# Enumerating Finite Braid Group Orbits on $SL_{2}(\mathbb{C})$-Character
Varieties

Amal Vayalinkal

###### Abstract

We analyze finite orbits of the natural braid group action on the character
variety of the $n$ times punctured sphere. Building on recent results relating
middle convolution and finite complex reflection groups, our work implements
Katz’s middle convolution to explicitly classify finite orbits in the
$SL_{2}(\mathbb{C})$-character variety of the punctured sphere. We provide
theoretical results on the existence of finite orbits arising from the
imprimitive finite complex reflection groups and formulas for constructing
such examples when they exist. In the primitive finite complex reflection
groups, we perform an exhaustive search and provide computational results. Our
contributions also include Magma computer code for middle convolution and for
computing the orbit under this action when it is known to be finite.

###### Contents

  1. 1 Introduction
    1. 1.1 Prior Work
  2. 2 Background
    1. 2.1 Middle Convolution
    2. 2.2 Finite Complex Reflection Groups
    3. 2.3 Representations and Character Varieties
  3. 3 Methods
    1. 3.1 Computing The Braid Group Orbit
    2. 3.2 Finding Nice Tuples
  4. 4 Imprimitive Reflection Groups
    1. 4.1 Nice 3-Tuples in $G(m,p,3)$
    2. 4.2 Nice 4-Tuples in $G(m,p,3)$
    3. 4.3 Nice 4-Tuples in $G(m,p,4)$
    4. 4.4 Nice 5-Tuples in $G(m,p,4)$
    5. 4.5 Nice $n$-Tuples in $G(m,p,n)$ for $n\geq 5$
  5. 5 Primitive Reflection Groups
    1. 5.1 Nice 3-Tuples in Rank 3 Groups
    2. 5.2 Nice 4-Tuples in Rank 3 Groups
    3. 5.3 Nice 4-Tuples in Rank 4 Groups
    4. 5.4 Nice 5-Tuples in Rank 4 Groups
  6. A Middle Convolution Tables For Nice 3-Tuples
  7. B Middle Convolution Tables of Nice 4-Tuples
  8. C Middle Convolution Tables for Nice 5-Tuples

##  1 Introduction

Dynamics of the natural braid group action on character varieties has been
studied for over a century. The growing interest in classifying the finite
orbits of this action has led to a wealth of new techniques and examples.
Inspired by Boalch [1] who explored finite complex reflection groups as a
source of finite orbits, Lam-Landesman-Litt [17] showed, up to mild
conditions, all finite orbits come from finite complex reflection groups via
Katz’s middle convolution operation [15]. With the goal of completing their
analysis, we search the finite complex reflection groups to enumerate all
possibilities and study them. We classify all finite orbits arising via middle
convolution from the finite complex reflection groups.

Let $\Sigma_{0,T+1}$ denote the $T$+1-punctured sphere. Given $T$ and $n$, a
rank $n$ representation of the fundamental group of $\Sigma_{0,T+1}$ can be
identified with a tuple of matrices $(A_{1},\dots A_{T+1})\in
GL_{n}(\mathbb{C})^{T+1}$ satisfying

$\displaystyle\prod_{i=1}^{T+1}A_{i}=\text{Id}$

The mapping class group of $\Sigma_{0,T+1}$, denoted
$\operatorname{Mod}_{0,T+1}$ has a well-defined action on such
representations. It is well known that $\operatorname{Mod}_{0,T+1}$ is
isomorphic to the spherical braid group on $T+1$ strands, denoted $B_{T+1}$.
This group has $T$ generators $\sigma_{1},\dots,\sigma_{T}$ and acts on
$T$+1-tuples in the following way:

$\sigma_{i}(A_{1},A_{2},\dots,A_{T},A_{T+1})=(A_{1},\dots,A_{i-1},A_{i}A_{i+1}A_{i}^{-1},A_{i},A_{i+2},\dots,A_{T+1})$

If we further consider such representations up to simultaneous conjugation,
this action descends to an action on the character variety i.e. on the
isomorphism classes of representations.

######  Definition 1.0.1.

Conjugacy classes of tuples $[A_{1},\cdots,A_{T+1}]$ with finite orbit under
this action are called MCG-finite and we say a tuple is irreducible if the
associated representation is irreducible.

Recently, Lam-Landesman-Litt [17] classified finite orbits of rank two tuples
with at least one matrix of infinite order. They proved that such Zariski
dense representations come from pullbacks of curves or arise via middle
convolution from finite complex reflection groups.

The first kind, “pullback” representations were classified by Diarra [8].

Finite irreducible complex reflection groups, which can be viewed as
generalizations of Coxeter groups, were classified by Shephard and Todd in
1954 as belonging to an infinite family $G(m,p,n)$ for each rank $n$ or being
one of 34 exceptional cases denoted $G_{3}$ to $G_{37}$ [9]. In view of this
classification, Lam-Landesman-Litt [17, Cor. 5.1.4] prove that for $T+1\geq
7$, MCG-finite representations do not come from groups of rank five or
greater. Their paper leaves open for analysis the infinite family of rank
three and four and most of the exceptional groups of rank three and four, as
well as the case when $T+1<7$, which we investigate.

Katz [15] originally introduced middle convolution in the language of perverse
sheaves and one of our contributions is implementing computer code to
explicitly compute the middle convolution given a tuple of matrices, following
the algebraic reformulation of Dettweiler-Reiter [6, Section 2]. We also
provide code for computing the braid group orbit for MCG-finite 4-tuples and
5-tuples in the $SL_{2}(\mathbb{C})$-character variety of the punctured
sphere. This computer code is publicly accessible online at the following
link: amalvayalin.github.io/research.html and we hope it encourages further
applications of this operation.

######  Definition 1.0.2.

A $\mathbf{nice}$ $T$-tuple is a tuple (up to simultaneous conjugation) from a
finite complex reflection group such that its middle convolution produces a
rank 2 MCG-finite $T+1$-tuple.

For the precise definition of when a tuple is nice, see Definition 3.0.1. An
explanation for how we get MCG-finite $T+1$-tuples from a $T$-tuple is
provided in Remark 3.0.2. Since middle convolution depends on a parameter
$\lambda$ (see Section 2.1), the same nice $T$-tuple can give rise to distinct
conjugacy classes of rank 2 MCG-finite $T+1$-tuples, as discussed in Remark
3.0.3.

With this definition in mind, our problem of studying rank 2 MCG-finite tuples
changes to studying nice tuples in the finite complex reflection groups. Using
properties of complex reflection groups, as well as prior work on this topic,
it remains to study nice $n$ and $n+1$-tuples in the rank $n=3,4$ primitive
groups, nice $n+1$-tuples in the imprimitive groups of rank $2<n\leq 5$ and
nice $n$-tuples in imprimitive groups of rank $n\geq 3$. More details on why
these are the only remaining cases is provided in Remark 3.0.4, Remark 4.0.5
and [17, Cor. 5.1.4].

In the imprimitive groups, we either prove no such nice tuples exist, or
provide a system of equations for an infinite family of solutions. We found
nice tuples in all the primitive reflection groups of rank three. Among the
rank four primitive reflection groups, all but $G_{29}$ and $G_{31}$ contain
nice tuples. For each nice tuple in the primitive groups, we provide
representatives (up to simultaneous conjugation and braid group orbit
equivalence) before and after middle convolution. We also list the size of the
subgroup generated by the rank 2 matrices, and in most cases we provide an
approximate ‘size’ for the braid group orbit.

######  Theorem 1.0.3.

There are no nice $n$-tuples in $G(m,p,n)$ for $n\geq 5$.

######  Theorem 1.0.4.

In the rank 3 imprimitive groups, an infinite family of nice $3$-tuples exist
in $G(m,1,3)$ and $G(m,m,3)$ and in each case we provide conditions on the
eigenvalues of the product of the matrices in the tuple. Nice $4$-tuples exist
in $G(m,p,3)$ for $m>1$ if and only if one of the following holds:

  1. 1.

$p=m$ and $m\neq 3$.

  2. 2.

$p=1$ and $\gcd(m,3)=1$.

  3. 3.

$p\neq 1,m$ and $p=3$.

In all three cases the parameter for middle convolution is a root of unity of
order $m$.

######  Theorem 1.0.5.

In the rank 4 imprimitive groups, nice $4$-tuples and nice $5$-tuples exist in
$G(m,p,4)$ if and only if $p=m$ and $m=2$ or $m=4$.

The proof of these theorems, as well as further conditions on the eigenvalues
of the product of the matrices and equations for how to construct such
examples are provided in Section 4.

In our computations for the primitive groups, nice tuples within each group
are partitioned into different $\mathbf{types}$ based on eigenvalues (see
Definition 3.2.1 and Remark 3.2.2).

######  Theorem 1.0.6.

For each primitive reflection group of rank three, we summarize the distinct
rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group
equivalence) produced by taking middle convolution of the different nice
tuples (up to type equivalence). We also note how many of the rank 2 tuples
generate an infinite subgroup.

  1. 1.

In $G_{23}$, there are three nice 3-tuples, resulting in six distinct MCG-
finite 4-tuples after middle convolution. Three of the six generate infinite
subgroups. This group also has one nice 4-tuple, which after middle
convolution produces one MCG-finite 5-tuple that generates an infinite
subgroup.

  2. 2.

In $G_{24}$, there is only one nice 3-tuple, which produced three distinct
MCG-finite 4-tuples after middle convolution, all generating infinite
subgroups. There does not exist any nice 4-tuples in this group.

  3. 3.

In $G_{25}$, there are four nice 3-tuples, resulting in eight distinct MCG-
finite 4-tuples after middle convolution. Five of these generate infinite
subgroups. This group also has three nice 4-tuples, which produced two
distinct MCG-finite 4-tuples after middle convolution. Only one of these
4-tuples generates an infinite subgroup.

  4. 4.

In $G_{26}$, there are three nice 3-tuples, resulting in seven distinct MCG-
finite 4-tuples after middle convolution. Five of these generate an infinite
subgroup. In this group there are no nice 4-tuples.

  5. 5.

In $G_{27}$, there are three nice 3-tuples, resulting in nine distinct MCG-
finite 4-tuples after middle convolution. Six of these generate an infinite
subgroup. This group also has one nice 4-tuple, which produced one MCG-finite
5-tuple after middle convolution. This 5-tuple does not generate an infinite
subgroup.

######  Theorem 1.0.7.

For each primitive reflection group of rank four, we summarize the distinct
rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group
equivalence) produced by taking middle convolution of the different nice
tuples (up to type equivalence). Only nice tuples in $G_{32}$ produces an MCG-
finite tuple that generates an infinite subgroup after middle convolution.

  1. 1.

In $G_{28}$, there is only one nice 4-tuple, which produces one MCG-finite
5-tuple after middle convolution.

  2. 2.

In $G_{29}$ there are no nice 4-tuples nor any nice 5-tuples.

  3. 3.

In $G_{30}$, there are three nice 4-tuples, each producing one distinct MCG-
finite 5-tuple after middle convolution. For each of these 5-tuples, after
tensoring with a character so the matrices are in $SL_{2}(\mathbb{C})$ we end
up with one matrix being identity, which means these tuples reduce to MCG-
finite 4-tuples instead. There are no nice 5-tuples in this group.

  4. 4.

In $G_{31}$ there are no nice 4-tuples nor any nice 5-tuples.

  5. 5.

In $G_{32}$ there are three nice 4-tuples, resulting in four distinct MCG-
finite 5-tuples after middle convolution. One of these 5-tuples generates an
infinite subgroup. After tensoring with a character so the matrices are in
$SL_{2}(\mathbb{C})$, for two of the remaining three 5-tuples, we get the
identity matrix, so these reduce to MCG-finite 4-tuples instead. This group
also contains at least two nice 5-tuples, producing at least one example of an
MCG-finite 6-tuple. The 6-tuple did not generate an infinite subgroup. The
search for nice 5-tuples in this group was too large computationally, so this
list may be incomplete.

###  1.1 Prior Work

MCG-finite representations correspond to algebraic solutions to nonlinear
differential equatons, such as the Painlevé VI equation and more generally the
Schlesinger equations [4]. Regarded as a nonlinear analogue of the
hypergeometric equation, the Painlevé VI (PVI) equation posed over 100 years
ago, has inspired much work on the study of finite orbits. Algebraic solutions
to the PVI equation correspond to irreducible MCG-finite representations when
$n=2$ and $T+1=4$ and motivates progress towards a nonlinear Schwartz list
[2]. Among the many advances made in this area, notable contributions on
finite orbits come from Hitchin [14], Dubrovin-Mazzocco [9], Boalch [1],
Kitaev [16] and an eventual computer-aided classification by Lisovy-Tykhyy
[19].

The work of Dubrovin-Mazzocco originally studied the PVI equation in
connection with reflection groups [9]. Expanding on this idea, solutions
constructed from finite reflection groups through special cases of middle
convolution, like the “Okamoto transform,” were explored by others in [1],
[20], [7]. More generally, a computer-aided classification was provided by
Tykhyy in [27], with explicit orbits listed when $T+1=4,5$ and conjectural
descriptions of the different possibilities for $T+1\geq 6$. We found exact
matches for orbits given in this list, except in two cases where the closest
match differs in just one choice of parameter. As both these examples almost
match with the same orbit they provided, we suspect this difference might be
due to a typo. Details on these two tuples are provided in Section 5.1.3 and
Section 5.1.4 and the differing coordinates are marked with an asterisk in the
tables.

The connection between finite complex reflection groups and middle
convolutions was further explored more recently in the work of Lam-Landesman-
Litt [17, Cor. 1.1.7] who proved that all MCG-finite tuples with at least one
matrix of infinite order are either pullback representations (classified in
[8]) or come from finite complex reflection groups via middle convolution.

Our work investigates the finite complex reflection groups for MCG-finite
tuples and examines their middle convolution in relation to the braid group
orbit. In the applicable cases, we also explore how the examples we find
relate to the classifications provided by Lisovy-Tykhyy [19] and Tykhyy [27].
Note that the analysis in these two papers exclude the tuples where all the
matrices share a common eigenvector; these correspond to non-irreducible cases
and were classified by Cousin-Moussard [11, Section 3].

##  2 Background

###  2.1 Middle Convolution

Defined by Katz [15], middle convolution is an invertible functor in the
category of perverse sheaves depending on some parameter
$\lambda\in\mathbb{C}^{\times}$ and is denoted $MC_{\lambda}$. Middle
convolution can also be viewed as a functor in the category of local systems
on the punctured sphere to itself, and with this interpretation, Katz proved
that all irreducible rigid local systems on the punctured sphere can be
constructed from rank 1 local systems via scaling or middle convolution.

Letting $k$ be a field and $F_{T}$ denote the free group on $T$ generators,
middle convolution can equivalently be understood as a functor on the category
of finite dimensional $k[F_{T}]$ modules to itself as described by Dettweiler-
Reiter [5, Prop. 2.6]. The relation to local systems on the $T+1$-punctured
sphere is by taking $k=\mathbb{C}$ and viewing a tuple of $T$ matrices as
defining the action of $\mathbb{C}[F_{T}]$ on the module i.e. a representation
of $\mathbb{C}[F_{T}]$. In this way irreducible tuples correspond to
irreducible representations. While middle convolution preserves irreducibility
[5, Cor. 3.6], it typically changes the dimension of the module (rank of the
local system on $T+1$-punctured sphere) and does not preserve basic properties
e.g. finiteness of the subgroup generated by the matrices (i.e. monodromy
group). Important to the study of finite orbits, one desirable property of
middle convolution is that it is equivariant with the action of the braid
group $B_{T}$ [5, Theorem 5.1]. One way to construct nontrivial finite orbits
is to take irreducible tuples generating a finite subgroup, hence finite braid
group orbit, and taking the middle convolution to (hopefully) get a tuple
generating an infinite group which will necessarily be irreducible and have
finite braid group orbit.

The purely algebraic analogue by Dettweiler-Reiter [5] gives an explicit
description of middle convolution in terms of tuples of matrices and relates
some properties of middle convolution to the matrices in the tuple.

######  Definition 2.1.1.

Given a tuple of $T$ matrices $\textbf{A}=[A_{1},A_{2},\dots,A_{T}]$ with
$A_{i}\in GL_{n}(\mathbb{C})$, the $\mathbf{product}$ of such a tuple is the
product of the matrices in the tuple, in the ordering given by the tuple. So
the product of the tuple $\mathbf{A}$ is $A_{1}A_{2}\cdots A_{T}$. The
$\mathbf{inverse}$ of the tuple A is the tuple
$[A_{T}^{-1},\dots,A_{1}^{-1}]$, denoted $\mathbf{A^{-1}}$. The
$\mathbf{inverse}$ $\mathbf{product}$ of a tuple is the matrix product of the
inverse tuple.

######  Definition 2.1.2.

Given a tuple of $T$ matrices $\textbf{A}=[A_{1},A_{2},\dots,A_{T}]$ with
$A_{i}\in GL_{n}(\mathbb{C})$, we denote the middle convolution of
$\mathbf{A}$ with parameter $\lambda\in\mathbb{C}^{\times}$ as
$MC_{\lambda}(\textbf{A})=[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}}]$.
Note that $MC_{\lambda}(\textbf{A})$ will again be a tuple of $T$ matrices,
well-defined up to simultaneous conjugation, with $\tilde{A_{i}}\in
GL_{n^{\prime}}(\mathbb{C})$ for some $n^{\prime}$ not necessarily equal to
$n$.

######  Remark 2.1.3.

Middle convolution does not preserve the product of the matrices as it is only
well-defined up to simultaneous conjugation.

Letting $V$ denote the underlying vector space associated to the
$\mathbb{C}[F_{T}]$ module defined by a tuple $\mathbf{A}$. Then the relation
between $n$ and $n^{\prime}$ is as follows.

######  Theorem 2.1.4.

Let $V$ denote the $\mathbb{C}[F_{T}]$-module defined by the tuple
$\textbf{A}=[A_{1},A_{2},\dots,A_{T}]$ and let
$\lambda\in\mathbb{C}^{\times}$.

  1. 1.

If $\lambda\neq 1$ then

$\dim(MC_{\lambda}(V))=\sum_{i=1}^{T}\rank(A_{i}-I)-(\dim(V)-\rank(\lambda\cdot
A_{1}A_{2}\cdots A_{T}-I))$

###### Proof.

Proofs of this statement, as well as further details and properties of the
middle convolution can be found in [6, Section 2] and [5, Section 2-5]. ∎

To get an MCG-finite $T+1$-tuple of rank 2, we want to find a $T$-tuple of
reflections (hence $\rank(A_{i}-I)=1$) such that their product has a
nontrivial eigenvalue of multiplicity $T-2$. This is why we search for nice
tuples, Definition 3.0.1 in Section 3.2.

Another property of middle convolution that will greatly simplify our
computations is the following.

######  Theorem 2.1.5.

(Dettweiler and Reiter, 2000) If $\mathbf{A}$ is an irreduble tuple of rank
greater than 1, then the modules $MC_{\lambda^{-1}}(\mathbf{A^{-1}})$ and
$MC_{\lambda}(\mathbf{A})^{-1}$ are isomorphic.

