is then a translation on the subspace $V_{\gamma}^{\perp}$.

For $A_{3}^{(1)}$ type $\beta$-system $\widetilde{W}_{\beta}=\langle s_{\beta_{0}},s_{\beta_{1}},s_{\beta_{2}},s_{\beta_{3}},\sigma_{1}\sigma_{2}\rangle$, quasi-translational element associated to the fundamental weight $h_{\beta_{1}}$ is given by the formula in Equation (34),

[TABLE]

Element $t_{h_{\beta_{1}}}$ acts like a translation on the $\beta$-system while permutes the two simple roots of $\gamma$-system:

[TABLE]

Simple roots of $\beta$-system have the form $\beta_{j}=\sum_{i=1}^{5}a^{i}_{j}\alpha_{i}$, and the corresponding coroots ${\beta}^{\vee}_{j}=\sum_{i=1}^{5}a^{i}_{j}{\alpha}^{\vee}_{i}$ (for $j=1,2,3$), where coefficients $a^{i}_{j}$ are given in Equation (50). Fundamental weights of the $\beta$-system are then given by

[TABLE]

Quasi-translations along the other directions of weight lattice of the $\beta$-system can be then given by

[TABLE]

That is we have,

[TABLE]

where $T_{1}T_{2}T_{3}T_{4}=1$.

3.3. A second variation on the $(A_{3}\times A_{1}\times A_{1})^{(1)}$

subsystem

For the same underlying $(A_{3}\times A_{1}\times A_{1})^{(1)}$ type $\beta-\eta-\gamma$ subroot system another subgroup $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{3}^{(1)})$ of $\widetilde{W}(D^{(1)}_{5})$ can be formed by considering the normalizer of the $A_{3}^{(1)}$ type $\beta$-system $W_{\beta}=\langle s_{\beta_{0}},s_{\beta_{1}},s_{\beta_{2}},s_{\beta_{3}}\rangle$ in $\widetilde{W}(D_{5}^{(1)})$. We have $N(W_{\beta})=N_{\beta}\ltimes W_{\beta}$, where $N_{\beta}=C(\beta)\rtimes\langle g^{\prime},\sigma_{1}\sigma_{2}\rangle$ is the setwise stabilizer of $\{\beta_{1},\beta_{2},\beta_{0},\beta_{3}\}$. The centralizer of $\beta$-system $C(\beta)$ is generated by reflections along roots of the $D_{5}^{(1)}$ root system which are orthogonal to the $\beta$-system. From our earlier discussion in Section 3.1 we know they form the $\eta$ and $\gamma$-systems given in Equation (50). Hence we have

[TABLE]

The $\langle g^{\prime},\sigma_{1}\sigma_{2}\rangle$ part of $N_{\beta}$ permutes the simple $\beta$-system, acting on it as diagram automorphisms. Moreover, it can be decomposed into elements that either act on $\eta-\beta$-system or $\gamma-\beta$-system only, so that a direct product of two extended Weyl groups for the $\eta$ and $\gamma$-systems given in Equation (67) can be formed. An inspection of Equation (53) tells us that $\sigma_{1}\sigma_{2}$, and $g^{\prime}\sigma_{1}\sigma_{2}$ are just such two elements. Hence normalizer of the $W_{\beta}$ in $\widetilde{W}(D_{5}^{(1)})$ is given by

[TABLE]

Two quasi-translations can be defined for the $\widetilde{W}\left(A_{1}^{(1)}\right)$ type $\eta$ and $\gamma$-systems in the normalizer above, associated with fundamental weights $h_{\eta_{1}}$ and $h_{\gamma_{1}}$, respectively. We have

[TABLE]

Their actions on $\gamma-\eta-\beta$-system are given respectively by

[TABLE]

and

[TABLE]

Elements $t_{h_{\gamma_{1}}}$ and $t_{h_{\eta_{1}}}$ act on $\beta$-system as permutations of order 4, thus they become translations after four iterations, that is $t_{h_{\gamma_{1}}}^{4}$ and $t_{h_{\eta_{1}}}^{4}$ are translations on the $D_{5}$ weight lattice.

