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\begin{document}
\title[]{Macdonald Representations
of Complex Reflection Groups}%
\author{Alun O Morris}%
\author{Patrick Mwamba}%
\address{Institute of Mathematical and Physical Sciences\\
University of Wales\\ Aberystwyth, Ceredigion SY23 3BZ, Wales, U.K.}%
\address{Department of Mathematics\\
University of Zambia\\
 Lusaka, Zambia}%
\email{alun@morus25.fsnet.co.uk}%

\thanks{}%
\subjclass{}%
\keywords{}%

%\date{}%
\dedicatory{To Alain Lascoux to celebrate his sixtieth birthday and his massive
contribution to algebraic combinatorics and representation theory}%
%\commby{}%
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\begin{abstract}

I G Macdonald (1972) introduced a unified approach to give many irreducible representations
of Weyl groups in terms of their root systems. This generalised to Weyl groups the earlier
well known constructions based on Young tableaux due to W Specht. These were interpreted
in terms of positive systems of subsystems of root systems.
A M Cohen (1976) extended the idea of root systems to complex reflection groups
giving explicitly  root systems for all dimensions greater than two.
M C Hughes (1980) had further extended his ideas to generalise the concepts of subsystems
and positive systems. These are now used to construct some irreducible representations
of complex reflection groups.

\end{abstract}

\maketitle

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\font\rms=cmr8
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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 52 \rms (2004), Article~B52g\hfill}
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\section{\bf Introduction.}

In a now classic paper, I G Macdonald \cite{mcd} gave a unified
construction of irreducible representations of Weyl groups in
terms of subsystems of their root systems. He further remarked
that the same construction could be extended to any finite Coxeter
group and its reflection subgroups. Later, G Lusztig and N
Spaltenstein \cite{lusspa} generalized his construction in a way
which means that it also can be extended to complex reflection
groups. In this paper, the main aim will be to extend Macdonald's
original construction to complex reflection groups in a more
direct way which will use the ideas on root systems defined for
these groups as explained below.

G C Shephard and J A Todd \cite{Sh&Todd} have given a complete
classification of the irreducible complex reflection groups, they
showed that they are either the infinite families, the cyclic
groups, the symmetric groups and the groups $G(m,p,n)$, where
$m,p,n$ are positive integers such that $p$ divides $m$, $m \geq
2$ and $p=1$ if $n=1$ and $34$ exceptional cases which we denote
$ST_{i}, \; 4 \leq i \leq 37$.

In the case of Weyl groups or finite Coxeter groups the theory is
well developed and documented, see for example, \cite{bour}. For
complex reflection groups however, some of the basic ideas are not
as well developed with no universally accepted analogues for such
fundamental concepts as root systems and their subsystems or a
length function; for more recent attempts for some of the
classical groups, see \cite{brma1}, \cite{brma2}, \cite{rs} and
\cite{s}. In addition, the most significant work on complex
reflection groups in recent years has originated  in the work of
M. Brou\'{e}, G. Malle et al, see for example, \cite{bm},
\cite{bmr}, which showed how important these groups are in a more
general context.


However, for our purpose, the concept of root systems introduced
by A M Cohen \cite{cohen} in his reclassification of the finite
irreducible complex reflection groups is far more useful. These
root systems have been developed further by M C Hughes
\cite{hughes1}, \cite{hughes2} and H Can \cite{can}, in
particular, their subsystems and positive systems, called primary
systems in this context.

Thus, in Section 2 of this paper we introduce all of the
preliminary material required on complex reflection groups and
their root systems mainly following \cite{cohen}. In Section 3, we
recall the ideas of M C Hughes and H Can on subsystems and primary
systems. In particular, in order to obtain the subsystems of these
root systems they have developed algorithms which show how to
construct extended Cohen-Dynkin diagrams (c.f. extended Dynkin
diagrams in the real case) in this context. In Tables 1 and 2 we
present in a systematic way the extended Cohen-Dynkin diagrams for
all the exceptional groups. In the final section, Section 4,
Macdonald modules are presented for complex reflection groups.
These are presented not in terms of subsystems but in terms of
reflection subgroups. In particular cases, it is seen that most of
the irreducible representations can be obtained in terms of
subsystems. However, as not all of the reflection subgroups appear
via subsystems, the results are presented in this more general
context. An additional reason is that some of the components
required in the proof have already appeared in the work of R
Stanley \cite{st} on determining the relative invariants of
complex reflection groups. Absolutely crucial for the proof is
Lemma 4.1, which generalizes the result in the real case that if
$a$ is a positive root a reflection $s_{a}$ permutes all the
positive roots except $a$ which is mapped onto its negative. The
section ends with some applications to the $2$-dimensional
reflection groups. In the Appendix, further applications are
given, where most of the irreducible representations of three
further groups are given, $ST_{24}, ST_{25}$ and $ST_{26}$ - this
work together with that for the $2$-dimensional groups, is mainly
the work of the second author, Patrick Mwamba, who has sadly died
since completing this work.