###### Proof.

Proved in [5, Theorem 5.5]. ∎

Our code for middle convolution uses the formulation given by Dettweiler-
Reiter in [6, Definition 2.1] as opposed to their formulation in [5, Section
2] which only differs by changing the order of blocks in the matrices. The
code requires a matrix group $W$ containing the matrices in the tuple
$\mathbf{A}$ and the base ring of $W$ should contain the parameter $\lambda$
used for middle convolution. Note that the final function $Compute\\_MC$ will
automatically check if our intermediate constructions act invariantly on the
subspaces $K$ and $L$, as defined in the notation of [6, Definition of 2.1].
The full computer code can be found online at
amalvayalin.github.io/research.html.

###  2.2 Finite Complex Reflection Groups

A real reflection can be identified as a matrix with one eigenvalue -1 and all
other eigenvalues are 1, and pseudo-reflections refers to matrices with one
eigenvalue a root of unity and all other eigenvalues 1. A finite complex
reflection group (FCRG) is a finite group generated by pseudo-reflections. As
we are only working with complex groups, the distinction between real or
complex eigenvalues is not relevant and henceforth reflections can refer to
real reflections or pseudo-reflections.

All irreducible finite complex reflection groups were classified by Shephard-
Todd [22], as belonging to an infinite family of imprimitive reflection groups
or being one of 34 exceptional primitive cases.

######  Definition 2.2.1.

The rank of a reflection group is the smallest dimension of the vector space
on which the group acts irreducibly. Every finite complex reflection group of
rank $n$ has a minimal generating set consisting of either $n$ or $n+1$
reflections. The group is said to be well-generated if it can be generated by
$n$ reflections.

As we want irreducible MCG-finite T+1-tuples, we know that our nice $T$-tuple
(see Defintion 3.0.1) must at least be of length $n$ to generate an
irreducible FCRG, so in each group we only need to search for $T$-tuples with
$T>n$. Among the rank three and four primitive irreducible groups, only
$G_{31}$ is not well-generated (see Cartan matrices in [18, Appendix Section
3]. An imprimitive group $G(m,p,n)$ is well-generated if $p=m$ or $p=1$.

Associated to each FCRG of rank $n$ is a set of $n$ invariants called the
degrees of the reflection group [22, Theorem 5.1], [18, Table D.3]. An
important property of these invariants relevant for our purposes is their
relation to the eigenvalues of the group elements. Note that we already know
the eigenvalues for any element of the group must be roots of unity since the
group is finite.

######  Theorem 2.2.2.

Let $W$ be a FCRG of rank $n$ and let $\\{d_{i}\\}_{i=1,\dots,n}$ denote the
degrees of $W$. Suppose $\lambda$ is some primitive $d$-th root of unity that
is an eigenvalue for some element of $W$. Then, $d\mid d_{i}$ for some degree
$d_{i}$. Moreover, the maximum dimension of the eigenspace corresponding to
$\lambda$ for any element of the group is the number of degrees $d_{i}$ that
is divisible by $d$, and for some element in $W$ the eigenspace corresponding
to $\lambda$ achieves this maximal dimension.

###### Proof.

Proved in [26, Theorem 3.4]. ∎

The theorem above will be helpful in our computations for the primitive groups
as we know how large our field needs to be to ensure the code finds all
eigenvalues.

Further properties specific to imprimitive reflection groups are described in
Section 4.

###  2.3 Representations and Character Varieties

Given a finitely generated group $\Gamma$, a representations of $\Gamma$ is a
homomorphism $\rho:\Gamma\longrightarrow G$ where $G<GL_{n}(\mathbb{C})$ is
some algebraic group. Considering the vector space $V$ the matrices
$\rho(\gamma)$ act on, $\rho$ is said to be irreducible if there is no nonzero
subspace $U\subsetneq V$ that is left invariant by the matrix $\rho(\gamma)$
for all $\gamma\in\Gamma$. Two representations $\rho$ and $\rho^{\prime}$ are
isomorphic if there exists some $g\in G$ such that
$\rho(\gamma)=g^{-1}\rho^{\prime}(\gamma)g$ for all $\gamma\in\Gamma$. The
character of a representation $\rho$ is the map
$\chi_{\rho}:\Gamma\longrightarrow\mathbb{C}$ given by
$\chi_{\rho}(\gamma)=\tr\rho(\gamma)$. Isomorphic representations have the
same character since trace is fixed under conjugation. The character variety
$\mathfrak{X}(\Gamma,G)$ parametrizes the isomorphism classes of
representations $\rho:\Gamma\longrightarrow G$ in terms of suitable traces.

In the case of the punctured sphere $\Sigma_{0,T+1}$, the fundamental group is
$\pi_{1}(\Sigma_{0,T+1})=\langle\gamma_{1},\dots,\gamma_{T+1}\mid\gamma_{1}*\gamma_{2}*\cdots*\gamma_{T+1}=1\rangle$.
Any representation of $\Gamma=\pi_{1}(\Sigma_{0,T+1})$ is identified by
$\rho(\gamma_{i})=M_{i}\in G$ such that $M_{1}\cdot M_{2}\cdots M_{T+1}=I$ is
the identity matrix. Two such representations $\rho$ and $\rho^{\prime}$ are
equivalent if the tuple $[\rho(\gamma_{1}),\dots,\rho(\gamma_{T+1})]$ is
related to $[\rho^{\prime}(\gamma_{1}),\dots,\rho^{\prime}(\gamma_{T+1})]$ by
simultaneous conjugation.

Taking $G=SL_{2}(\mathbb{C})$, we can consider the
$SL_{2}(\mathbb{C})$-character variety of $\pi_{1}(\Sigma_{0,T+1})$.

Given a tuple of of matrices $[M_{1},M_{2},\dots,M_{T+1}]$, let

$\displaystyle t_{i}=\Tr(M_{i})\qquad t_{ij}=\Tr(M_{i}M_{j})\qquad
t_{ijk}=\Tr(M_{i}M_{j}M_{k})$ for $1\leq i<j<k\leq T+1$

Since $\pi_{1}(\Sigma_{0,T+1})=F_{T}$ we have a natural inclusion
$\mathfrak{X}(\pi_{1}(\Sigma_{0,T+1}),SL_{2}(\mathbb{C}))\subseteq\mathfrak{X}(F_{T},SL_{2}(\mathbb{C}))$.
This means for our purposes, $M_{T+1}$ will always be $(M_{1}M_{2}\cdots
M_{T})^{-1}$ so $t_{T+1}=\Tr(M_{1}M_{2}\cdots M_{T})$ as all the matrices
$M_{i}$ will be in $SL_{2}(\mathbb{C})$ where matrices have the same trace as
their inverse.

######  Theorem 2.3.1.

Let $[M_{1},M_{2},\dots,M_{T+1}]\in SL_{2}(\mathbb{C})^{T+1}$ such that $\Pi
M_{i}=I$. Then up to simultaneous conjugation, any such $(T+1)$-tuple can be
identified by the following trace coordinates:

  1. 1.

For $T=3$: $(t_{1},t_{2},t_{3},t_{12},t_{13},t_{23},t_{4})$

  2. 2.

For $T=4$:
$(t_{1},t_{2},t_{3},t_{4},t_{12},t_{13},t_{14},t_{23},t_{24},t_{34},t_{123},t_{124},t_{134},t_{234},t_{5})$

###### Proof.

For proof and further discussion of character varieties of the punctured
sphere see [12, Section 4] and [21, Section 5]. ∎

Such trace coordinates are also sometimes referred to as the
$\mathbf{signature}$ of the tuple.

##  3 Methods

For the rest of this written work, fix $\zeta_{d}=e^{\frac{2\pi i}{d}}$ as a
primitive $d$-th root of unity.

Since the product of $[A_{1},\cdots,A_{T+1}]$ must be the identity, we know
any MCG-finite tuple is determined by the first $T$ matrices, up to
simultaneous conjugation. As middle convolution does not preserve products
(see Remark 2.1.3), our idea is to take middle convolution of the first $T$
matrices to get $2\times 2$ matrices and then induce a representation of
$\pi_{1}(\Sigma_{0,T+1})$ by taking the $T+1$-th matrix to be the inverse
product. The condition for middle convolution being rank 2, (stated in Theorem
2.1.4) depends on the dimension of the subspace fixed by, or equivalently the
multiplicity of the eigenvalue 1 of each matrix in the tuple and the
eigenvalues of their product. For much better control on the former, we
consider reflections since they only have one nontrivial eigenvalue, of
multiplicity 1.

Thus, for each irreducible finite complex reflection group $W$ of rank $n$, we
want to find all irreducible $T$-tuples of reflections
$\textbf{A}=[A_{1},A_{2},\dots,A_{T}]$ in $W$ such that for a suitable choice
of parameter $\lambda$, $MC_{\lambda}(\textbf{A})$ has rank 2. Given $T$, the
formula given in Theorem 2.1.4 tells us that the product $\Pi_{i=1}^{T}A_{i}$
of reflections in A should have a nontrivial eigenvalue $\lambda^{-1}$ of
multiplicity $T-2$ so that
$MC_{\lambda}(\textbf{A})=[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}}]$
is a $T$-tuple of rank 2. We give a name to tuples satisfying these
conditions.

######  Definition 3.0.1.

A $T$-tuple $\textbf{A}=[A_{1},A_{2},\dots,A_{T}]$ of matrices from a finite
complex reflection group $W$ is called $\mathbf{nice}$ if A satisfies the
following:

  1. 1.

All $A_{i}$ are reflections in $W$

  2. 2.

$W=\langle A_{1},A_{2},\dots,A_{T}\rangle$

  3. 3.

$(A_{1}A_{2}\cdots A_{T})^{-1}$ has an eigenvalue $\lambda\neq 1$ of
multiplicity $T-2$.

The condition of generating the whole group is to ensure that we are not
“over-counting” as some reflection groups contain other reflection groups as
subgroups, and we only want to study irreducible MCG-finite tuples.

######  Remark 3.0.2.

Given a nice $T$-tuple $\mathbf{A}=[A_{1},A_{2},\dots,A_{T}]$, we know there
exists some $\lambda\neq 1$ such that
$MC_{\lambda}(\mathbf{A})=[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}}]$
is a $T$-tuple of rank 2. taking the $T+1$-th matrix to be

$\widetilde{A_{T+1}}=(\Pi_{i=1}^{T}\tilde{A_{i}})^{-1}$

we get a $T+1$-tuple
$[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}},\widetilde{A_{T+1}}]$ whose
product is the identity. As $\mathbf{A}$ comes from a finite group we know it
has finite braid group orbit, and since middle convolution is equivariant with
the braid group action (see Section 2.1), we know that
$[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}},\widetilde{A_{T+1}}]$ must
be MCG-finite. Moreover, if $\\{A_{1},A_{2},\dots,A_{T}\\}$ generates an
irreducible group, then as middle convolution preserves irreducibility, we
know our $T+1$-tuple is irreducible as well. Thus, we have used a nice
$T$-tuple to produce an irreducible MCG-finite $T+1$-tuple.

######  Remark 3.0.3.

For some nice $T$-tuples, it is possible that multiple choices of $\lambda$
have multiplicity $T-2$ and hence the different choices of parameter $\lambda$
gives rise to possibly different rank 2 MCG-finite $T+1$-tuples.

######  Remark 3.0.4.

The eigenvalue condition of multiplicity $T-2$ tells us that in any complex
reflection group of rank $n$, for $T>n+2$ the middle convolution will not be
rank 2 so there are no nice $T$-tuples. By Definition 2.2.1, since we need at
least $n$ reflections to generate the whole group, we know there are no nice
tuples when $T<n$. For $T=n+2$, the product of the $T$ matrices would be a
scalar matrix (as there must be an eigenvalue of multiplicity $n$) which we
can take to be the identity, reducing to the $T=n+1$ case. So for each
$n\in\\{3,4\\}$, we only search the primitive reflection groups for nice
tuples of length $T=n$ or $T=n+1$.

###  3.1 Computing The Braid Group Orbit

The braid group can be presented as
$B_{T+1}=\langle\sigma_{1},\dots,\sigma_{T}\mid\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\rangle$
where the first relation holds for $1\leq i\leq T-3$ and the second for
$i-j\geq 2$. A generator $\sigma_{i}$ acts on $T+1$-tuples in the following
way:

$\sigma_{i}(M_{1},M_{2},\dots,M_{T},M_{T+1})=(M_{1},\dots,M_{i-1},M_{i}M_{i+1}M_{i}^{-1},M_{i},M_{i+2},\dots,M_{T+1})$

Let $F_{+}$ denote the set of all words in positive powers of these
generators. Further, let $Orb(B_{T+1},x)$ denote the orbit of some tuple $x$
under the action of $B_{T+1}$ and $Orb(F_{+},x)$ denote the set of images of
$x$ under the action of words in $F_{+}$.

######  Lemma 3.1.1.

For a tuple $x$ of matrices, $Orb(B_{T+1},x)$ is finite if and only if
$Orb(F_{+},x)$ is finite.

###### Proof.

Since $F_{+}\subset B_{T+1}$, one direction is obvious. For the other
direction first note that for any fixed $g\in B_{T+1}$, if the set
$\\{g^{i}x:i\geq 0\\}$ is finite, then we can rewrite $g^{-1}x=g^{n}x$ for
some $n\geq 0$, so we get that $\\{g^{i}x:i\in\mathbb{Z}\\}$ must be finite.
Further, since the elements of $B_{T+1}$ act one at a time, it is enough to
study how words in positive powers of the generators act on $x$. Thus, if
$Orb(F_{+},x)$ is finite, then the braid group orbit must be finite as well. ∎

To help with computing the braid group orbit, we need a way to capture the
equivalence relation of tuples being related by simultaneous conjugation.
Using character varieties, we can write down trace coordinates for our tuple,
which we know how to do from Theorem 2.3.1 when the matrices are in
$SL_{2}(\mathbb{C})$.

######  Definition 3.1.2.

Given a nice tuple $\mathbf{A}=[A_{1},A_{2},\dots,A_{T}]$, let
$MC_{\lambda}(\mathbf{A})=[\tilde{A_{1}},\tilde{A_{2}},\dots,\tilde{A_{T}}]$
denote the middle convolution. If
$(a_{1},a_{2},\dots,a_{T+1})\in(\mathbb{C}^{\times})^{T+1}$ is such that
$a_{i}\tilde{A_{i}}\in SL_{2}(\mathbb{C})$ for $i=1,\dots T$ and
$a_{T+1}((\tilde{A_{1}}\tilde{A_{2}}\cdots\tilde{A_{T}})^{-1})\in
SL_{2}(\mathbb{C})$ with $\Pi^{T+1}a_{i}=1$, the tuple

$\mathbf{\hat{A}}=[a_{1}\tilde{A_{1}},a_{2}\tilde{A_{2}},\dots,a_{T+1}(\tilde{A_{1}}\tilde{A_{2}}\cdots\tilde{A_{T}})^{-1}]$

is said to be $\mathbf{induced}$ by $MC_{\lambda}(\mathbf{A})$ or by
$\mathbf{A}$ if the parameter $\lambda$ is clear. $\mathbf{\hat{A}}$ is a
tuple in $(SL_{2}(\mathbb{C}))^{T+1}$ which multiplies to the identity.

When we say trace coordinate of a tuple $MC_{\lambda}(\mathbf{A})$ we will
always mean the trace coordinates of the induced tuple $\mathbf{\hat{A}}$
unless otherwise specified.

We want to compute the orbit of induced tuples $\mathbf{\hat{A}}$ up to
simultaneous conjugation. This is equivalent to computing the orbit of the
point associated to $\mathbf{\hat{A}}$ in the character variety
$\mathfrak{X}(\pi_{1}(\Sigma_{0,T+1}),SL_{2}(\mathbb{C}))$. Note that since
all of our tuples come from FCRG and middle convolution is equivariant with
the action of the braid group we already know all the orbits are finite.

The code for the algorithm described below can be found online at
https://amalvayalin.github.io/research.html.

Given a tuple $\mathbf{M}=[M_{1},\dots,M_{T+1}]$ of
$SL_{2}(\mathbb{C})^{T+1}$, such that their product is the identity, the braid
group $B_{T+1}$ orbit is computed as follows.

We first compute the signature of $\mathbf{M}$, and then apply each of the $T$
generators to the tuple $\mathbf{M}$. If $\sigma_{i}(\mathbf{M})$ has a new
signature, then we store the tuple in $To\\_Test$ and the signature in
$Signatures$ and add $\mathbf{M}$ to $Tested$. We then apply $\sigma_{i}$
again to the (atmost $T$) new tuples in $To\\_Test$ we got from the action on
$\mathbf{M}$, once again storing only tuples which have new signatures. We
repeat this process until the sets $To\\_Test$ and $Tested$ are the same. The
function returns the set $Tested$, which should contain only one
representative per distinct signature, and can optionally return the set
$Signatures$ as well. Note that here our definition of signature is the trace
coordinates, which differs from the definition used in Tykhyy’s classification
in terms of residues [27].

###  3.2 Finding Nice Tuples

Nice $T$-tuples correspond to MCG-finite $T+1$-tuples in the following way:
given a nice tuple $[A_{1},\dots,A_{T}]$, we get a corresponding
representation $[A_{1},\dots,A_{T},A_{T+1}]$ of the $T+1$-punctured sphere if
we take $A_{T+1}=(A_{1}A_{2}\cdots A_{T})^{-1}$. Thus, it is enough to search
each complex reflection group for nice tuples.

By Theorem 2.2.2, an eigenvalue of multiplicity $T-2$ for an element in $W$
must be a $d$-th root of unity for $d$ dividing at least $T-2$ degrees of $W$.
Any reflection group can be constructed in Magma by providing the generating
reflections; the irreducible finite complex reflection groups can be easily
constructed in Magma by providing the Shephard-Todd number or the
corresponding parameters for $G(m,p,r)$. In most cases Magma generates the
group $W$ via the reflection representation over the trace field of $W$ [10,
Table 1, Table 2].

One issue is that not all characteristic polynomials will split over the trace
field, as Magma computes eigenvalues for matrices over the field of entries.
Since our search depends on the eigenvalues of the product of matrices in the
tuple(s), we must change the base field of $W$ to the smallest cyclotomic
field containing all the possible eigenvalues. The largest this extension will
be $\mathbb{Q}(\zeta_{d})$, where $d$ is the least common multiple of all the
degrees of $W$, but in certain cases smaller fields might suffice. For large
$d$ computation time is typically slower, but we have expanded the field to
the LCM of the degrees unless otherwise stated. For example, in the case of
$W=G_{32}$, the degrees are $\deg(W)=\\{12,18,24,30\\}$ and so
$\operatorname{lcm}(\deg(W))=360$ which slows down computation time
considerably. For this group, we prove in Section 5.3.3 that it is enough to
search for 4-tuples over $\mathbb{Q}(\zeta_{12})$ by showing any eigenvalue of
multiplicity 2 must be a 12-th root of unity.

Before outlining the steps of our search, we define some terms.

######  Definition 3.2.1.

Given a tuple of matrices $\mathbf{A}=[A_{1},A_{2},\dots,A_{T}]$, the
$\mathbf{type}$ of $\mathbf{A}$ is determined by the eigenvalues of
$(A_{1}A_{2}\cdots A_{T})^{-1}$ counted with multiplicity. Tuples whose
inverse product has the same eigenvalues with multiplicity are said to be of
the same type. An $\mathbf{exemplar}$ of a type is a choice of nice tuple of
that type. Given a tuple of some type A, the type of the inverse tuple is
called the $\mathbf{inverse}$ $\mathbf{type}$ of A and the two types together
make an $\mathbf{inverse}$ $\mathbf{pair}$.