4. Normalizer theory in the context of a $\widetilde{W}(D_{5}^{(1)})$ $q$-Painlevé equation

4.1. Takenawa’s embedding of KNY’s $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$

system

We see that it corresponds exactly to the normalizer $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}\ltimes{W}(A_{1}^{(1)})$ in $\widetilde{W}(D_{5}^{(1)})$ given in Equation (55), where we have chosen a conjugation of the $(A_{3}\times A_{1}\times A_{1})^{(1)}$ type subroot system of $\widetilde{W}(D_{5}^{(1)})$ to coincide with that of [17]. The element $\phi$ which gives rise to discrete Painlevé equation [17, Thm 3.2] corresponds to $t_{h_{\eta_{1}}}$ given by Equation (56), an element of infinite order in the $N_{\gamma}$ part of the normalizer, which acts permutatively on the simple $\gamma$-system given in Equation (58). The fact that $\phi$ iterated twice becomes a translation is due to the fact that it acts on $\gamma$-system as a permutation of order 2. The $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}$ symmetry of the KNY system is recovered by considering only the subspace orthogonal to the $\gamma$-system, which is spanned by the $\left(A_{1}\times A_{3}\right)^{(1)}$ type $\eta-\beta$-system.

4.2. Okubo-Suzuki system $n=1$ case, $(A_{3}\times A_{1}\times A_{1})^{(1)}$

In 2018, Okubo and Suzuki proposed a new $(A_{2n+1}\times A_{1}\times A_{1})^{(1)}$ type generalisation of Sakai’s $\widetilde{W}(D_{5}^{(1)})$ $q$-PVI equation from the framework of Cluster algebra, which we refer as the OS-system. It contains four previously known generalisations of the $q$-PVI equation [11]:

• •

$T_{1}$: $q$-Drinfeld-Sokolov system $q$-$P_{(n+1,n+1)}$ [16],

• •

$T_{2}$: Sakai’s $q$-Garnier system [13],

• •

$T_{3}$: Nagao-Yamada’s variation of the $q$-Garnier system [9],

• •

$T_{4}$: Tsuda’s $q$-UC hierarchy [18].

However, if one looks at the defining relations for the Weyl group symmetries of OS system given in [11, Thm 2.1], it can be seen that we have in fact the Weyl group $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{2n+1}^{(1)})$. For the case $n=1$, the $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{3}^{(1)})$ type system, as we have shown in Section 3.3, can be embedded inside a $\widetilde{W}(D_{5}^{(1)})$ symmetry group. In fact, we have shown that it arises exactly as the normalizer of $A_{3}^{(1)}$ type $\beta$-system $W_{\beta}=\langle s_{\beta_{0}},s_{\beta_{1}},s_{\beta_{2}},s_{\beta_{3}}\rangle$ in $\widetilde{W}(D_{5}^{(1)})$.

Now we are in the position to analyse in detail the nature of the four directions $T_{i}$ $(1\leq i\leq 4)$ in terms of the properties of the two normalizers $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{3}^{(1)})$ and $\widetilde{W}\left(A_{1}\times A_{3}\right)^{(1)}\ltimes{W}(A_{1}^{(1)})$ with an underlying subroot system $(A_{3}\times A_{1}\times A_{1})^{(1)}$ of $\widetilde{W}(D_{5}^{(1)})$. In particular, we describe the four directions in terms of the fundamental weights $h_{i}$ ($1\leq i\leq 5$) of $\widetilde{W}(D_{5}^{(1)})$, given by Equation (13).

The correspondence between our notations and those in [11] is given as follows,

[TABLE]

The four directions $T_{i}$ ($1\leq i\leq 4$) given in [11] are then given by,

[TABLE]

First, let us look at the $T_{1}$ direction. We have

[TABLE]

In general, elements such as $t_{h_{\gamma_{1}}}$ or $t_{h_{\eta_{1}}}$ are not translations, since they act on $\beta$-system as permutations of order four as shown in Equations (70) and (71). However, composition $t_{h_{\gamma_{1}}}t_{h{\eta_{1}}}$ is a translation since the permutative actions of $t_{h_{\gamma_{1}}}$ and $t_{h_{\eta_{1}}}$ on $\beta$-system are exactly inverse of each other so applying one after another cancel out the permutative effect and we obtain indeed a translation on $\gamma-\eta-\beta$-system,

[TABLE]

or on the simple system of $D_{5}^{(1)}$,

[TABLE]