The use of types allows us to partition a set of nice tuples based on the
eigenvalues of the inverse product. Since there are many nice tuples of the
same type, we choose exemplars for each type to perform the in-depth
computations on. Note that the choice of exemplar for any type is arbitrary
and it may be possible that different exemplars result in different orbits. To
ensure this does not happen, we go back through all the tuples for each type
and make sure that it belongs to the braid group orbit we compute for the
middle convolution of the exemplar. More details on the braid group
computations can be found in Section 3.1.

In some groups we get types that are inverse pairs. In these cases we only
provide the computations for one exemplar from each pair. To ensure the
inverse type would not produce a new orbit, we check that the inverse tuple of
the middle convolution lies in the orbit of the middle convolution of the
exemplar; this is enough by the relationship between inverse tuples and
inverse parameters given in Theorem 2.1.5. The code for how we checked each
type is available at https://amalvayalin.github.io/research.html.

######  Remark 3.2.2.

Aside from a few exceptional cases that were computationally too large, we
verified that for each choice of the parameter $\lambda$, the rank 2 MCG-
finite tuples coming from nice tuples of the same type are in the same braid
group orbit, as are exemplars from the inverse type. In this sense, types can
be viewed as ‘almost’ equivalence classes for nice tuples up to inverse pairs.

After coercing the matrix group into a larger field if necessary, the general
process for searching for $T$-tuples is as follows.

  1. 1.

Fix a representative $Q_{i}$ for each conjugacy class of reflections in $W$
and fix an ordering $[Q_{1},\cdots,Q_{n}]$.

  2. 2.

Given $T$, construct the list $M$ of distinct sets of $T$ reflections in $W$,
using the ordering of $Q$ to eliminate some tuples equivalent up to
simultaneous conjugacy.

  3. 3.

Search $M$ for all sets such the matrices generate $W$ (means the tuple is
irreducible) and the product of the matrices, for some order of multiplying,
has an eigenvalue of multiplicity $T-2$. Each such set is then organized into
a tuple based on the order of multiplication and stored in a list X of
irreducible tuples.

  4. 4.

For each nice tuple $x\in X$, compute the eigenvalues of the inverse product
of $x$ and partition $X$ into different types corresponding to the different
sets of eigenvalues of the inverse product.

  5. 5.

Choose exemplars $\mathbf{A},\mathbf{B}$… for each type and compute the middle
convolution
$MC_{\lambda_{1}}(\mathbf{A}),MC_{\lambda_{2}}(\mathbf{A}),MC_{\lambda}(\mathbf{B})$…
for suitable choice(s) of $\lambda$.

  6. 6.

Compute the subgroup of $GL_{2}(\mathbb{C})$ generated by
$MC_{\lambda}(\mathbf{A}),MC_{\lambda}(\mathbf{B}),\dots$

  7. 7.

For each middle convolution of the exemplars, find a suitable character
$[a_{1},\dots,a_{T+1}]$ and construct the induced tuple $\mathbf{\hat{A}}$

  8. 8.

Compute the braid group orbit of $\mathbf{\hat{A}}$ and identify the
$SL_{2}(\mathbb{C})$ subgroup generated by the matrices of $\mathbf{\hat{A}}$

  9. 9.

Check the other exemplars and inverse tuple do not produce new orbits

  10. 10.

Use the tuples in the braid group orbit to compute the parameter and residue
signatures of the full equivalence class as defined in the previous
classifications to try and match our examples to theirs.

We recorded all the details for our exemplars and the middle convolution
computations for each of the primitive reflection groups of rank 3 and 4. The
middle convolution tables are listed in the Appendix. The types and exemplars
are listed in the section for each group, as is the comparison tables with the
other classifications. More details on what we use to compare with the
existing classifications of [19] and [27] are given in Sections 5.1 and 5.2 .

##  4 Imprimitive Reflection Groups

Imprimitive reflection groups $G(m,p,n)$ are parameterized by $m,p,n\geq 1$
with $p\mid m$. These groups can be viewed as $n\times n$ generalized
permutation matrices, which each row and column containing only one non-zero
entry which is either 1 or an $\frac{m}{p}$-th root of unity. For $m>1$ and
$(m,p,n)\neq(2,2,2)$ these groups are always irreducible, meaning they act
irreducibly on $\mathbb{C}^{n}$ (see [3, Section 2]).

An element $A$ of $G(m,p,n)$ can be written as
$A=[a_{1},a_{2},\dots,a_{n}\mid\sigma]$ for $\sigma\in S_{n}$ and $p\mid
a_{1}+a_{2}\dots+a_{n}$. This element can be identified with the matrix where
the $(i,\sigma(i))$-th entry of $A$ is $\zeta_{m}^{a_{i}}$ and all the other
entries are 0 [25, Section 2.3]. Note that all these $a_{i}$ should be read
modulo $m$ since we are working with powers of an $m$-th root of unity.
Additionally, there is a natural surjective homomorphism
$\varphi:G(m,p,n)\longrightarrow S_{n}$ given by sending $A\mapsto\sigma$.

One observation that follows from the definition is that for any $p$,
$G(m,m,n)\leq G(m,p,n)$ and $G(m,p,n)\leq G(m,1,n)$ as subgroups.

Just as in the primitive case, the degrees of imprimitive groups tell us about
the possible eigenvalues. Writing $q=\frac{m}{p}$, we have:

$\displaystyle\deg(G(m,p,n))$ $\displaystyle=\\{m,2m,\dots,(n-1)m,qn\\}$ (1)
$\displaystyle|ZG(m,p,n)|$ $\displaystyle=q\gcd(p,n)$ (2)

where $ZG(m,p,n)$ is the center of $G(m,p,n)$ and a cyclic subgroup.

Reflections in the group $G(m,p,n)$ will be of one of the following two types:

  1. 1.

Type 1: These are elements such that $\sigma=(ij)$ is a transposition with
$a_{k}\equiv 0\mod m$ for $k\neq i,j$ and $a_{i}+a_{j}\equiv 0\mod m$. All
reflections of Type 1 are of order 2 and lie in the subgroup $G(m,m,n)$ of
$G(m,p,n)$. Using the notation from [25, Section 2.3] we write a Type 1
reflection as $s(i,j;a_{i})$ with the understanding that this is the same as
$s(j,i;-a_{i})$.

  2. 2.

Type 2: These are elements such that $\sigma=e$ and for some $i$,
$a_{i}\not\equiv 0\mod m$ and $a_{k}\equiv 0\mod m$ for all $k\neq i$. These
matrices are diagonal matrices of order $m/\gcd(m,a_{i})$ and will be denoted
$s(i;a_{i})$. Type 2 matrices can only exist when $p<m$.

######  Theorem 4.0.1.

Let $S$ be a set of reflections with minimal possible cardinality such that
$S$ generates $G(m,p,n)$. Then $S$ satisfies the following conditions:

  1. 1.

When $p=1$, $S$ contains $n-1$ reflections of Type 1 and one reflection of
Type 2 of order $m$

  2. 2.

When $p=m$, $S$ contains $n$ reflections of Type 1

  3. 3.

When $p\neq 1,m$, $S$ contains $n$ reflections of Type 1 and one reflection of
Type 2 of order $\frac{m}{p}$

###### Proof.

Stated in [25, Lemma 3.2] with proofs in [23, Subsection 2.10 and Lemma 2.1]
and [24, Lemma 2.2]. ∎

######  Theorem 4.0.2.

For any reflection $r\in G(m,p,n)$, the homomorphism
$\varphi:G(m,p,n)\longrightarrow S_{n}$, one of the following must be true:

  1. 1.

$\varphi(r)$ is a transposition and $r$ has order 2

  2. 2.

$\varphi(r)$ is the identity.

###### Proof.

This is true by the definition of reflections and the definition of the
homomorphism. For a more thorough examination of related properties, we refer
to [3, Proposition 2.2]. ∎

Given any set of reflections generating an irreducible imprimitive group the
permutations associated to the reflections in the set must generate a
transitive subgroup of $S_{n}$, or else the group cannot be irreducible. Note
that in the case where the associated permutations are transpositions (i.e.
the reflections are Type 1), the only transitive subgroup of $S_{n}$ generated
by transpositions is $S_{n}$ itself [18, Lemma 2.13].

A subgroup of a reflection group generated by reflections is called a
reflection subgroup. We modify some notation from [28] to state relevant
theorems for identifying reflection subgroups in the imprimitive groups. Let
$X=\\{s(a_{h},a_{h+1};b_{k})\\}$ be a set of reflections in $G(m,m,n)$. We can
associate to $X$ the graph $\Gamma_{X}$ with nodes $\\{a_{i}\\}$ and edges
$(a_{h},a_{h+1})$ for $s(a_{h},a_{h+1};b_{k})$ in $X$. Letting $\langle
X\rangle$ denote the reflection group generated by the reflections in $X$, we
see that this group is irreducible if and only if the graph $\Gamma_{X}$ is
connected. Suppose $\Gamma_{X}$ is such that the graph contains exactly one
cycle, given by $r$ edges corresponding to $\\{s(a_{h},a_{h+1};b_{h})\\}$ for
$1\leq h\leq r$ for some $2\leq r\leq n$. Then we define the parameter
$\delta(X)$ to be the absolute value of $\sum_{1}^{r}b_{h}$. If $p\neq m$, we
say $X$ is a nice reflection set of $G(m,p,n)$ if $X$ contains exactly one
reflection of Type 2.

######  Remark 4.0.3.

Note that $s(a_{h},a_{h+1};b_{h})=s(a_{h+1},a_{h};-b_{h})$, so if $X$ is such
that the graph contains only one cycle of length $r$, then the cycle
corresponds to some permutation $(a_{1}a_{2}\cdots a_{r})$ (in cycle notation)
and the signs of $b_{h}$ must be chosen accordingly when computing
$\delta(X)$. In particular, if the cycle is of length two, i.e. given as a
double edge between two nodes corresponding to reflections
$s(a_{1},a_{2};b_{1})$ and $s(a_{1},a_{2};b_{2})$ then
$\delta(X)=|b_{1}-b_{2}|$.

######  Theorem 4.0.4.

With the notation as defined above, we have the following relations for the
reflection subgroup generated by $X$.

  1. 1.

Let $X$ be reflection set of $G(m,m,n)$ such that the graph $\Gamma_{X}$ is
connected and contains exactly one cycle. Then $\langle X\rangle=G(m,m,n)$ if
and only if the integer $\delta(X)$ is coprime to $m$.

  2. 2.

Let $X$ be a nice reflection set of $G(m,p,n)$ with $s(a_{1};b)$ the one
reflection of Type 2, such that $\Gamma_{X}$ is connected with $n_{1}$ nodes
and exactly one cycle. Then $\langle
X\rangle=G(\frac{m}{m_{1}},\frac{\gcd(b,m)}{m_{1}},n_{1})$ where
$m_{1}=\gcd(b,m,\delta(X))$.

  3. 3.

Let $X$ be a nice reflection set of $G(m,p,n)$ with $s(a_{1};b)$ the unique
reflection of Type 2 such that the graph $\Gamma_{X}$ is a connected tree with
$n_{1}$ nodes. Then $\langle X\rangle=G(\frac{m}{\gcd(b,m)},1,n_{1})$

###### Proof.

The first is proven in [23, Thm. 2.19] and the second is proven in [28,
Theorem 3.10]. Note that in both these papers they refer to cycles as circles.
The third statement is proven in [28, Theorem 3.9], where they define rooted
trees to capture the presence of Type 2 reflections. We refer to these paper
for more details on the graph theoretic interpretation of generating sets and
their connection to reflection subgroups of the imprimitive reflection groups.
∎

######  Remark 4.0.5.

Among the imprimitive groups, we have to check for $n$-tuples and $n+1$-tuples
in $G(m,p,n)$. By Theorem 4.0.1, the only possible groups that can contain
nice $n$-tuples are the well-generated groups $G(m,1,n)$ and $G(m,m,n)$. We
prove there are no nice $n$-tuples in either of these groups when $n\geq 5$ in
Section 4.5. By results in [17, Lemma 5.3.2], we know that $n+1$-tuples can
only exist in $G(m,p,n)$ for $n<5$. This work will fully analyze the existence
of nice $n+1$-tuples for $2<n<5$.

###  4.1 Nice 3-Tuples in $G(m,p,3)$

In this case we are seeking nice tuples $[r_{1},r_{2},r_{3}]$ from $G(m,p,3)$
such that their product has a nontrivial eigenvalue of multiplicity 1. Note
that since $\langle r_{1},r_{2},r_{3}\rangle=G(m,p,3)$ we know that $p=1$ or
$p=m$ are the only possibilities as these are the only well-generated
imprimitive groups (see Definition 2.2.1)

######  Lemma 4.1.1.

In $G=G(m,1,3)$ for a nice tuple $[r_{1},r_{2},r_{3}]$ with $r_{1},r_{2}$ of
Type 1 and $r_{3}$ of Type 2 with nontrivial eigenvalue $\zeta_{m}^{\prime}$
of order $m$, letting $A=r_{1}r_{2}r_{3}$, the eigenvalues of $A$ must be the
distinct cube roots of $\zeta_{m}^{\prime}$.

###### Proof.

Say we have three reflections

$r_{1}r_{2}r_{3}=A$

for some $A\in G$ with Eigv(A) = $\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$
with some $\lambda_{i}\neq 1$. Note that since the $r_{i}$ generate $G$,
exactly two of them are Type 1 corresponding to distinct transpositions and
the third is Type 2 with order $m$, say with nontrivial eigenvalue
$\zeta_{m}^{\prime}$. Using the homomorphism $\varphi:G\longrightarrow S_{3}$,
we know that $\varphi(A)$ is even and since it cannot be the identity it must
be a $3$-cycle. Since $A$ is a generalized permutation matrix, we know
$\tr(A)=0$ and we get the following equalities must be true:

$\displaystyle\zeta_{m}^{\prime}$
$\displaystyle=\det(r_{1}r_{2}r_{3})=\det(A)=\lambda_{1}\lambda_{2}\lambda_{3}$
(3) $\displaystyle 0$
$\displaystyle=\tr(A)=\lambda_{1}+\lambda_{2}+\lambda_{3}$ (4)

Since $\lambda_{i}$ are all roots of unity, we know their sum being zero means
the argument differs by $\frac{2\pi}{3}$. Further writing
$A=[a_{1},a_{2},a_{3}|(123)]$, without loss of generality, gives us the
matrix:

$A=\begin{pmatrix}0&\zeta_{m}^{a_{1}}&0\\\ 0&0&\zeta_{m}^{a_{2}}\\\
\zeta_{m}^{a_{3}}&0&0\end{pmatrix}$

Thus we can compute the characteristic polynomial as
$p_{A}(x)=-(x^{3}-\zeta_{m}^{a_{1}+a_{2}+a_{3}})$ and since the constant term
of the characteristic polynomial is the determinant of $A$, we get that
$\zeta_{m}^{\prime}=\zeta_{m}^{a_{1}+a_{2}+a_{3}}$. Thus $\lambda_{i}$ must be
the distinct cube roots of $\zeta_{m}^{\prime}$. ∎

######  Theorem 4.1.2.

There are nice 3-tuples in $G(m,1,3)$ for all $m>1$.

###### Proof.

The reflections in the tuple $[s(2,3;1),s(1,2;1),s(3;1)]$ will generate
$G(m,1,3)$ by Theorem 4.0.4. By the preceding lemma, the product will have
eigenvalues corresponding to the three distinct cube roots of $\zeta_{m}$, all
of which cannot be 1. ∎

######  Theorem 4.1.3.

In $G=G(m,m,3)$, if we have $r_{1}r_{2}r_{3}=A$ for reflections $r_{i}$
generating $G$, then the eigenvalues of $A$ are
$\\{\lambda^{-2},\lambda,-\lambda\\}$

###### Proof.

Since all the $r_{i}$ are of Type $1$, using $\varphi:G\longrightarrow S_{3}$
we see that $\varphi(A)$ must be a transposition, say $(ij)\in S_{3}$. Writing
$A=[a_{1},a_{2},a_{3}\mid(ij)]$ we see that $\tr(A)=\zeta_{m}^{a_{k}}$ where
$k\neq i,j$ and $\det(A)=-1$. Letting
$\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$ denote the eigenvalues of $A$, we
get that

$\displaystyle\zeta_{m}^{a_{k}}$
$\displaystyle=\lambda_{1}+\lambda_{2}+\lambda_{3}$ $\displaystyle-1$
$\displaystyle=\lambda_{1}\lambda_{2}\lambda_{3}$

The characteristic polynomial of $A$ is
$p_{A}(x)=-(x^{2}-\zeta_{m}^{a_{i}+a_{j}})(x-\zeta_{m}^{a_{k}})$. Let
$\lambda_{k}=\zeta_{m}^{a_{k}}$ so then we get that $\lambda_{i}=-\lambda_{j}$
since these are the roots of $x^{2}-\zeta_{m}^{a_{i}+a_{j}}$. Using the
determinant equation we get $1=\lambda_{i}^{2}\lambda_{k}$ hence
$\lambda_{i}^{2}=\lambda_{k}^{-1}=\zeta_{m}^{-a_{k}}=\zeta_{m}^{a_{i}+a_{j}}$
so both $\lambda_{i}$ and $\lambda_{j}$ square to the inverse of
$\lambda_{k}$. ∎

######  Corollary 4.1.4.

Nice 3-tuples exist in $G(m,m,3)$ for all $m>1$.

###### Proof.

In this case we can consider reflections
$[s(2,3;b_{1}),s(1,2;b_{2}),s(1,2;b_{3})]$ with $b_{2}$ and $b_{3}$ chosen so
that $|b_{2}-b_{3}|$ is coprime to $m$. Taking $b_{1}=1,b_{2}=1$ and
$b_{3}=0$, we get $\delta(X)=|b_{2}-b_{3}|$ is coprime to $m$ and so by
Theorem 4.0.4, we know these reflections will generate $G(m,m,3)$. The
characteristic polynomial of the product $r_{1}r_{2}r_{3}=A$ will in this case
be $p_{A}(x)=-(x-\zeta_{m})(x^{2}-\zeta_{m}^{-1})$. Since the roots of the
quadratic will be distinct, we know we will have at least two distinct
eigenvalues, so the only worry is if the eigenvalues might be
$\\{\zeta_{m},\zeta_{m},1\\}$ as then there is no nontrivial eigenvalue of
multiplicity 1. If 1 is a root of the quadratic, then we know
$\zeta_{m}^{-1}=1$ and $\zeta_{m}=1$ which is not possible. Hence, such nice
tuples always exist. ∎

###  4.2 Nice 4-Tuples in $G(m,p,3)$

We are seeking nice tuples of four reflections from $G(m,p,3)$ such that their
product has an nontrivial eigenvalue $\lambda$ of multiplicity 2. This can be
written as the identity

$\displaystyle r_{1}r_{2}r_{3}r_{4}=\lambda r_{5}^{-1}$ (5)

where the $r_{i}$ generate $G(m,p,3)$ and $r_{5}$ is some reflection, not
necessarily in $G(m,p,3)$.

######  Lemma 4.2.1.

In the notation of equation 5 above, for any such nice tuple, it must be that
$r_{5}^{-1}$ is in $G(m,1,3)$ and $\lambda$ is an $m$-th root of unity.
Moreover, in any $G(m,p,3)$, (including $p=1$ and $p=m$) it must be that
exactly four of these reflections are Type 1 and only one is of Type 2.