We see that the element $(t_{\gamma_{1}}t_{\eta_{1}})^{-1}$ is a translation by

[TABLE]

To describe $T_{i}$ ($i=2,3,4$) we make use of elements of normalizer $\left(\widetilde{W}(A_{1}^{(1)})\times\widetilde{W}(A_{3}^{(1)})\right)\ltimes{W}(A_{1}^{(1)})$ given in Equation (55). In particular, the quasi-translational elements in $\widetilde{W}(A_{3}^{(1)})$ part of the normalizer. We have

[TABLE]

Quasi-translational element $t_{-h_{\beta_{3}}}$, defined in Equation (66), acts on the ${W}(A_{1}^{(1)})$ type $\gamma$-system as a permutation of order 2. Therefore, applied twice we have a translation on $\gamma-\eta-\beta$-system,

[TABLE]

or equivalently on the simple system of $D_{5}^{(1)}$,

[TABLE]

In particular, we have

[TABLE]

where we have used Equations (62) and (7).

For $T_{3}$ direction we have

[TABLE]

Quasi-translational element $t_{h_{\beta_{1}}}$ is defined in Equation (60), whose action on the $\gamma-\eta-\beta$-system is given by Equation (61).

On composition with $s_{\gamma_{1}}$, which acts on the $\gamma-\eta-\beta$-system by

[TABLE]

we obtain a translation on the $\gamma-\eta-\beta$-system

[TABLE]

or equivalently on the simple system of $D_{5}^{(1)}$ as

[TABLE]

The element $s_{\gamma_{1}}t_{-h_{\beta_{1}}}$ is a translation by

[TABLE]

Lastly for $T_{4}$ direction we have

[TABLE]

Since both elements $t_{h_{\beta_{1}}}$ and $t_{h_{\beta_{3}}-h_{\beta_{2}}}$, given by Equations (60) and (65) respectively, permute the simple $\gamma$-system: $\gamma_{0}\leftrightarrow\gamma_{1}$, applying one after the other cancels out the effect of the permutation thus gives us a translation. Its action on the $\gamma-\eta-\beta$-system is given by,

[TABLE]

and on the simple system of $D_{5}^{(1)}$,

[TABLE]

Element $t_{h_{\beta_{1}}}t_{h_{\beta_{3}}-h_{\beta_{2}}}$ is a translation on the $D_{5}^{(1)}$ weight lattice by

[TABLE]

Remark 2**.**

In general, from a $(D_{2n+1}\times A_{1}\times A_{1})^{(1)}$ subsystem of a $D_{2n+3}^{(1)}$ root system, by consider either the normalizer of a $D_{2n+1}^{(1)}$ or an $A_{1}^{(1)}$ type subsystem two subgroups $\widetilde{W}\left(A_{1}\times A_{1}\right)^{(1)}\ltimes{W}(D_{2n+1}^{(1)})$ and $\widetilde{W}\left(D_{2n+1}\times A_{1}\right)^{(1)}\ltimes{W}(A_{1}^{(1)})$ can be respectively formed. For the $n=1$ case these two symmetry groups coincide with the OS-system and Takenawa’s embedding in Sakai’s $\widetilde{W}\left(D_{5}^{(1)}\right)$ $q$-PVI equation. The normalizer procedure can be used to obtain generalisations of Painlevé equations as subsystems systems with bigger symmetry groups. Such a system was in fact given by Masuda in his $D_{n}^{(1)}$ generalisation of Sakai’s $\widetilde{W}\left(D_{5}^{(1)}\right)$ $q$-PVI equation [8].

5. Conclusion

In the present work, we explained the nature and properties of quasi-translational elements arising in the study of Painlevé equations based on theory of normalizers of Coxeter groups. In particular, we showed that the quasi-translational nature of such elements resulted from the fact that they act permutatively on some reflection subgroups of the original group. The order of permutation decides the number of iteration for which they become actual translations in the weight lattice. Quasi-translations arise under different guises in a variety of contexts in the theory of discrete integrable equations. For example, in “symmetrisation” procedures of asymmetric discrete Painlevé equations known as projective reductions [6, 15]; from reductions of partial difference equations [1]; or as elements that give rise to discrete equations which govern evolutions of Schramm’s circle patterns [5], these we plan to discuss in subsequent publications.