###### Proof.

Note that since $\lambda$ is an eigenvalue of multiplicity 2 for some matrix
in $G(m,1,3)$ it must be that the order of $\lambda$ divides at least 2
degrees in $\deg G(m,1,3)=\\{m,2m,3m\\}$. Thus the order of $\lambda$ divides
$m$ and so $\lambda I$ is in the center of $G(m,1,3)$. This tells us that
$r_{5}$ is always in $G(m,1,3)$. To generate any of $G(m,p,3)$ we know the set
$\\{r_{i}\\}_{i\leq 5}$ must contain at least 2 reflections of Type 1 by
Theorem 4.0.1. For any $p$, rearranging the equation and using the
homomorphism $\varphi:G(m,1,3)\longrightarrow S_{3}$ we get that
$\varphi(r_{1}r_{2}r_{3}r_{4}r_{5})=\varphi(\lambda I)=e$. Since $e$ is even
we know there must be exactly 2 or 4 reflections of Type 1. The product of two
transpositions can only be the identity if they are the same transposition and
we need at least two distinct transpositions to generate a transitive subgroup
of $S_{3}$, hence there are exactly 4 reflections of Type 1 and one reflection
of Type 2 for any $G(m,p,3).$ ∎

Note that this does not prove that $\lambda$ is primitive $m$-th root, however
in the next three theorems we will conclude this is indeed always the case. Up
to rearranging the equation 5, we may assume $r_{5}^{-1}$ is the unique
reflection of Type 2 and label its nontrivial eigenvalue $\zeta$ which we know
must be some $m$-th root of unity, not necessarily primitive.

######  Remark 4.2.2.

To study the tuples in the cases where they might exist, we first study the
number of ways to factorize the identity in $S_{3}$ into a transitive product
of 4 transpositions. In $S_{3}$ any two distinct transpositions will generate
a transitive subgroup which must be all of $S_{3}$ so fixing the first
transposition as $(12)$ we only need to find, up to simultaneous conjugacy,
tuples of 3 transpositions that multiply to $(12)$. Using computer search to
find such conjugacy classes of 3-tuples, then adjoining with $(12)$ on the
left yields four options:

  1. 1.

$[(12),(12),(23),(23)]$

  2. 2.

$[(12),(23),(12),(13)]$

  3. 3.

$[(12),(23),(23),(12)]$

  4. 4.

$[(12),(13),(23),(13)]$

Note that these tuples are multiplied starting at the leftmost transposition,
although that does not matter in this case. Converting these tuples to Type 1
reflections we can compute the product for each option.

  1. 1.

$s(1,2;b_{1})s(1,2;b_{2})s(2,3;b_{3})s(2,3;b_{4})=\begin{pmatrix}\zeta_{m}^{b_{1}-b_{2}}&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{2}+b_{3}-b_{4}}&0\\\
0&0&\zeta_{m}^{-b_{3}+b_{4}}\end{pmatrix}$

  2. 2.

$s(1,2;b_{1})s(2,3;b_{2})s(1,2;b_{3})s(1,3;b_{4})=\begin{pmatrix}\zeta_{m}^{b_{1}+b_{2}-b_{4}}&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{3}}&0\\\
0&0&\zeta_{m}^{-b_{2}-b_{3}+b_{4}}\end{pmatrix}$

  3. 3.

$s(1,2;b_{1})s(2,3;b_{2})s(2,3;b_{3})s(1,2;b_{4})=\begin{pmatrix}\zeta_{m}^{b_{1}+b_{2}-b_{3}-b_{4}}&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{4}}&0\\\ 0&0&\zeta_{m}^{-b_{2}+b_{3}}\end{pmatrix}$

  4. 4.

$s(1,2;b_{1})s(1,3;b_{2})s(2,3;b_{3})s(1,3;b_{4})=\begin{pmatrix}\zeta_{m}^{b_{1}+b_{3}-b_{4}}&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{2}-b_{3}}&0\\\
0&0&\zeta_{m}^{-b_{2}+b_{4}}\end{pmatrix}$

Equations for $\\{b_{i}\\}$ can be found by setting each of these matrices
equal to $\lambda r_{5}^{-1}$:

$\displaystyle\begin{pmatrix}\lambda&0&0\\\ 0&\lambda&0\\\
0&0&\lambda\zeta\end{pmatrix}$

for some root $\zeta$ with order dividing $m$. The choice of putting $\zeta$
in the third row is arbitrary and $\zeta$ could have been placed in any
diagonal entry.

######  Lemma 4.2.3.

For any nice 4-tuple in $G(m,p,3)$, $p\neq 1$ it must be that either $\lambda$
or $\lambda\zeta$ in the notation of Remark 4.2.2 has the same order as
$\zeta_{m}^{\delta(X)}$, where $X$ is any generating reflection set given by
the nice 4-tuple.

###### Proof.

By Lemma 4.2.1, given any nice 4-tuple in $G(m,p,3)$ satisfying equation 5, we
can rearrange all the Type 1 reflections to be one of the factorizations given
in Remark 4.2.2 up to simultaneous conjugation. As any reflection set $X$
generating $G(m,p,3)$ $p\neq 1$ will only need at least three Type 1
reflections, we see that for each of the four factorizations above, there are
multiple ways to choose $X$ so that the associated graph only contains one
cycle. For example, the first factorization gives us the possibilities:
$X=\\{s(1,2;b_{1}),s(1,2;b_{2}),s(2,3;b_{i})\\}$ for $i=3,4$ and
$X=\\{s(1,2;b_{i}),s(2,3;b_{3}),s(2,3;b_{4})\\}$ for $i=1,2$. So we know
$\delta(X)$ must be of the form

$\delta(X)=\begin{cases}|b_{1}-b_{2}|\\\ |b_{3}-b_{4}|\end{cases}$

By the equations each factorization must satisfy, we know
$\zeta_{m}^{b_{1}-b_{2}}=\lambda$ and $\zeta_{m}^{-b_{3}+b_{4}}=\lambda\zeta$
so one of these must have the same order as $\zeta_{m}^{\delta(X)}$. The
argument is similar for the third factorization since the formula for
$\delta(X)$ only differs in the indices. The possible $\delta(X)$ values from
the second and fourth factorizations only differ in the indices, so we only
justify the result for the second factorization. In this case the options for
$\delta(X)$ are

$\delta(X)=\begin{cases}|b_{1}+b_{2}-b_{4}|\\\ |b_{1}-b_{3}|\\\
|b_{2}+b_{3}-b_{4}|\end{cases}$

Up to taking absolute value, the first two options must equal $\lambda$ and
the third must be $\lambda\zeta$ so one of these has the same order as
$\zeta_{m}^{\delta(X)}$. ∎

######  Theorem 4.2.4.

Nice 4-tuples exist in $G(m,m,3)$ if and only if $m\neq 3$. For $m\neq 3$, we
have:

  1. 1.

If $\gcd(m,3)=1$ then $order(\lambda)=order(\zeta)=m$

  2. 2.

If $\gcd(m,3)=3$ then $order(\lambda)=m=3order(\zeta)$ with $order(\zeta)\neq
1$.

###### Proof.

To generate $G(m,m,3)$, we know that our nice 4-tuple must contain a
generating reflection set $X$ such that $\delta(X)$ is coprime to $m$, which
means either $\lambda$ or $\lambda\zeta$ has order $m$, by Lemma 4.2.3. If
$\lambda$ does not have order $m$, then $\lambda\zeta$ has order $m$, but
taking the determinant of equation 5, we know that $1=\lambda^{3}\zeta$ so
$\lambda^{2}$ has order $m$ which means $\lambda$ has order $m$. Thus,
$\lambda$ is always a primitive $m$-th root. Moreover, we know $m\neq 3$ since
the determinant tells us $\lambda^{3}=\zeta^{-1}$ and $\zeta\neq 1$ as it is
the nontrivial eigenvalue of the Type 2 reflection $r_{5}^{-1}$.

If $\gcd(m,3)=1$ then we know $\lambda^{3}=\zeta^{-1}$ is also a primitive
$m$-th root and hence so is $\zeta$. Using factorization 1, we can take the
tuple $[s(1,2;1),s(1,2;0),s(2,3;2),s(2,3;0)]$, so $\lambda=\zeta_{m}$ and
$\zeta=\zeta_{m}^{-3}$. This tuple generates $G(m,m,3)$ by Theorem 4.0.4 since
taking $X=\\{s(1,2;1),s(1,2;0),s(2,3;2)\\}$ gives $\delta(X)=1$ which is
coprime to $m$.

If $\gcd(m,3)=3$ then $\lambda^{3}$ has order $m/3$ and thus so too will
$\zeta$. Again taking the tuple as above will work, since $\delta(X)=1$ will
still be coprime to $m$ and the product will give $\lambda=\zeta_{m}$ and
$\zeta=\zeta_{m}^{-3}$ which now is not a primitive $m$-th root. ∎

######  Theorem 4.2.5.

Nice 4-tuples exist in $G(m,1,3)$ if and only if $\gcd(m,3)$ = 1. In this case
the order of $\lambda$ is $m$.

###### Proof.

Let $[r_{2},r_{3},r_{4},r_{5}]$ be a nice 4-tuple in $G(m,1,3)$. Let $r_{5}$
be the unique reflection of Type 2 with nontrivial eigenvalue $\zeta^{-1}$ of
order $m$, so we can write $\zeta^{-1}=\zeta_{m}^{-b}$ with $(-b,m)=1$. Let
$d=order(\lambda)$. We know $d\mid m$ since $\lambda I$ is in the center
$ZG(m,1,3)$. Taking determinants we get the equation

$\lambda^{3}=\zeta^{-1}$

so $m\mid d$ since $(\zeta^{-1})^{d}=1$, hence $d=m$. Since $\zeta^{-1}$ is a
primitive $m$-th root, so too is $\lambda^{3}$ and therefore we know
$\gcd(m,3)=1$.

Let $m$ be such that $\gcd(m,3)=1$, so we know $m-3$ is coprime to $m$. Taking
$b=m-3$, we can consider the tuple $[s(1,2;0),s(2,3;1),s(2,3;-1),s(3;m-3)]$.
Taking $X=\\{s(1,2;b_{2}),s(2,3;b_{3}),s(3;-b)\\}$ we see by Theorem 4.0.4
that $\langle X\rangle=G(m,1,3)$ since $\gcd(m-3,m)=1$. Taking the product, we
get $r_{2}r_{3}r_{4}r_{5}=\lambda r_{1}^{-1}$ for some Type 1 reflection
$r_{1}=s(1,2;-1)$ with $\lambda=\zeta_{m}$. ∎

######  Theorem 4.2.6.

Suppose $p\neq 1$ and $p\neq m$. Then there exists a nice $4$-tuple in
$G(m,p,3)$ if and only if $p=3$. In this case $\lambda$ must be of order $m$.

###### Proof.

Let $[r_{2},r_{3},r_{4},r_{5}]$ be a nice 4-tuple in $G(m,p,3)$. Letting
$r_{5}$ denote the unique reflection of Type 2, we know it must have order
$q=p^{-1}m$ so we can denote its nontrivial eigenvalue as
$\zeta=\zeta_{q}^{b}=\zeta_{m}^{pb}$ with $\gcd(b,m)=1$. Further, we have the
product $r_{2}r_{3}r_{4}r_{5}=\lambda r_{1}^{-1}$ with $r_{1}\in G(m,m,3)\leq
G(m,p,3)$, hence $\lambda I$ is in the center $ZG(m,p,3)$. Letting
$d=order(\lambda)$ we see that since $|ZG(m,p,3)|=q\gcd(p,3)$, we must have
that $d\mid q\gcd(p,3)$. Taking determinants, we get $\lambda^{3}=\zeta^{-1}$
and hence $q\mid d$ and $d\mid 3q$. If $d\nmid q$ then since $d\neq 3$
(becasue $\zeta^{-1}\neq 1$), and $d\mid q\gcd(p,3)$ we must have that
$\gcd(p,3)=3$. This gives us two cases:

  1. 1.

$d=q$

  2. 2.

$d=3q$ and $\gcd(p,3)=3$

Recall that by Theorem 4.0.4, for a reflection set $X$ to generate $G(m,p,3)$,
we need $\gcd(pb,m)=p$ and $\gcd(pb,\delta(X),m)=1$.

In the first case $d=q$ so we can write $\lambda=\zeta_{q}^{a}=\zeta_{m}^{p}a$
and $\zeta=\zeta_{q}^{b}=\zeta_{m}^{pb}$ with $(b,q)=1=(a,q)$. By Lemma 4.2.3,
we see that

$\gcd(\delta(X),m)=\begin{cases}\gcd(pa,m)\\\ \gcd((a+b)p,m)\end{cases}$

Since neither of these satisfy $\gcd(pb,\delta(X),m)=1$, it is not possible
for a nice 4-tuple in $G(m,p,3)$ to have $d=q$.

Suppose now that $p=3k$ so $m=3kq$. Again we can write
$\zeta=\zeta_{q}^{b}=\zeta_{m}^{pb}$ with $\gcd(b,q)=1$, and this time since
$d=3q$ we write $\lambda=\zeta_{m}^{ka}$ with $\gcd(a,m)=1$. In this case the
possible options for $\delta(X)$ gives us

$\gcd(\delta(X),m)=\begin{cases}\gcd(ka,m)\\\ \gcd(ka+pb,m)\end{cases}$

In either case we get $\gcd(pb,\delta(X),m)=k$ so we conclude $k=1$. This also
gives us that $\lambda$ must be of order $m$. Thus, we conclude nice 4-tuples
can only exist in $G(m,3,3)$. In this case, we can write
$\lambda=\zeta_{m}^{a}$ with $(a,m)=1$ and we still have
$\zeta=\zeta_{m}^{3b}$ with $\gcd(b,m)=1$ for $q=p^{-1}m$. We can take our
4-tuple to be $[s(1,2;0),s(2,3;1),s(2,3;-1),s(3;m-3)]$, noting that since
$3\mid m$ we also have that $\gcd(m-3,m)=3$. Here $\delta(X)=2$ so
$\gcd(m-3,2,m)=1$ and we conclude this tuple is indeed nice. ∎

######  Remark 4.2.7.

Note that in all 3 cases described above, we will have to take middle
convolution with parameter $\lambda^{-1}$ since our equations use the tuple
product and not the inverse product.

###  4.3 Nice 4-Tuples in $G(m,p,4)$

We are seeking nice tuples satisfying

$r_{1}r_{2}r_{3}r_{4}=A$

with $r_{i}$ a reflection in $G(m,p,4)$ and $A$ has a nontrivial eigenvalue of
multiplicity 2. Again the only possibilities are $p=1$ and $p=m$ since the
group must be generated by 4 reflections.

######  Theorem 4.3.1.

In $G=G(m,1,4)$ there are no nice tuples of 4 reflections.

###### Proof.

Suppose we had $r_{1}r_{2}r_{3}r_{4}=A$ such that $r_{i}$ generate $G$ and $A$
has eigenvalue $\lambda\neq 1$ of multiplicity 2. Since $\\{r_{i}\\}$ generate
$G$, we know exactly three of them are of Type 1 and exactly one is of Type 2
with nontrivial eigenvalue an $m$-th root of unity, say $\zeta_{m}^{\prime}$.
Note that since the three Type 1 reflections should correspond to
transpositions that generate all of $S_{4}$, they must be three distinct
transpositions with each of $\\{1,2,3,4\\}$ showing up at least once. This
means their product cannot be a transposition. Using $\varphi:G\longrightarrow
S_{4}$ we get that $\varphi(A)=\sigma$ for some 4-cycle $\sigma$ since the
permutation must be odd and cannot be a transposition. Writing
$A=[a_{1},a_{2},a_{3},a_{4}\mid\sigma]$ we can compute that the character
polynomial of $A$ $p_{A}(x)$ must be

$p_{A}(x)=x^{4}-\zeta_{m}^{a_{1}+a_{2}+a_{3}+a_{4}}$

This gives us that all the eigenvalues of $A$ are the distinct 4th roots of
$\zeta_{m}^{a_{1}+a_{2}+a_{3}+a_{4}}$ which means $A$ cannot have an
eigenvalue of multiplicity 2. ∎

In the case of $G(m,m,4)$, we know for any nice 4-tuple, we must have that all
the reflections are of Type 1. This means they all map to transpositions in
$S_{4}$ and hence the permutation associated to $A$ must be even. In $S_{4}$
the only options are $e$, the disjoint product of transpositions and a three
cycle. We would need at least 6 transpositions to write $e$ as the transitive
product of transpositions, so we only have two choices. Without loss of
generality, if $A$ corresponds to disjoint transpositions, we take it to be
$(12)(34)$ and if $A$ corresponds to a 3-cycle we take it to be $(132)$.

######  Remark 4.3.2.

The number of transitive ordered factorizations of $(12)(34)$ into four
transpositions in $S_{4}$ is 96 [13]. Using a computer, we find that there are
12 factorizations up to simultaneous conjugation. We can reduce these 12 even
further by eliminating factorizations that are related by the action of one of
the generators of $B_{4}$, since this means they lie in the same braid group
orbit. These preliminary computations give us three equivalence classes of
factorizations, (which may still lie in the same braid group orbit) which we
write with representatives:

  1. 1.

$[(12),(34),(13),(13)]$

  2. 2.

$[(13),(13),(12),(34)]$

  3. 3.

$[(13),(24),(23),(14)]$

Choosing general reflections in $G(m,m,4)$ associated to these factorization
and computing the products we get:

  1. 1.

$s(1,2;b_{1})\cdot s(3,4;b_{2})\cdot s(1,3;b_{3})\cdot
s(1,3;b_{4})=\begin{pmatrix}0&\zeta_{m}^{b_{1}}&0&0\\\
\zeta_{m}^{-b_{1}+b_{3}-b_{4}}&0&0&0\\\ 0&0&0&\zeta_{m}^{b_{2}}\\\
0&0&\zeta_{m}^{-b_{2}-b_{3}+b_{4}}&0\end{pmatrix}$

  2. 2.

$s(1,3;b_{1})\cdot s(1,3;b_{2})\cdot s(1,2;b_{3})\cdot
s(3,4;b_{4})=\begin{pmatrix}0&\zeta_{m}^{b_{1}-b_{2}+b_{3}}&0&0\\\
\zeta_{m}^{-b_{3}}&0&0&0\\\ 0&0&0&\zeta_{m}^{-b_{1}+b_{2}+b_{4}}\\\
0&0&\zeta_{m}^{-b_{4}}&0\end{pmatrix}$

  3. 3.

$s(1,3;b_{1})\cdot s(2,4;b_{2})\cdot s(2,3;b_{3})\cdot
s(1,4;b_{4})=\begin{pmatrix}0&\zeta_{m}^{b_{1}-b_{3}}&0&0\\\
\zeta_{m}^{b_{2}-b_{4}}&0&0&0\\\ 0&0&0&\zeta_{m}^{-b_{1}+b_{4}}\\\
0&0&\zeta_{m}^{-b_{2}+b_{3}}&0\end{pmatrix}$

######  Remark 4.3.3.

Similarly, we can consider transitive ordered factorizations of $(132)$ into
four transpositions in $S_{4}$. Up to simultaneous conjugation and the action
of a generator of $B_{4}$, we again get three equivalence classes of
factorizations (which may not be distinct):

  1. 1.

$[(14),(14),(12),(23)]$

  2. 2.

$[(14),(14),(13),(12)]$

  3. 3.

$[(14),(14),(23),(13)]$

These factorizations give us the following matrices:

  1. 1.

$s(1,4;b_{1})\cdot s(1,4;b_{2})\cdot s(1,2;b_{3})\cdot
s(2,3;b_{4})=\begin{pmatrix}0&0&\zeta_{m}^{b_{1}-b_{2}+b_{3}+b_{4}}&0\\\
\zeta_{m}^{-b_{3}}&0&0&0\\\ 0&\zeta_{m}^{-b_{4}}&0&0\\\
0&0&0&\zeta_{m}^{-b_{1}+b_{2}}\end{pmatrix}$

  2. 2.

$s(1,4;b_{1})\cdot s(1,4;b_{2})\cdot s(1,3;b_{3})\cdot
s(1,2;b_{4})=\begin{pmatrix}0&0&\zeta_{m}^{b_{1}-b_{2}+b_{3}}&0\\\
\zeta_{m}^{-b_{4}}&0&0&0\\\ 0&\zeta_{m}^{-b_{3}+b_{4}}&0&0\\\
0&0&0&\zeta_{m}^{-b_{1}+b_{2}}\end{pmatrix}$

  3. 3.

$s(1,4;b_{1})\cdot s(1,4;b_{2})\cdot s(2,3;b_{3})\cdot
s(1,3;b_{4})=\begin{pmatrix}0&0&\zeta_{m}^{b_{1}-b_{2}+b_{4}}&0\\\
\zeta_{m}^{b_{3}-b_{4}}&0&0&0\\\ 0&\zeta_{m}^{-b_{3}}&0&0\\\
0&0&0&\zeta_{m}^{-b_{1}+b_{2}}\end{pmatrix}$

######  Lemma 4.3.4.

In $G=G(m,m,4)$ if we had $r_{1}r_{2}r_{3}r_{4}=A$ with $r_{i}$ reflections
generating $G$ and $A$ has a nontrivial eigenvalue $\lambda$ of multiplicity 2
then the eigenvalues of $A$ are one of the following:

  1. 1.

$\\{\lambda,\lambda,-\lambda,-\lambda\\}$ for $\lambda$ a 4-th root of unity

  2. 2.

$\\{\lambda,\lambda,\lambda_{1},\lambda_{2}\\}$ for $\lambda$ a 4-th root of
unity and $\lambda_{i}\neq\lambda$ are the two distinct cube roots of
$\lambda^{-1}$ other than $\lambda$.

###### Proof.

We know all the $r_{i}$ are of Type 1, so using $\varphi:G\longrightarrow
S_{4}$ we know that $\varphi(A)$ must be an even permutation, so it is either
a double transposition or a 3-cycle. Let the eigenvalues of $A$ be
$\\{\lambda,\lambda,\lambda_{1},\lambda_{2}\\}$. If $\varphi(A)$ is a double
transposition then we see that

$\displaystyle 0$ $\displaystyle=\tr(A)=2\lambda+\lambda_{1}+\lambda_{2}$
$\displaystyle 1$ $\displaystyle=\det(A)=\lambda^{2}\lambda_{1}\lambda_{2}$

Rearranging the equation for trace gives us that

$1=|\lambda|=\frac{|\lambda_{1}+\lambda_{2}|}{2}=\frac{|\lambda_{1}|+|\lambda_{2}|}{2}=1$

By properties of the triangle inequality, we get that $\lambda_{1}$ and
$\lambda_{2}$ are colinear, hence $\lambda_{1}=\pm\lambda_{2}$. However, if
$\lambda_{1}=-\lambda_{2}$ then the trace equation means $\lambda=0$ which is
not possible, hence $\lambda_{1}=\lambda_{2}$. Now the trace equation tells us
that $2(\lambda+\lambda_{1})=0$, hence $\lambda_{1}=-\lambda$. Next, we can
use the determinant equation to see that $1=\lambda^{4}$ and so $\lambda$ must
be a 4th root of unity. This gives us that the eigenvalues of $A$ are as
described in case 1).

Further, when $\varphi(A)$ is a double transposition, say
$A=[a_{1},a_{2},a_{3},a_{4}\mid(12)(34)]$, the characteristic polynomial is
$p_{A}(x)=(x^{2}-\zeta_{m}^{a_{1}+a_{2}})(x^{2}-\zeta_{m}^{a_{3}+a_{4}})$.
However, since we know the roots of these polynomials agree, we see that
$\zeta_{m}^{a_{1}+a_{2}}=\zeta_{m}^{a_{3}+a_{4}}$, but we already knew that
$\zeta_{m}^{a_{1}+a_{2}}=\zeta_{m}^{-a_{3}-a_{4}}$ since $m\mid
a_{1}+a_{2}+a_{3}+a_{4}$. Thus we conclude that $\zeta_{m}^{a_{1}+a_{2}}=\pm
1$ and so $\zeta_{m}^{a_{1}}=\pm\zeta_{m}^{-a_{2}}$ and similarly,
$\zeta_{m}^{a_{3}}=\pm\zeta_{m}^{-a_{4}}$ where the signs must agree for both
pairs.
If $\varphi(A)$ is a 3-cycle, we write $A=[a_{1},a_{2},a_{3},a_{4}\mid(132)]$,
since taking a different 3-cycle just changes the indices. The determinant
condition stays the same as above, however our equation for trace is now:

$\zeta_{m}^{a_{4}}=\tr(A)=2\lambda+\lambda_{1}+\lambda_{2}$

Computing the characteristic polynomial we get
$p_{A}(x)=(x-\zeta_{m}^{a_{4}})(x^{3}-\zeta_{m}^{a_{1}+a_{2}+a_{3}})$ and
since the cube roots of $\zeta_{m}^{a_{1}+a_{2}+a_{3}}$ are all distinct, we
know that for $\lambda$ to have multiplicity two, it must be one of the cube
roots and $\lambda=\zeta_{m}^{a_{4}}$. Since
$\zeta_{m}^{a_{1}+a_{2}+a_{3}}=\zeta_{m}^{-a_{4}}=\lambda^{-1}$, we see that
$\lambda,\lambda_{1}$ and $\lambda_{2}$ must be the three distinct cube roots
of $\lambda^{-1}$. Now we have that $\lambda^{3}=\lambda^{-1}$ and hence
$\lambda^{4}=1$, proving that the eigenvalues of $A$ are as described in case
2). ∎

######  Theorem 4.3.5.

Nice 4-tuples exist in $G(m,m,4)$ iff $m=2$ or $m=4$. In either case,
$\lambda$ is of order $m$.

###### Proof.

Say we had a nice tuple in $G(m,m,4)$ such that $r_{1}r_{2}r_{3}r_{4}=A$.
Suppose first that $A$ corresponds to a three-cycle, then our nice 4-tuple is
equivalent to one of the factorizations in Remark 4.3.3. For all three
factorizations, by the previous Lemma, we know that
$\zeta_{m}^{-b_{1}+b_{2}}=\lambda$ and by Theorem 4.0.4, we know that
$\delta(X)=|b_{1}-b_{2}|$ must be coprime to $m$. This means $\lambda$ is a
primitive $m$-th root and by Lemma 4.3.4 we know $m=4$ or $m=2$ in this case.

If $A$ corresponds to a double transposition, then our nice 4-tuple is
equivalent to one of the factorizations in Remark 4.3.2 and we may write
$A=[a_{1},a_{2},a_{3},a_{4}\mid(12)(34)]$. From the first factorization and
the proof of Lemma 4.3.4, we see that $\pm
1=\zeta_{m}^{a_{1}+a_{2}}=\zeta_{m}^{b_{3}-b_{4}}=\lambda^{2}$. In this case
$\delta(X)=|b_{3}-b_{4}|$ which we know must be coprime to $m$ by Theorem
4.0.4, we get that $\lambda^{2}$ is a primitive $m$-th root so $m=2$. The same
argument will also give that $m$ must be 2 if our nice tuple is equivalent to
the second factorization in Remark 4.3.2. For the third possible
factorization, we see that the cycle has four edges and so
$\delta(X)=|b_{1}-b_{3}+b_{2}-b_{4}$ since the cycle corresponds to the
permutation $(1324)$. But again we have that $\pm
1=\zeta_{m}^{a_{1}+a_{2}}=\zeta_{m}^{b_{1}+-b_{3}+b_{2}-b_{4}}$ so
$\lambda^{2}$ is a primitive $m$-th root so $m=2$. If $m=4$, then we can take
$r_{1}=s(1,4;-1),r_{2}=s(1,4;0),r_{3}=s(1,2;-1),r_{4}=s(2,3;0)$. Then
$\delta(X)=1$ so these reflections generate $G(4,4,4)$. Further, we have

$A=\begin{pmatrix}0&0&-1&0\\\ i&0&0&0\\\ 0&1&0&0\\\ 0&0&0&i\end{pmatrix}$

which has eigenvalue $i$ with multiplicity 2. If $m=2$, then we can take
$r_{1}=s(1,4;1),r_{2}=s(1,4;0),r_{3}=s(1,2;0),r_{4}=s(2,3;0)$ so again
$\delta(X)=1$ and generates $G(2,2,4)$. In this case

$A=\begin{pmatrix}0&0&-1&0\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&-1\end{pmatrix}$

which has eigenvalue -1 with multiplicity 2. ∎

Note that when taking the middle convolution for such tuples, the parameter
should be $\lambda^{-1}$ since these equations are for the tuple product and
not the inverse product.

###  4.4 Nice 5-Tuples in $G(m,p,4)$

Searching for 5-tuples $G(m,p,4)$ so that the middle convolution will be rank
2 means we want the product to have an eigenvalue of multiplicity 3. A
$4\times 4$ matrix with an eigenvalue of multiplicity 3 can be viewed as a
scalar times a reflection, so we want to explore the existence of
$[r_{1},r_{2},r_{3},r_{4},r_{5}]$ such that

$r_{1}r_{2}r_{3}r_{4}r_{5}=\lambda r_{6}^{-1}$

where $\lambda\neq 1$ and $r_{6}$ is some reflection, not necessarily in
$G(m,p,4)$.

######  Lemma 4.4.1.

If we have 5 reflections generating $G=G(m,p,4)$ such that

$\displaystyle r_{1}r_{2}r_{3}r_{4}r_{5}=\lambda r_{6}^{-1}$ (6)

then it must be that

  1. 1.

$r_{6}\in G$ and $G=G(m,m,4)$ with $order(\lambda)\mid m$

  2. 2.

$\lambda^{4}=1$

###### Proof.

Note first that $G(m,m,4)\leq G(m,p,4)\leq G(m,1,4)$ so $r_{i}\in G(m,1,4)$
for $i<6$ and so $\lambda r_{6}^{-1}\in G(m,1,4)$. Since this matrix has an
eigenvalue of multiplicity 3, we know $d=order(\lambda)$ is such that $d$
divides 3 of the degrees in $\deg G(m,1,4)=\\{m,2m,3m,4m\\}$. This tells us
that $d\mid m$ as if it did not then $d\mid 3m-2m=m$. Since $\lambda$ has
order dividing $m$, $\lambda I$ is in the center of $G(m,1,4)$ and thus we can
conclude $r_{6}\in G(m,1,4)$, though not yet necessarily in $G(m,p,4)$. This
changes our problem into studying 6 reflections in $G(m,1,4)$ satisfying

$r_{1}r_{2}r_{3}r_{4}r_{5}r_{6}=\lambda I$

Next we use the homomorphism $\varphi:G(m,1,4)\longrightarrow S_{4}$. By
Theorem 4.0.2, we know
$\varphi(r_{1}r_{2}r_{3}r_{4}r_{5}r_{6})=\varphi(\lambda I)=e$ and since $e$
is even, we know an even number of $\varphi(r_{i})$ must be transpositions.

To conclude all 6 are transpositions, note that $\\{r_{i}\\}_{i<6}$ are
supposed to generate $G(m,p,4)$ which means $\\{\varphi(r_{i})\\}_{i<6}$ must
generate a transitive subgroup of $S_{4}$, hence $S_{4}$ itself. This removes
the possibility of just 2 transpositions since we need at least 3 distinct
transpositions to generate $S_{4}$.

While it is possible to generate $S_{4}$ with 4 transpositions, our equation
also requires that their product is the identity which is not possible with at
least 3 distinct transpositions (as can be verified by hand).

Thus we know all 6 reflections must be Type 1 and so $\\{r_{i}\\}_{i<6}$ can
only generate $G(m,m,4)$, we conclude $G=G(m,m,4)$.

Using that all reflections of Type 1 have nontrivial eigenvalue -1, taking
determinant gives us that

$1=(-1)^{6}=\det(r_{1}r_{2}r_{3}r_{4}r_{5}r_{6})=\det(\lambda I)=\lambda^{4}$

so we know lambda must be a 4th root of unity. Further, using that
$order(\lambda)\in\\{2,4\\}$ must divide at least 3 degrees in $\deg
G(m,m,4)=\\{m,2m,3m,4\\}$ we conclude that $order(\lambda)\mid m$. ∎

######  Remark 4.4.2.

To study such tuples in $G(m,m,4)$ we first find the ways to decompose $e$ in
$S_{4}$ into a transitive product of 6 transpositions up to conjugacy. To make
such a list, note that we can assume the first two transpositions are not
disjoint, as if they are we can swap them around. This tells us their product
is a 3-cycle and up to conjugation we may assume it is $(132)$. We then found
all the 4-tuples of transpositions that multiple to $(123)$ up to simultaneous
conjugation and generate all of $S_{4}$. Adjoining these options to the
$(12),(13)$ on the left, we get 27 options for transitive factorizations of
the identity , and using computer code to partially compute the braid group
orbit we were able to reduce the list further to two options (not necessarily
distinct). The tuples in $S_{4}$ are:

  1. 1.

$[(12),(13),(12),(23),(24),(24)]$

  2. 2.

$[(12),(13),(24),(24),(12),(23)]$

Using the Type 1 notation for the reflections in $G(m,m,4)$ as
$r_{i}=s(j_{i},k_{i};b_{i})$ we can take the matrix product. The first tuple
gives us the equation

$\displaystyle\begin{pmatrix}\zeta_{m}^{b_{1}-b_{3}}&0&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{2}-b_{4}+b_{5}-b_{6}}&0&0\\\
0&0&\zeta_{m}^{-b_{2}+b_{3}+b_{4}}&0\\\
0&0&0&\zeta_{m}^{-b_{5}+b_{6}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\lambda&0&0&0\\\ 0&\lambda&0&0\\\
0&0&\lambda&0\\\ 0&0&0&\lambda\end{pmatrix}$

and the second tuple gives us the equation

$\displaystyle\begin{pmatrix}\zeta_{m}^{b_{1}+b_{3}-b_{4}-b_{5}}&0&0&0\\\
0&\zeta_{m}^{-b_{1}+b_{2}-b_{6}}&0&0\\\
0&0&\zeta_{m}^{-b_{2}+b_{5}+b_{6}}&0\\\
0&0&0&\zeta_{m}^{-b_{3}+b_{4}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\lambda&0&0&0\\\ 0&\lambda&0&0\\\
0&0&\lambda&0\\\ 0&0&0&\lambda\end{pmatrix}$

######  Theorem 4.4.3.

Nice 5-tuples exist in $G(m,m,4)$ if and only if $m=2$ or $m=4$ and in either
case $\lambda$ is of order $m$.

###### Proof.

Suppose we had a nice 5-tuple in $G(m,m,4)$, so by the previous lemma we know
$2\mid m$. We know it must be equivalent to one of the two factorizations in
Remark 4.4.2. Considering the first factorization and looking at all
transitive subsets of 4 reflections we can choose from the 6, we get the
following possibilities for $\delta(X)$:

$\displaystyle\delta(X)=\begin{cases}|b_{1}-b_{3}|\\\ |b_{6}-b_{5}|\\\
|b_{4}+b_{3}-b_{2}|\\\ |b_{4}+b_{1}-b_{2}|\end{cases}$

The first three of these are such that $\zeta_{m}^{\delta(X)}$ equals
$\lambda$ up to the absolute value on $\delta(X)$, so if any of those are
coprime to $m$ then $\lambda$ is of order $m$. So the only way $m\neq 2,4$ is
if the only option for $\delta(X)$ that is coprime to $m$ is
$|b_{4}+b_{1}-b_{2}|$. Writing $m=2^{s}k$ with $(k,2)=1$ and $s\geq 1$, we can
write $\lambda=\zeta_{m}^{k(s-j)}$ for $j=1,2$ depending on the order of
$\lambda$. The equations for $b_{i}$ modulo $m$ then becomes:

$\displaystyle b_{1}-b_{3}\equiv k2^{s-j}$ $\displaystyle-
b_{1}+b_{2}-b_{4}+b_{5}-b_{6}\equiv k2^{s-j}$ $\displaystyle-
b_{2}+b_{3}+b_{4}\equiv k2^{s-j}$ $\displaystyle-b_{5}+b_{6}\equiv k2^{s-j}$

The second and fourth equation give us $-b_{1}+b_{2}-b_{4}\equiv 2k2^{s-j}$
which is not coprime to $m$. Thus, this option for $\delta(X)$ does not work
so one of the first three options are coprime to $m$ and hence $\lambda$ is of
order $m$. Since $\lambda$ is a 4-th root of unity, we know $m=2$ or $m=4$.

A similar argument will give the same conclusion for the second factorization.
For $m=4$, consider the tuple
$[s(1,2;1),s(1,3;0),s(1,2;0),s(2,3;1),s(2,4;1)]$. Then $\delta(X)=1$ and so
these reflections will generate $G(4,4,4)$. The product of this tuple will be
$i\cdot s(2,4;2)$ as this corresponds to the first factorization. For $m=2$,
we can take the tuple $[s(1,2;1),s(1,3;0),s(1,2;0),s(2,3;1),s(2,4;0)]$. Then
$\delta(X)=1$ and so these reflections will generate $G(2,2,4)$. The product
of this tuple will be $-1\cdot s(2,4;1)$ as this corresponds to the first
factorization. ∎

Note that with the examples above, we will have to take middle convolution
with parameter $\lambda^{-1}$ since our equations are for the tuple product
and not the inverse product.

###  4.5 Nice $n$-Tuples in $G(m,p,n)$ for $n\geq 5$

Recall that the only possible groups when nice $n$-tuples can exist are when
$p=1$ and $p=m$ since these are the only well-generated imprimitive groups
(See Definition 2.2.1).

Fact: A connected graph of $n$ vertices and $n-1$ edges must be a tree.

######  Claim 4.5.1.

The product of $n-1$ transpositions is an $n$-cycle if and only if the graph
associated to a set of transposition in $S_{n}$ is a tree, where the edge
$(a,b)$ is only present if $(ab)$ is one of the $n-1$ transpositions.

###### Proof.

By induction: if the product of $n-1$ transpositions is the product of a
disjoint $k$-cycle and an $(n-k)$-cycle, then each of these correspond to
disjoint trees with $k-1$ and $(n-k-1)$ edges respectively, so the remaining
edge must connect the two trees, resulting in a tree with $n-1$ edges, hence
the product is an $n$-cycle. ∎

######  Theorem 4.5.2.

There are no nice $n$-tuples in $G(m,1,n)$ for $n\geq 5$.

###### Proof.

In $G(m,1,n)$ any nice $n$-tuple consists of $n-1$ reflections of Type 1 and
one reflection of Type 2, say $r_{n}$. Suppose we had $r_{1}r_{2}\cdots
r_{n}=A$ with $[r_{1},\dots,r_{n}]$ a nice tuple. Then by the claim(s) above,
the graph associated to the transpositions $\\{\varphi(r_{i})\\}$, with
$\varphi(r_{n})=e$ must be a tree since the graph is connected by transitivity
and therefore $\varphi(A)$ is an $n$-cycle. Thus the characteristic polynomial

$char_{A}(x)=(-1)^{n}(x^{n}-\zeta)$

for some $m$-th root of unity $\zeta$ and therefore has $n$ distinct roots so
no $A$ has no eigenvalues of multiplicity $n-2$ for $n>3$. ∎

For $G(m,m,n)$ we need the additional observations provided below.

######  Claim 4.5.3.

Let $\sigma=(x_{1}x_{2},\cdot x_{n})$ be an $n$-cycle in $S_{n}$ and
$\tau=(y_{1}y_{2})$ be some transposition in $S_{n}$ with $y_{1}<y_{2}$.
Letting $c$ denote the distance between $y_{1}$ and $y_{2}$ in $\sigma$ modulo
the length of $\sigma$, we have that the product $\sigma\cdot\tau$ is the
product of a disjoint $c$-cycle and a disjoint $(n-c)$-cycle.

###### Proof.

Since $\sigma$ fixes no points, it is enough to show that multiplying with
$\tau$ cannot fix two points in $\\{1,\dots,n\\}$. Letting $y_{1}=x_{i}$ and
$y_{2}=x_{j}$ with $i<j$ then we get

$y_{1}\mapsto x_{j+1}\mapsto x_{j+2}\mapsto\dots\mapsto x_{i-1}\mapsto
x_{i}=y_{1}$

where the subscripts are read modulo $n$. The length of this cycle is
precisely $j-i\mod n$. Doing a similar process for $y_{2}$ we get a cycle of
length $n-(j-i)\mod n$. ∎

Note that since $\sigma$ is written in disjoint cycle notation the ‘distance’
between, i.e. number of digits between $y_{1}$ and $y_{2}$ read in the order
of multiplication (e.g. left to right if multiplication is left to right) is
well defined.

######  Corollary 4.5.4.

In the notation of the claim above, if $n>2$, then the product $\sigma\tau$
has at most one fixed point, with this occurring only if $y_{1}$ and $y_{2}$
are adjacent in the cycle $\sigma$.

###### Proof.

If $y_{1}$ and $y_{2}$ are adjacent then $c=1$, so we are left with just an
$n-1$ cycle which has exactly 1 fixed point when $n-1>1$ and 2 fixed points
when $n=2$. ∎

######  Theorem 4.5.5.

There are no nice $n$-tuples in $G(m,m,n)$ for $n\geq 5$.

###### Proof.

In $G(m,m,n)$, any nice tuple $[r_{1},r_{2},\dots,r_{n}]$ consists of $n$
reflections of Type 1. Again by transitivity, we know the graph associated to
$\\{\varphi(r_{i})\\}$ is connected and since we only need $n-1$ edges to make
a connected graph on $n$ vertices, we know there is some transposition we can
remove so that the associated graph is still connected. By cyclic permutation
of the $r_{i}$ via simultaneous conjugation, we can assume the redundant
transposition corresponds to $r_{n}$, so $\varphi(A\cdot r_{n}^{-1})$ is an
$n$-cycle by the claim above. This tells us that $\varphi(A)$ is the product
of an $n$ cycle and a transposition. By the proofs above, we know $\varphi(A)$
is either an $(n-1)$-cycle or the product of a $c$-cycle and an $(n-c)$-cycle.
So either the characteristic polynomial is

$char_{A}(x)=(-1)^{n}(x^{n-1}-\zeta_{1})(x-\zeta_{2})$

or

$char_{A}(x)=(-1)^{n}(x^{c}-\zeta_{1})(x^{n-c}-\zeta_{2})$

We see that in both cases an eigenvalue of $A$ can have atmost multiplicity 2,
so when $n>4$, there are no eigenvalues of multiplicity $n-2$, hence no such
nice tuples can exist in $G(m,m,n)$. ∎

##  5 Primitive Reflection Groups

For each rank $r\in\\{3,4\\}$, we investigate the existence of nice $T$-tuples
for $T=r$ and $T=r+1$ in the primitive reflection groups, denoted $G_{i}$ for
$i=23,24,\dots,32$. For each group and each choice of $T$, we provide a table
of the different types (up to inverse pairs), including the label for the
type, the residues of the eigenvalues, as well as the nontrivial eigenvalues
of the exemplar tuple under ‘Tuple Eigenvalues’. We also list the chosen
exemplars $\mathbf{A}=[A_{1},A_{2},\dots,A_{T}]$ for each type. For each
exemplar we compute the middle convolution with the suitable choice(s) of
parameter $\lambda$ and this data is available in the appendices A, B, C.
These tables also include, for each choice of eigenvalue $\lambda$, the
residue of $\lambda$ in the ‘$\xi$’ column as well as the notation used in
Magma to identify the root of unity under ‘$\lambda$’. We list the $2\times 2$
matrices composing $MC(\mathbf{A})=[M_{1},\dots,M_{T}]$ and record the order
of the $GL_{2}(\mathbb{C})$ subgroup generated by the these matrices under ‘S.
Size’. Note that a zero in this column means the subgroup is infinite.

To get the induced tuple $\mathbf{\hat{A}}$ from $MC_{\lambda}(\mathbf{A})$
many choices of character will work, so we provide the one we use under the
column ‘Character’ or sometimes abbreviated as ‘Char.’. In some cases, we may
need to go to a larger field to construct a suitable character. We then
compute and record the braid group orbit size of the induced tuple (under
action of the positive powers of the braid group generators) and record it
under ‘O. Size’. More information about how we compute the braid group orbit
is provided in Section 3.1. For examples where the braid group orbits are the
same size, we check if the signature of one lies in the orbit of the other.

######  Definition 5.0.1.

We say two braid group orbits are $\mathbf{equivalent}$ if the signature of
one of the induced tuples lies in the orbit of another induced tuple.

This is one reason why our braid group code optionally can return the list of
signatures in each orbit. Note that this differs from the broader notion of
equivalence in other literature such as Tykhyys’s classification [27]. Note
that the work of Lisovy-Tykhyy [19] and Tykhyy [27] only consider the case
where matrices in the tuple do not have a shared eigenvector. In our case,
since nice tuples give an irreducible representation and middle convolution
preserves irreducibility, we know all our tuples cannot share a common
eigenvector. Thus, we expect to identify which tuples in these classifications
come from the complex reflection groups via middle convolution. Details on
what parameters we use to compare our orbits with these classifications is
provided below at the start of Section 5.1 and 5.2.

For each distinct orbit we find, we provide data in the comparison tables on
how our orbits relate to other classifications, as well as identify the finite
$SL_{2}(\mathbb{C})$ subgroups generated by the tuples. We list the subgroups
of $SL_{2}(\mathbb{C})$ in the Magma notation, given by $\langle
order,number\rangle$ where order is the order of the finite group and number
is the numbering of the group in the Small Groups Database. The numbering used
by Magma is the same as that in the Small Group database implemented in GAP
(see https://magma.maths.usyd.edu.au/magma/handbook/text/779, accessed on July
7th, 2024) and can be readily found online for all the groups we see. For
infinite groups, the order is listed as ‘0’, following the notation used in
Magma. The notation $SL(2,k)$ means $2\times 2$ matrices of determinant 1 over
the finite field of $\mathbb{F}_{k}$.

###  5.1 Nice 3-Tuples in Rank 3 Groups

Nice tuples in this case will be tuples of 3 reflections generating the whole
group such that their inverse product has some nontrivial eigenvalue of
multiplicity 1. After middle convolution, our the induced tuple will
correspond to $SL_{2}(\mathbb{C})$ local systems on the 4-punctured sphere and
we can compare our braid group orbits to those in the literature. We compute
the parameters $(\omega_{X},\omega_{Y},\omega_{Z},\omega_{4})$ in the notation
of [19] and found that the $\omega$ parameters for each of our orbits match
(up to permutation) with one listed in the work of Lisovy-Tykhyy. Since
Tykhyy’s classification [27] further refines the original classification by
Lisovy-Tykhy [19], each tuple of $\omega$ parameters gives multiple orbits
that Tykhyy identifies uniquely in terms of residues of the trace coordinates
$(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$ and
$(\sigma_{12},\sigma_{23},\sigma_{13},\sigma_{24})$ for some representative.
In the nice 3-tuple case, using the $\omega$ parameters, we are able to easily
identify how our orbit size relates the orbit size listed by Tykhyy for each
$\mathcal{M}^{(4)}$ (in Tykhyy’s notation) and narrow down the possibilities
for which of Tykhyy’s orbits our tuple corresponds to. In Tykhyy’s
classification, tuples are considered equivalent up to: multiplying any two
matrices by -1, complex conjugation of all elements of all matrices,
cyclically permuting the matrices in the tuple, and up to taking inverses and
reversing the order of the tuple [27]. We perform all these operations and
compute the residue signatures for the matrices in each orbit to find one that
matches the example provided in Tykhyy’s work. In most cases we found exact
matches, and in other cases we provide the closest match, marking the
differing coordinate with an asterisk (see Section 5.1.3 and 5.1.4).

Note that we only list this data for orbits that are considered different by
our definition and these may end up being the same orbit in the Lisovy-Tykhyy
or Tykhyy classifications ([19], [27]). For each orbit we compute, we provide
the identifying label used in both the classifications, as well as our orbit
size and the orbit size they provided (which is the same in both works).

####  5.1.1 $G_{23}$

The group $G_{23}$ of order 120 is also denoted $W(H_{3})$ with degrees
$\deg(G_{23})=\\{2,6,10\\}$ [18, Table D.3]. All reflections in this group are
conjugate with nontrivial eigenvalue -1. For nice 3-tuples in $G_{23}$, the
computation returned three possible sets of eigenvalues for the inverse
product.

$G_{23}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1)$ |  $\\{\frac{1}{2},\frac{1}{10},\frac{9}{10}\\}$
B | $(-1,-1,-1)$ |  $\left\\{\frac{1}{2},\frac{3}{10},\frac{7}{10}\right\\}$
C | $(-1,-1,-1)$ |  $\\{\frac{1}{2},\frac{1}{6},\frac{5}{6}\\}$
Table 1:

With $q=\zeta_{5}$, we provide one example of a nice tuple such that the
inverse product has eigenvalues corresponding to each type.

For Type A in $G_{23}$ one exemplar is:

$\displaystyle A_{1}$ $\displaystyle=\begin{pmatrix}1&0&0\\\ 0&1&1\\\
0&0&-1\end{pmatrix}\quad A_{2}=\begin{pmatrix}0&-1&0\\\ -1&0&0\\\
-q^{3}-q^{2}+1&-q^{3}-q^{2}+1&1\end{pmatrix}\quad
A_{3}=\begin{pmatrix}1&0&0\\\ q^{3}+q^{2}&-1&-1\\\ 0&0&-1\end{pmatrix}$

Similarly for Type B, an exemplar is:

$\displaystyle B_{1}=\begin{pmatrix}1&0&0\\\ 0&1&1\\\ 0&0&-1\end{pmatrix}\quad
B_{2}=\begin{pmatrix}1&-q^{3}-q^{2}&-q^{3}-q^{2}\\\ 0&0&-1\\\
0&-1&0\end{pmatrix}\quad B_{3}=\begin{pmatrix}1&0&0\\\ q^{3}+q^{2}&-1&-1\\\
0&0&1\end{pmatrix}$

For Type C an exemplar is:

$\displaystyle C_{1}=\begin{pmatrix}1&0&0\\\ 0&1&1\\\ 0&0&-1\end{pmatrix}\quad
C_{2}=\begin{pmatrix}1&-q^{3}-q^{2}&0\\\ 0&-1&0\\\ 0&1&1\end{pmatrix}\quad
C_{3}=\begin{pmatrix}0&-1&0\\\ -1&0&0\\\
-q^{3}-q^{3}+1&-q^{3}-q^{2}+1&1\end{pmatrix}$

The data for all three types for $G_{23}$ is compiled in Table A. For all
three types, choosing a different exemplar would not have resulted in a new
orbit. The two orbits of size 10 arising from Type A are the same, as are the
two orbits of size 10 from Type B. However, these two orbits are distinct
across the two types in the sense that their signatures did not show up in the
others’ orbit. For Type C, the two orbits of size 18 are the same. Information
about how these 6 distinct orbits relate to the other classifications is
provided in Table 2. In this group we also find nice 4-tuples, details
provided in Section 5.2.1

$\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{23}$ Type A Comparisons
-1 | $(3-\sqrt{5},3-\sqrt{5},3-\sqrt{5},\frac{7\sqrt{5}-11}{2})$ | $16$ | $(1,1,1,\frac{1}{5})$ | $(\frac{3}{5},\frac{3}{5},\frac{3}{5},\frac{1}{3})$ | 60 | 10 | 40 | 0
$-q^{3}$ | $(3-\sqrt{5},3-\sqrt{5},3-\sqrt{5},\frac{7\sqrt{5}-11}{2})$ | 16 | $(\frac{3}{5},\frac{2}{5},\frac{3}{5},\frac{2}{5})$ | $(\frac{2}{5},\frac{2}{5},\frac{1}{3},\frac{3}{5})$ | 57 | 10 | 10 |  $\langle 120,5\rangle$
$G_{23}$ Type B Comparisons
-1 | $(3+\sqrt{5},3+\sqrt{5},3+\sqrt{5},-\frac{7\sqrt{5}+11}{2})$ | 17 | $(1,1,1,\frac{3}{5})$ | $(0,\frac{1}{5},\frac{1}{3},\frac{1}{3})$ | 58 | 10 | 40 | 0
$-q^{4}$ | $(3+\sqrt{5},3+\sqrt{5},3+\sqrt{5},-\frac{7\sqrt{5}+11}{2})$ | 17 | $(\frac{4}{5},\frac{4}{5},\frac{4}{5},\frac{4}{5})$ | $(\frac{1}{5},0,\frac{1}{3},\frac{1}{3})$ | 59 | 10 | 10 |  $\langle 120,5\rangle$
$G_{23}$ Type C Comparisons
-1 | $(2,2,2,-1)$ | 31 | $(1,\frac{1}{3},0,0)$ | $(\frac{1}{5},1,\frac{2}{5},\frac{2}{5})$ | 99 | 18 | 72 | 0
$z^{5}$ | $(2,2,2,-1)$ | 31 | $(\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3})$ | $(0,\frac{1}{5},\frac{3}{5},\frac{3}{5})$ | 96 | 18 | 18 |  $\langle 120,5\rangle$
Table 2: Here $q=\zeta_{5},z=\zeta_{30}$ and all three subgroups of order 120
in $SL_{2}(\mathbb{C})$ are isomorphic to $SL(2,5)$

####  5.1.2 $G_{24}$

The group $G_{24}$ of order 336 is also denoted $W(J_{3}(4))$ with degrees
$\deg(G_{24})=\\{4,6,14\\}$ [18, Table D. 3]. All reflections in this group
are conjugate with nontrivial eigenvalue -1. Looking at the degrees, we first
expand the base field to be the cyclotomic field of degree 84. There is one
inverse pair of nice 3-tuples in this group, so we only provide the details
for one in Table 3.

$G_{24}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1)$ |  $\\{\frac{1}{14},\frac{9}{14},\frac{11}{14}\\}$
Table 3:

Letting $z=\zeta_{7}$, an exemplar of Type A is:

$\displaystyle A_{1}=\begin{pmatrix}1&-z^{4}-z^{2}-z&-z^{4}-z^{2}-z\\\
\>0&0&-1\\\ \>0&-1&0\end{pmatrix}\quad A_{2}=\begin{pmatrix}1&1&0\\\
\>0&-1&0\\\ \>0&1&1\end{pmatrix}\quad A_{3}=\begin{pmatrix}-1&0&0\\\
\>1&1&0\\\ \>z^{4}+z^{2}+z&0&1\end{pmatrix}$

The middle convolution data for $G_{24}$ is provided in Table A. These three
distinct orbits are the only orbits from Type A and we checked that in all
three cases, the inverse of the exemplars do not produce any new orbits. The
comparison data for $G_{24}$ is provided in 4.

$\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{24}$ Type A Comparisons
$-z^{4}$ | $(1,1,1,0)$ | $8$ | $(\frac{4}{7},\frac{4}{7},\frac{3}{7},\frac{1}{7})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3})$ | 28 | 7 | 28 | 0
$-z$ | $(1,1,1,0)$ | 8 | $(\frac{1}{7},\frac{5}{7},\frac{1}{7},\frac{1}{7})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{1}{2})$ | 30 | 7 | 28 | 0
$-z^{2}$ | $(1,1,1,0)$ | 8 | $(\frac{2}{7},\frac{2}{7},\frac{4}{7},\frac{2}{7})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{1}{2})$ | 29 | 7 | 28 | 0
Table 4: Here $z=\zeta_{7}$

####  5.1.3 $G_{25}$

The group $G_{25}$ of order 648 is also denoted $W(L_{3})$ with degrees
$\deg(G_{25})=\\{6,9,12\\}$ [18, Table D.3]. $G_{25}$ has two conjugacy
classes, consisting of matrices with nontrivial eigenvalue $w$ or $w^{2}$
respectively, for $w=\zeta_{3}$. Expanding the base field to be the cyclotomic
field of order 36, we get 8 possible types, which actually split into 4 pairs
of inverses. We only list one set of inverse eigenvalues for each pair in
Table 5.

$G_{25}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(w,w,w^{2})$ |  $\\{\frac{5}{6},\frac{1}{6},\frac{2}{3}\\}$
B | $(w,w^{2},w^{2})$ |  $\left\\{\frac{1}{9},\frac{7}{9},\frac{4}{9}\right\\}$
C | $(w,w,w)$ |  $\\{\frac{11}{12},\frac{5}{12},\frac{2}{3}\\}$
D | $(w,w,w)$ |  $\\{\frac{1}{3},\frac{5}{6},\frac{5}{6}\\}$
Table 5: Here $w=\zeta_{3}$ denotes a 3rd root of unity

With $w=\zeta_{3}$ the exemplars for each type are provided below. Exemplar of
Type A:

$\displaystyle A_{1}=\begin{pmatrix}1&0&0\\\ 0&1&w^{2}\\\
0&0&w\end{pmatrix}\quad A_{2}=\begin{pmatrix}w&0&0\\\ w^{2}&1&0\\\
0&0&1\end{pmatrix}\quad A_{3}=\begin{pmatrix}1&w&0\\\ 0&w^{2}&0\\\
0&w&1\end{pmatrix}$

Exemplar of Type B:

$\displaystyle B_{1}=\begin{pmatrix}1&0&0\\\ 0&1&w^{2}\\\
0&0&w\end{pmatrix}\quad B_{2}=\begin{pmatrix}-w&1&w^{2}\\\ 1&-w^{2}&1\\\
w^{2}&1&-w\end{pmatrix}\quad B_{3}=\begin{pmatrix}1&w&0\\\ 0&w^{2}&0\\\
0&w&1\end{pmatrix}$

Exemplar of Type C:

$\displaystyle C_{1}=\begin{pmatrix}1&-w^{2}&0\\\ 0&w&0\\\
0&-w^{2}&1\end{pmatrix}\quad C_{2}=\begin{pmatrix}w&0&0\\\ w^{2}&1&0\\\
0&0&1\end{pmatrix}\quad C_{3}=\begin{pmatrix}1&0&0\\\ -w^{2}&w&-w^{2}\\\
0&0&1\end{pmatrix}$

Note that for Type D, there is only two choices of eigenvalue to take midlde
convolution with and only one of them will result in $2\times 2$ matrices.
Exemplar of Type D:

$\displaystyle D_{1}=\begin{pmatrix}1&-w^{2}&-w\\\ 0&-w^{2}&1\\\
0&-w&0\end{pmatrix}\quad D_{2}=\begin{pmatrix}w&0&0\\\ w^{2}&1&0\\\
0&0&1\end{pmatrix}\quad D_{3}=\begin{pmatrix}0&-w&w^{2}\\\ 0&1&0\\\
-w^{2}&-1&-w^{2}\end{pmatrix}$

The details for the middle convolution of each type is provided in Table 29.
For all three types, different exemplars do not produce new orbits nor do
their inverses. For Type A, the two orbits of size 12 are distinct, as are all
three orbits of size 36 from Type B. For Type C, we had to extend to a field
containing $\zeta_{72}$ to construct the character. The two orbits of size 4
from Type C are the same. Including the one from Type D, we get 8 distinct
orbits from $G_{25}$. The comparison data is provided in Table 6. Comparing
with the Lisovy-Tykhyy paper, all the $\omega$ parameters seemed to correspond
to the family of orbits listed in Lemma 39 in [19], so we have provided the
specific values for the variables that work in each case. When comparing with
Tykhyy’s classification, there was only one orbit (from Type B) that we could
not find an exact match for. We listed the closest match, which seemed to be
Orbit 5 in [27] and distinguish the different coordinate with an asterisk. In
this group we also find nice 4-tuples, details provided in Section 5.2.2

$\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{25}$ Type A Comparisons
$-w$ | $(0,0,3,-2)$ | Type II, $X^{\prime}=2,X^{\prime\prime}=1$ | $(\frac{1}{6},\frac{1}{2},\frac{1}{6},\frac{1}{2})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{3},0)$ | 3 | 2 | 12 |  $\langle 24,4\rangle$
$-w^{2}$ | $(0,0,3,-2)$ | Type II, $X^{\prime}=2,X^{\prime\prime}=1$ | $(\frac{1}{6},\frac{1}{2},\frac{1}{6},\frac{1}{2})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{3},0)$ | 3 | 2 | 12 |  $\langle 24,4\rangle$
$w^{2}$ | $(0,0,3,-2)$ | Type II, $X^{\prime}=2,X^{\prime\prime}=1$ | $(1,\frac{1}{3},1\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{2},0,\frac{1}{3})$ | 4 | 2 | 24 | 0
$G_{25}$ Type B Comparisons
$z^{4}$ | (2, -1, -1, -1) | Type III, $\omega=-1$ | $(\frac{4}{9},\frac{2}{9},\frac{2}{9},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{2}{3})$ | 5 | 3 | 36 | 0
$-z^{10}$ | (2, -1, -1, -1) | Type III, $\omega=-1$ | $(\frac{8}{9},\frac{4}{9},\frac{4}{9},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{2}{3}^{*})$ | $5^{*}$ | 3 | 36 | 0
$z^{16}$ | (2, -1, -1, -1) | Type III, $\omega=-1$ | $(\frac{2}{9},\frac{1}{9},\frac{1}{9},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{1}{3})$ | 5 | 3 | 36 | 0
$G_{25}$ Type C Comparisons
$-z^{9}+z^{3}$ | $(4,4,4,-8)$ | Type IV, $\omega=4$ | $(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4})$ | $(\frac{1}{3},\frac{1}{3},\frac{1}{3},0)$ | 6 | 4 | 4 |  $\langle 48,28\rangle$
$-z^{6}$ | $(4,4,4,-8)$ | Type IV, $\omega=4$ | $(0,0,0,\frac{1}{2})$ | $(\frac{1}{3},\frac{1}{3},\frac{1}{3},0)$ | 7 | 4 | 4 | 0
$G_{25}$ Type D Comparisons
$w$ | $(3,3,3,-5)$ | Type I, $X=Y=Z=1$ | $(\frac{2}{3},\frac{2}{3},0,\frac{1}{3})$ | $(\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3})$ | 2 | 1 | 4 |  $\langle 24,3\rangle$
Table 6: Here $w=\zeta_{3},z=\zeta_{36}$. The correct signature for Tykhyy’s
orbit $5$ would have $3(\frac{4}{9})\equiv\frac{1}{3}$ as $\sigma_{24}$ but
such a signature was not in the orbit, we have provided the closest match. The
subgroups $\langle 24,4\rangle,\langle 48,28\rangle,\langle 24,3\rangle$
correspond to the dicyclic group of order 24 (binary dihedral group), binary
octahedral group, and $SL(2,3)$ respectively.

####  5.1.4 $G_{26}$

The group $G_{26}$ of order 1296 is also denoted $W(M_{3})$ with degrees
$\deg(G_{26})=\\{6,12,18\\}$ [18, Table D.3]. In $G_{26}$ there are three
conjugacy classes of reflections, corresponding to nontrivial eigenvalues in
$\\{\zeta_{3},\zeta_{3}^{2},-1\\}$. In this group the are 6 sets of
eigenvalues for the inverse product, which correspond to 3 inverse pairs, so
we only list one from each pair.

$G_{26}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(w,w^{2},-1)$ |  $\\{\frac{1}{12},\frac{7}{12},\frac{5}{6}\\}$
B | $(w,w,-1)$ |  $\left\\{\frac{5}{18},\frac{11}{18},\frac{17}{18}\right\\}$
C | $(w^{2},w^{2},-1)$ |  $\\{\frac{1}{6},\frac{1}{6},\frac{5}{6}\\}$
Table 7: Here $w=\zeta_{3}$

The exemplars for this group are all provided over $\mathbb{Q}(\zeta_{36})$
with $z=\zeta_{36}$.

Here is an exemplar of Type $A$:

$\displaystyle A_{1}=\begin{pmatrix}1&0&0\\\ -1&2\cdot z^{6}&z^{6}\\\
-z^{6}+2&-3\cdot z^{6}&-z^{6}\end{pmatrix}\quad
A_{2}=\begin{pmatrix}0&z^{6}&0\\\ 1&-z^{6}+1&0\\\
z^{6}-2&z^{6}+1&1\end{pmatrix}\quad A_{3}=\begin{pmatrix}1&0&0\\\ 0&1&1\\\
0&0&-1\end{pmatrix}$

An exemplar for Type B:

$\displaystyle B_{1}=\begin{pmatrix}z^{6}-1&2\cdot z^{6}&z^{6}\\\ 0&1&0\\\
0&0&1\end{pmatrix}\quad B_{2}=\begin{pmatrix}0&z^{6}-1&z^{6}-1\\\ z^{6}&2&1\\\
-z^{6}-1&z^{6}-2&z^{6}-1\end{pmatrix}\quad B_{3}=\begin{pmatrix}1&0&0\\\
0&1&1\\\ 0&0&-1\end{pmatrix}$

An exemplar for Type C:

$\displaystyle C_{1}=\begin{pmatrix}1&0&0\\\ -z^{6}&z^{6}-1&z^{6}-1\\\
z^{6}+1&-3\cdot\left(z^{6}+1\right)&-2\cdot\left(z^{6}-1\right)\end{pmatrix}\quad
C_{2}=\begin{pmatrix}1&-z^{6}+1&-z^{6}+1\\\ 0&1&0\\\
0&-z^{6}-1&-z^{6}\end{pmatrix}\quad C_{3}=\begin{pmatrix}1&0&0\\\ 0&1&1\\\
0&0&-1\end{pmatrix}$

In $G_{26}$ choice of different exemplars nor the inverses for any type
produce new orbits. The middle convolution data for this group can be found in
Table 30. Further, all three orbits from Type A are distinct, as are all three
orbits from Type B. When comparing with Tykhyy’s classification, there was
only one orbit (from Type B) that we could not find an exact match for. We
listed the closest match, which seemed to be Orbit 5 in [27] and distinguish
the differing coordinate with an asterisk.

In this group we also find nice 4-tuples, details provided in Section 5.2.3

$\lambda$ | $\omega_{X,Y,Z,4}$ | Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their O. Size |  Our O. Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{26}$ Type A Comparisons
$z^{3}$ | $(0,\sqrt{3},0,0)$ | Type II, $X^{\prime}=0,X^{\prime\prime}=\sqrt{3}$ | $(\frac{5}{12},\frac{1}{4},\frac{5}{12},\frac{3}{4})$ | $(\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{1}{2})$ | 4 | 2 | 24 | 0
$-z^{3}$ | $(0,-\sqrt{3},0,0)$ | Type II, $X^{\prime}=0,X^{\prime\prime}=-\sqrt{3}$ | $(\frac{1}{4},\frac{1}{12},\frac{1}{4},\frac{11}{12})$ | $(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{5}{6})$ | 4 | 2 | 24 | 0
$-z^{6}+1$ | $(0,-z^{9}+2z^{3},0,0)$ | Type II, $X^{\prime}=0,X^{\prime\prime}=-z^{9}+2z^{3}$ | $(\frac{1}{6},\frac{1}{2},\frac{2}{3},\frac{1}{2})$ | $(\frac{1}{2},\frac{1}{2},,\frac{5}{6},\frac{1}{2})$ | 3 | 2 | 24 |  $\langle 24,4\rangle$
$G_{26}$ Type B Comparisons
$z^{10}$ | $(-z^{9}+2z^{3},-z^{9}+2z^{3},2,-1)$ | Type III, $\omega=-z^{9}+2z^{3}$ | $(\frac{7}{9},\frac{7}{18},\frac{7}{18},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{5}{6}^{*})$ | $5^{*}$ | 3 | 36 | 0
$-z^{4}$ | $(z^{9}-2z^{3},z^{9}-2z^{3},2,-1)$ | Type III, $\omega=z^{9}-2z^{3}$ | $(\frac{1}{9},\frac{1}{18},\frac{1}{18},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{1}{6})$ | 5 | 3 | 36 | 0
$-z^{16}$ | $(-z^{9}+2z^{3},-z^{9}+2z^{3},2,-1)$ | Type III, $\omega=-z^{9}+2z^{3}$ | $(\frac{5}{9},\frac{5}{18},\frac{5}{18},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{1}{6})$ | 5 | 3 | 36 | 0
$G_{26}$ Type C Comparisons
$-z^{6}+1$ | $(0,0,2,-1)$ | Type III, $\omega=0$ | $(1,\frac{1}{2},\frac{1}{2},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{1}{2})$ | 5 | 3 | 12 |  $\langle 12,1\rangle$
Table 8: Here $w=\zeta_{3},z=\zeta_{36}$. The correct signature for Tykhyy’s
orbit $5$ would have $3(\frac{7}{18})\equiv\frac{1}{6}$ as $\sigma_{24}$ but
such a signature was not in the orbit so we provide the closest match. The
subgroups $\langle 24,4\rangle$ and $\langle 12,1\rangle$ are the dicyclic
groups of order 24 and 12 respectively.

####  5.1.5 $G_{27}$

This group is of order 2160 and is also denoted $W(J_{3}(5))$. This group only
has one conjugacy class of reflections with nontrivial eigenvalue -1 and
$\deg(G_{27})=\\{6,12,30\\}$. The original base ring of this group in Magma is
the cyclotomic field of order 15, and we expanded the base ring to the
cyclotomic filed of order 60 to ensure all the eigenvalues are found. We found
three distinct inverse pair types in this group.

$G_{27}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1-1)$ |  $\\{\frac{2}{60},\frac{38}{60},\frac{50}{60}\\}$
B | $(-1,-1-1)$ |  $\left\\{\frac{5}{60},\frac{35}{60},\frac{50}{60}\right\\}$
C | $(-1,-1,-1)$ |  $\\{\frac{10}{60},\frac{34}{60},\frac{46}{60}\\}$
Table 9:

Writing $z=\zeta_{60}$ and setting $u=-z^{10}-z^{8}+z^{2}$ and
$v=z^{16}+z^{4}$, we get that $z^{50}v=-z^{14}+z^{6}+z^{4}$ so we can write
the exemplar for Type A as:

$\displaystyle A_{1}=\begin{pmatrix}z^{50}v&0&z^{20}\\\ -v+1&1&v+1\\\
v&0&-z^{50}v\end{pmatrix}\quad A_{2}=\begin{pmatrix}0&0&-z^{20}\\\
u+z^{10}+2&1&v+z^{10}\\\ z^{10}&0&0\end{pmatrix}\quad
A_{3}=\begin{pmatrix}1&1&0\\\ 0&-1&0\\\ 0&1&1\end{pmatrix}$

Using the same notation, we can write the Type B exemplar as:

$\displaystyle B_{1}=\begin{pmatrix}-1&u&u\\\ v+1&-u&-u-1\\\
-v-1&u+1&u+2\end{pmatrix}\quad B_{2}=\begin{pmatrix}z^{50}v&0&z^{20}\\\
-v+1&1&v+1\\\ v&0&-z^{50}v\end{pmatrix}\quad B_{3}=\begin{pmatrix}1&1&0\\\
0&-1&0\\\ 0&1&1\end{pmatrix}$

Similarly, the Type C exemplar is:

$\displaystyle C_{1}=\begin{pmatrix}z^{50}v&0&z^{20}\\\ -v+1&1&v+1\\\
v&0&-z^{50}v\end{pmatrix}\quad C_{2}=\begin{pmatrix}1&1&0\\\ 0&-1&0\\\
0&1&1\end{pmatrix}\quad C_{3}=\begin{pmatrix}1&0&0\\\ -v-1&u+1&u+2\\\
v+1&-u&-u-1\end{pmatrix}$

In $G_{27}$ for all three types, different exemplars do not produce new
orbits. Moreover the inverse types do not produce new orbits and we have
checked that these 9 orbits are distinct. The middle convolution details for
Type A, B and C are listed in Table 33, 33, and 33 respectively. The
comparison data is provided in Table 10. In this group we also find nice
4-tuples, details provided in Section 5.2.4

$\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{27}$ Type A Comparisons
$z^{2}$ | $(\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\sqrt{5}-1)$ | 26 | $(\frac{8}{15},\frac{7}{15},\frac{4}{5},\frac{7}{15})$ | $(\frac{2}{5},\frac{2}{5},\frac{1}{2},\frac{1}{2})$ | 78 | 15 | 60 | 0
$-z^{8}$ | $(\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\sqrt{5}-1)$ | 26 | $(\frac{13}{15},\frac{1}{5},\frac{2}{15},\frac{2}{15})$ | $(\frac{3}{5},\frac{2}{5},\frac{1}{2},\frac{1}{2})$ | 87 | 15 | 60 | 0
$-z^{20}$ | $(\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\frac{3-\sqrt{5}}{2},\sqrt{5}-1)$ | 26 | $(\frac{1}{3},\frac{1}{3},\frac{3}{5},\frac{1}{3})$ | $(\frac{3}{5},\frac{3}{5},\frac{1}{2},\frac{1}{2})$ | 81 | 15 | 60 |  $\langle 120,5\rangle$
$G_{27}$ Type B Comparisons
$z^{5}$ | (1, 1, 1, 1) | 39 | $(\frac{7}{12},\frac{5}{12},\frac{7}{12},\frac{1}{4})$ | $(\frac{2}{5},\frac{1}{2},\frac{1}{3},\frac{1}{2})$ | 113 | 24 | 96 | 0
$-z^{5}$ | (1, 1, 1, 1) | 39 | $(\frac{3}{4},\frac{1}{12},\frac{1}{12},\frac{1}{12})$ | $(\frac{1}{2},\frac{3}{5},\frac{1}{3},\frac{1}{2})$ | 114 | 24 | 96 | 0
$-z^{20}$ | (1, 1, 1, 1) | 39 | $(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{3})$ | $(\frac{1}{2},\frac{2}{5},\frac{1}{2},\frac{1}{3})$ | 112 | 24 | 96 |  $\langle 120,5\rangle$
$G_{27}$ Type C Comparisons
$z^{10}$ | $(\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},-1-\sqrt{5})$ | 27 | $(\frac{4}{5},\frac{2}{3},\frac{1}{3},\frac{1}{3})$ | $(\frac{1}{5},\frac{4}{5},\frac{1}{2},\frac{1}{2})$ | 84 | 15 | 60 |  $\langle 120,5\rangle$
$-z^{4}$ | $(\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},-1-\sqrt{5})$ | 27 | $(\frac{14}{15},\frac{14}{15},\frac{3}{5},\frac{1}{15})$ | $(\frac{1}{5},\frac{4}{5},\frac{1}{2},\frac{1}{2})$ | 83 | 15 | 60 | 0
$-z^{16}$ | $(\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},\frac{3+\sqrt{5}}{2},-1-\sqrt{5})$ | 27 | $(\frac{4}{15},\frac{4}{15},\frac{2}{5},\frac{4}{15})$ | $(\frac{1}{5},\frac{1}{5},\frac{1}{2},\frac{1}{2})$ | 82 | 15 | 60 | 0
Table 10: Here $z=\zeta_{60}$. All the finite subgroups corresponds to
$SL(2,5)$

###  5.2 Nice 4-Tuples in Rank 3 Groups

Nice tuples in this group are 4-tuples of reflections generating the whole
group such that the inverse product has a nontrivial eigenvalye of
multiplicity 2. If $\mathbb{Q}(\zeta)$ denotes the trace field for some FCRG
$W$, and we have a matrix $A$ with eigenvalues
$\\{\lambda,\lambda,\lambda_{1}\\}$, then we know
$\tr(A)=2\lambda+\lambda_{1}\in\mathbb{Q}(\zeta)$. If we further knew that
$\lambda\in\mathbb{Q}(\zeta)$ then we can conclude that
$\lambda_{1}\in\mathbb{Q}(\zeta)$ and so we do not need to extend the base
field to ensure all characteristic polynomials of such matrices $A$ split. By
Theorem 2.2.2 we know that any such $\lambda$ of multiplicity 2 must have
order dividing at least two of the degrees of $W$. Studying degrees, we see
that in $G_{23}$ and $G_{24}$, $\lambda$ must have order dividing 2, i.e.
$\lambda=-1$ and so we do not need to expand the base ring. Similarly, for
$G_{25},G_{26},G_{27}$ such $\lambda$ must have order dividing 6, which means
we do not have to expand the base ring in these cases either. Our search found
nice 4-tuples in all the primitive rank 3 groups except for $G_{24}$.

In comparing with the existing classifications, we can only compare with
Tykhyy’s $\mathcal{M}^{(5)}$ orbits [27] since our 4-tuples gives us local
systems on the 5-punctured sphere. Unfortunately, without the $\omega$
parameters from the work of Lisovy-Tykhyy [19] like in the 3-tuple cases, it
is more difficult to relate the our orbit sizes to Tykhyy’s. We still computed
all the signatures for the different possible equivalent representatives for
each orbit and attempted to find a match, which was successful.

####  5.2.1 $G_{23}$

We only found one type of nice tuple.

$G_{23}$ 4-Tuple Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1,-1)$ |  $\\{\frac{1}{2},\frac{1}{2},0\\}$
Table 11:

Letting $q=\zeta_{5}$, here is an exemplar for Type A:

$\displaystyle A_{1}=\begin{pmatrix}q^{3}+q^{2}&q^{3}+q^{2}&q^{3}+q^{2}\\\
-q^{3}-q^{2}+1&-q^{3}-q^{2}+1&-q^{3}-q^{2}\\\
q^{3}+q^{2}&-1&0\end{pmatrix}\quad$ $\displaystyle
A_{2}=\begin{pmatrix}-q^{3}-q^{2}+1&-q^{3}-q^{2}+1&-q^{3}-q^{2}\\\
q^{3}+q^{2}&q^{3}+q^{2}&q^{3}+q^{2}\\\ -1&q^{3}+q^{2}&0\end{pmatrix}$
$\displaystyle A_{3}=\begin{pmatrix}-q^{3}-q^{2}&1&0\\\
q^{3}+q^{2}&q^{3}+q^{2}&0\\\ -q^{3}-q^{2}&-q^{3}-q^{2}+1&1\end{pmatrix}\quad$
$\displaystyle A_{4}=\begin{pmatrix}-1&0&0\\\ -q^{3}-q^{2}&1&0\\\
0&0&1\end{pmatrix}$

The middle convolution details can be found in Table B. This type is its own
inverse. For this tuple we could not compute the orbit size as it was too
large, but we were able to find a signature in the orbit (up to Tykhyy’s
equivalence relations) that agreed with one listed in Tykhyy’s paper. Because
we cannot compute the full braid group orbit, we also cannot check if
different exemplars would result in a different orbit after middle
convolution. We provide the comparison details in Table 12.

Type | $\lambda$ | $\theta_{1,2,3,4,5}$ | $\sigma_{12,23,34,45,51,13,24}$ |  Tykhyy $\\#$ |  T Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---
$G_{23}$ Type Comparisons
A | $-1$ | $(1,1,1,1,1)$ | $(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{5},\frac{1}{5})$ | 8 | 192 | - |  0
Table 12:

####  5.2.2 $G_{25}$

We found 6 different types which split into 3 inverse pairs, so we only
provide the details for one type from each pair in Table 13.

$G_{25}$ 4-Tuple Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(z,z^{2},z^{2},z^{2})$ |  $\\{\frac{1}{3},\frac{1}{3},0\\}$
B | $(z^{2},z^{2},z^{2},z^{2})$ |  $\left\\{\frac{1}{3},\frac{1}{3},\frac{2}{3}\right\\}$
C | $(z^{2},z,z,z^{2})$ |  $\\{\frac{2}{3},\frac{1}{6},\frac{1}{6}\\}$
Table 13:  Here $z=\zeta_{3}$

For all the exemplars listed below, $z=\zeta_{3}$.
Here is an exemplar for Type A:

$\displaystyle A_{1}=\begin{pmatrix}z&0&0\\\ z^{2}&1&0\\\
0&0&1\end{pmatrix}\quad A_{2}=\begin{pmatrix}-z&1&-z\\\ 0&1&0\\\
z&z^{2}&0\end{pmatrix}\quad A_{3}=\begin{pmatrix}1&z&0\\\ 0&z^{2}&0\\\
0&z&1\end{pmatrix}\quad A_{4}=\begin{pmatrix}-z&1&z^{2}\\\ 1&z+1&1\\\
z^{2}&1&-z\end{pmatrix}$

Here is an exemplar for Type B:

$\displaystyle B_{1}=\begin{pmatrix}-z&1&0\\\ z+1&0&0\\\
1&z&1\end{pmatrix}\quad B_{2}=\begin{pmatrix}-z&1&-z\\\ 0&1&0\\\
z&z^{2}&0\end{pmatrix}\quad B_{3}=\begin{pmatrix}0&z^{2}&z\\\ 0&1&0\\\
-z&1&-z\end{pmatrix}\quad B_{4}=\begin{pmatrix}1&z&0\\\ 0&z^{2}&0\\\
0&z&1\end{pmatrix}$

Here is an exemplar for Type C:

$\displaystyle C_{1}=\begin{pmatrix}1&z&1\\\ 0&0&z+1\\\
0&1&-z\end{pmatrix}\quad C_{2}=\begin{pmatrix}z&0&0\\\ z^{2}&1&0\\\
0&0&1\end{pmatrix}\quad C_{3}=\begin{pmatrix}z+1&-1&z+1\\\ 0&1&0\\\
z^{2}&-z&0\end{pmatrix}\quad C_{4}=\begin{pmatrix}-z&1&0\\\ z+1&0&0\\\
1&z&1\end{pmatrix}$

The middle convolution details for $G_{25}$ can be found in Table B. The
orbits for Type A and Type B are the same. All three types do not split into
new orbits and their inverses do not produce new orbits. The comparison
details for the two distinct orbits can be found in Table 14.

Type | $\lambda$ | $\theta_{1,2,3,4,5}$ | $\sigma_{12,23,34,45,51,13,24}$ |  Tykhyy $\\#$ |  T Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---
$G_{25}$ Type Comparisons
A | $z$ | $(0,0,0,0,\frac{2}{3})$ | $(0,0,\frac{1}{3},0,\frac{1}{2},0,\frac{1}{3})$ | 3 | 9 | 45 | 0
C | $z+1$ | $(\frac{1}{2},\frac{1}{6},\frac{2}{3},\frac{1}{6},\frac{1}{2})$ | $(\frac{1}{2},\frac{5}{6},\frac{5}{6},\frac{1}{2},1,\frac{1}{2},\frac{1}{3})$ | 1 | 4 | 120 |  $\langle 24,4\rangle$
Table 14: Here $z=\zeta_{3}$. The subgroup of order 24 is the dicyclic group.

####  5.2.3 $G_{26}$

In this group we found two inverse pairs of nice tuples and we provide the
details for one type from each pair in Table 15.

$G_{26}$ 4-Tuples Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,z,z^{2},z^{2})$ |  $\\{\frac{1}{6},\frac{1}{6},\frac{1}{2}\\}$
B | $(z,z^{2},z,-1)$ |  $\\{\frac{1}{6},\frac{1}{6},\frac{5}{6}\\}$
Table 15:  Here $z=\zeta_{3}$

With the notation $z=\zeta_{3}$, here is an examplar for Type A:

$\displaystyle A_{1}=\begin{pmatrix}z&z+2&1\\\ 0&1&0\\\
z+2&z^{2}-z&-z\end{pmatrix}\quad A_{2}=\begin{pmatrix}z&0&0\\\ z^{2}&1&0\\\
0&0&1\end{pmatrix}\quad A_{3}=\begin{pmatrix}-z+1&2z^{2}&z^{2}\\\
z^{2}&-1&-1\\\ 0&0&1\end{pmatrix}\quad A_{4}=\begin{pmatrix}-z&-1&-1\\\
z&z+2&z+1\\\ z^{2}-z&z^{2}-1&z^{2}\end{pmatrix}$

The exemplar for Type B:

$\displaystyle B_{1}=\begin{pmatrix}1&z^{2}&0\\\ 0&z&0\\\
0&-z+1&1\end{pmatrix}\quad B_{2}=\begin{pmatrix}-z&-1&0\\\ z^{2}&0&0\\\
2z+1&z+2&1\end{pmatrix}\quad B_{3}=\begin{pmatrix}z&0&0\\\ z+1&1&0\\\
0&0&1\end{pmatrix}\quad B_{4}=\begin{pmatrix}1&z^{2}-z&-z\\\ 0&z^{2}&z^{2}\\\
0&2z+1&z+1\end{pmatrix}$

The middle convolution details for $G_{26}$ can be found in Table 38. Neither
of these types split into multiple orbits nor do their inverse types produce
new orbits. The finite group generated by the matrices in $SL_{2}(\mathbb{C})$
are the same for both groups, Magma identifies this as $SmallGroup<24,4>$
which corresponds to the dicyclic group of order 24.

Type | $\lambda$ | $\theta_{1,2,3,4,5}$ | $\sigma_{12,23,34,45,51,13,24}$ |  Tykhyy $\\#$ |  T Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---
$G_{26}$ Type Comparisons
A | $z+1$ | $(\frac{1}{2},\frac{1}{6},\frac{1}{6},\frac{1}{3},\frac{1}{2})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{1}{2},\frac{1}{2})$ | 1 | 4 | 120 |  $\langle 24,4\rangle$
B | $z+1$ | $(\frac{1}{2},\frac{1}{6},\frac{2}{3},\frac{1}{6},\frac{1}{2})$ | $(\frac{1}{2},\frac{5}{6},\frac{5}{6},\frac{1}{2},1,\frac{1}{2},\frac{1}{3})$ | 1 | 4 | 240 |  $\langle 24,4\rangle$
Table 16: Here $z=\zeta_{3}$. The subgroup of order 24 is the dicyclic group.

####  5.2.4 $G_{27}$

In this group we found one inverse pair, and we provide the details for one
type of the pair.

$G_{27}$ 4-Tuples Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1,-1)$ |  $\\{\frac{5}{6},\frac{5}{6},\frac{1}{3}\\}$
Table 17:  Here $z=\zeta_{15}$

Here is an exemplar of Type A, where $z=\zeta_{15}$, $u=z^{4}+z$,
$v=z^{7}-z^{3}+z^{2}-1$:

$\displaystyle A_{1}$ $\displaystyle=\begin{pmatrix}u&u-1&-1\\\ -u-1&u&-v-u\\\
0&0&1\end{pmatrix}\quad A_{2}=\begin{pmatrix}-1&0&0\\\ 1&1&0\\\
-u&0&1\end{pmatrix}\quad
A_{3}=\begin{pmatrix}u+2v&v+z^{10}&-u+z^{5}\left(z^{5}-1\right)\\\ 0&1&0\\\
-u-1&-v-u-1&-2v-u\end{pmatrix}$ $\displaystyle A_{4}$
$\displaystyle=\begin{pmatrix}-v-u-z^{5}&-v-z^{5}&u+1\\\
v+u+z^{5}&v&-u+z^{10}\\\ -v-u+1&-v-z^{10}&u-z^{10}\end{pmatrix}$

The middle convolution details can be found in Table 38. In this case we could
not compute the orbit size since it was too long. Similar to $G_{23}$, we were
able to compute some signatures for tuples in the orbit and we found a match
with a signature listed in Tykhyy’s classification.

Type | $\lambda$ | $\theta_{1,2,3,4,5}$ | $\sigma_{12,23,34,45,51,13,24}$ |  Tykhyy $\\#$ |  T Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---
$G_{27}$ Type Comparisons
A | $-z^{5}$ | $(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3})$ | $(0,\frac{1}{5},\frac{1}{3},\frac{1}{3},\frac{1}{5},\frac{3}{5},\frac{1}{3})$ | 87 | 432 | - |  $\langle 120,5\rangle$
Table 18: Here $z=\zeta_{15}$. The subgroup of order 120 is $SL(2,5)$.

###  5.3 Nice 4-Tuples in Rank 4 Groups

In this case we are looking for nice tuples of 4 reflections generating the
whole group such that their inverse product has a nontrivial eigenvalue of
$T-2=2$. Of the five rank 4 reflection groups, we found nice 4-tuples in
$G_{28}$, $G_{30}$ and $G_{32}$. In the case of $G_{32}$, the field containing
all the eigenvalues would be $\mathbb{Q}(\zeta_{360})$ which is much too large
to perform the computations over. For $G_{32}$, in Section 5.3.3 we justify
why it is enough to search for such tuples over $\mathbb{Q}(\zeta_{12})$.

After taking the middle convolution, multiplying with a suitable character and
adjoining the inverse product, our tuples become $SL_{2}(\mathbb{C})$ local
systems on the 5-punctured sphere. We found that in some cases, one of the 5
matrices in the local system ends up being the identity, which means our tuple
comes from some local system on the 4-punctured sphere. In the case where we
have 4 nontrivial matrices, we can compute the $\omega$ parameters in the
notation of Lisovy-Tykhyy [19] and identify the corresponding
$\mathcal{M}^{(4)}$ orbit in Tykhyy’s paper [27]. When all 5 of the matrices
are nontrivial, we can only compare with Tykhyy’s $\mathcal{M}^{(5)}$ orbit
signatures from [27].

####  5.3.1 $G_{28}$

The group $G_{28}$ of order 1152 is also denoted $W(F_{4})$ with degrees
$\deg(G_{28})=\\{2,6,8,12\\}$ [18, Table D.3]. All reflections in $G_{28}$
have nontrivial eigenvalue -1, and form two conjugacy classes of reflections.
We only found one type of nice 4-tuples in this group, with two possible
choices of $\lambda$ parameters for middle convolution.

$G_{28}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1,-1)$ |  $\\{\frac{1}{6},\frac{1}{6},\frac{5}{6},\frac{5}{6}\\}$
Table 19:

An exemplar for such a tuple of Type A is:

$\displaystyle A_{1}=\begin{pmatrix}2&3&4&2\\\ -1&-2&-4&-2\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}\quad A_{2}=\begin{pmatrix}1&2&2&0\\\ 0&-1&-2&0\\\
0&0&1&0\\\ 0&1&1&1\end{pmatrix}\quad A_{3}=\begin{pmatrix}-1&-2&-4&-2\\\
2&3&4&2\\\ -1&-1&-1&-1\\\ 0&0&0&1\end{pmatrix}\quad
A_{4}=\begin{pmatrix}-1&0&0&0\\\ 1&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$

A different choice of exemplar does not produce a new orbit and the two orbits
computed above are the same, so we only get one new orbit from this group. The
middle convolution details can be found in Table 38. After putting the
matrices in $SL_{2}(\mathbb{C})$, we see that the inverse product of the
middle convolution of this tuple is the identity, so this tuple comes from a
local system on the 4-punctured sphere. As in the case of 3-tuples, we provide
the details comparing with the results of both Lisovy-Tykhyy and Tykhyy ([19],
[27]) in Table 20. The $\omega$ parameters appear to correspond to Orbit 31,
but in fact this orbit is of Type III, as described in Lemma 39 in [19].

$\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---
$G_{28}$ Type A Comparisons
$z^{4}$ | $(2,2,2,-1)$ | Type III, $\omega=2$ | $(\frac{2}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3})$ | $(\frac{1}{2},\frac{1}{3},\frac{1}{2},1)$ | 5 | 3 | 9 |  $\langle 24,3\rangle$
Table 20: Here $z=\zeta_{24}$ and the subgroup of order 24 is $SL(2,3)$.

####  5.3.2 $G_{30}$

The group $G_{30}$ of order 14400 is also denoted $W(H_{4})$ with degrees
$\deg(G_{30})=\\{2,12,20,30\\}$ [18, Table D.3]. All reflections in $G_{30}$
have nontrivial eigenvalue -1 and are conjugate to each other. Looking at the
degrees, we extend the base field to $\mathbb{Q}(\zeta_{60})$. We found three
distinct types of nice 4-tuples, the details are in Table 21.

$G_{30}$ Types
---
Type | Tuple Eigenvalues | Residues of Inverse Product
A | $(-1,-1,-1,-1)$ |  $\\{\frac{3}{10},\frac{3}{10},\frac{7}{10},\frac{7}{10}\\}$
B | $(-1,-1,-1,-1)$ |  $\left\\{\frac{1}{6},\frac{1}{6},\frac{5}{6},\frac{5}{6}\right\\}$
C | $(-1,-1,-1,-1)$ |  $\\{\frac{1}{10},\frac{1}{10},\frac{9}{10},\frac{9}{10}\\}$
Table 21: Here $z=\zeta_{60}$

For all of the exemplars below $z=\zeta_{60}$ and $u=z^{14}-z^{6}-z^{4}$. An
exemplar of type $A$ is given by:

$\displaystyle A_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&1\\\
0&0&0&-1\end{pmatrix}\quad A_{2}=\begin{pmatrix}1&0&0&0\\\ 2u-1&2u-1&u-1&-1\\\
-2u+1&-2u+2&-u+2&1\\\ -u+1&-2u&-u&-u\end{pmatrix}$ $\displaystyle
A_{3}=\begin{pmatrix}u&u-1&u-1&-1\\\ -2u+1&-2u+2&-2u+1&-u\\\ 2u-1&2u-1&2u&u\\\
-u&-u&-u&-u\end{pmatrix}\quad A_{4}=\begin{pmatrix}-u+1&-u+1&-u+1&-u\\\
0&1&0&0\\\ u&u-1&u&u\\\ -1&u&u&0\end{pmatrix}$

An exemplar of type B is given by:

$\displaystyle B_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&1\\\
0&0&0&-1\end{pmatrix}\quad B_{2}=\begin{pmatrix}0&-1&-1&-1\\\ -u&-u+1&-u&-u\\\
0&0&1&0\\\ u-1&u-1&u-1&u\end{pmatrix}$ $\displaystyle
B_{3}=\begin{pmatrix}1&-u&-u&-u\\\ 0&0&-1&-1\\\ 0&0&1&0\\\
0&-1&-1&0\end{pmatrix}\quad B_{4}=\begin{pmatrix}-2u+2&-3u+2&-2u+1&-u+1\\\
2u-1&3u-1&2u-1&u-1\\\ -u&-u+1&-u+1&1\\\ u&u-1&u&0\end{pmatrix}$

An exemplar of type C is given by:

$\displaystyle C_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&1\\\
0&0&0&-1\end{pmatrix}\quad C_{2}=\begin{pmatrix}1&0&0&0\\\
-u+1&-2u+1&-u+1&-u\\\ u-1&2u&u&u\\\ 2u-1&2u-2&2u-1&u\end{pmatrix}$
$\displaystyle C_{3}=\begin{pmatrix}u-1&2u-1&2u&u\\\ -3u+1&-3u+3&-2u+2&-u+1\\\
3u-1&3u-2&2u-1&u-1\\\ 0&0&0&1\end{pmatrix}\quad
C_{4}=\begin{pmatrix}0&-1&-1&-1\\\ -u&-u+1&-u&-u\\\ 0&0&1&0\\\
u-1&u-1&u-1&u\end{pmatrix}$

The middle convolution details for each type is presented in Table 40. For all
three types, different exemplars do not produce new orbits and since each type
is their own inverse type, these are all the orbits from this group. Further,
the two orbits produced from Type A are distinct, as are the two from Type C.
The two orbits from Type B are actually the same. In total $G_{30}$ gives us 5
different orbits.

For all 5 orbits, once we put the matrices in $SL_{2}(\mathbb{C})$, we see
that the last matrix is the identity, hence these actually corresponds to
orbits of 4 matrices multiplying to the identity. We compared these to the
existing classification(s) by Lisovy-Tykhyy and Tykhyy ([19], [27]) as we did
for the rank 3 complex reflection groups. Within each type, the two orbits are
equivalent to the same orbit of $\mathcal{M}^{(4)}$ in Tykhyy’s
classification, so we only provide the details once in Table 22.

Type | $\lambda$ | $\omega_{X,Y,Z,4}$ |  Lisovy-Tykhyy $\\#$ | $\theta_{1,2,3,4}$ | $\sigma_{12,23,13,24}$ |  Tykhyy $\\#$ |  Their Orbit Size |  Our Orbit Size |  $SL_{2}(\mathbb{C})$ Subgroup
---|---|---|---|---|---|---|---|---|---
$G_{30}$ Type Comparisons