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\markboth{\SMALL ALUN O MORRIS AND PATRICK MWAMBA}{\SMALL 
MACDONALD REPRESENTATIONS OF COMPLEX REFLECTION GROUPS}
%
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\section{\bf Preliminaries.}\label{prel}

In this section, the basic definitions and notation required later
are given following the approach in \cite{cohen}, \cite{hughes1}.

Let $V$ be a complex vector space of dimension $n$. A {\it
reflection} in $V$ is a linear transformation of $V$ of finite
order with exactly $(n-1)$ eigenvalues equal to 1. A {\it
reflection group} $G$ in $V$ is a group generated by reflections
in $V$. There exists a unitary inner product $(\; ,\;)$ on $V$
invariant under $G$. A reflection group $G$ is said to be
$r$-dimensional if the dimension of the subspace $V^{G}$ of points
fixed by $G$ is $n-r$. If $W \subset V $ is a subspace, denote by
$G_{W}$ the subgroup of those elements of $G$ which fix the
elements of $W$; $G_{W}$ is itself a reflection group. The group
$G$ is irreducible if the restriction to a $G$-invariant
complement of $V^{G}$ in $V$ is irreducible.

A (unitary) {\it root} of a reflection in $V$ is an eigenvector (of
length 1) corresponding to the unique eigenvalue not equal to 1
of the reflection. A (unitary) {\it root} of $G$ is a (unitary) root
of a reflection in $G$. Let $s$ be a reflection in $V$ of order $m>1$. There exists
$a \in {V}, a\neq 0$ and a primitive $m$-th root of unity $\zeta$
such that $s_{a,m}x = x-(1-\zeta)(x,a)(a,a)^{-1}a$
for all $x \in V$, where $s= s_{a,m}$. If $t$ is any unitary
transformation of $V$, we have
$ts_{a,m}t^{-1}=s_{ta,m}$.
Define ${\theta}_{G}:V\rightarrow {\mathbb {N}}$ by ${\theta}_{G}(a)=
|G_{W}|$, where $W=<a>^{\bot}, a \in V$. The number ${\theta}_{G}(a)$
is called the {\it  order} of $a$ (with respect to $G$).

Each reflection $s_{a,m}$ fixes a unique reflecting hyperplane
$$ H_{s_{a,m}} = \{x \in V \; | \; (x,a) = 0 \} = \{x \in V \; | \; s_{a,m}x=x
\}.$$ Clearly $H_{s_{\zeta^{i}a,m}} = H_{s_{a,m}}^{i} =
H_{s_{a,m}}, \; 0 \leq i \leq m-1.$ For each of the reflecting
hyperplanes $H$, choose a functional $\alpha_{H} \in V^{*}$, the
dual space of $V$, such that ker$(\alpha_{H}) = H$, that is, $H =
\{x \in V \; | \; \alpha_{H}(v) = (x,a) \}$. For convenience, we
sometimes denote a reflection which generates the cyclic group
$G_{H}$ of order $e_{H}$ by $s_{H}$.  The collection of all
reflecting hyperplanes for the group $G$ is an arrangement
$\mathcal{A}$ on which the group $G$ acts in a natural way. Let
$\mathcal{C}$ denote the set of $G$-orbits under this action, thus
$\mathcal{A} = \cup_{C \in \mathcal{C}}C$. The order $e_{H}$
depends on the orbit $C = G.H \in \mathcal{C}$; thus whenever it
is convenient we write $e_{C}$ in place of $e_{H}$ and $\zeta_{H}
= \zeta_{C}$ for the corresponding primitive $e_{C}$-th root of
unity.

Just as for real reflection groups, root systems can be equally
useful in the complex case. These are now presented following
\cite{cohen}.
 (i) A {\it vector graph}
is a pair $(B,\theta)$, where $B$ is a non-empty finite subset of
${\mathbb {C}}^{\infty}$, the vector space with standard unitary
inner product $(\,,\,)$, such that for all $a,b \in B, |(a,b)|=1$
if and only if $a=b$ and ${\theta}$ is a map from $B$ to ${\mathbb
{N}}\setminus\{1\}$. We say that $B$ is the set of {\it vertices}
and $\theta(a)$, for $a \in B$, is the {\it order} of $a$. A
vector graph $(B,w)$ is represented by a directed value graph by
assigning to each element $a \in B$ a node $a$ with weight
$\theta(a)$ and if $(a,b) \neq 0,1$ a directed edge from $a$ to
$b$ with weight $(a,b)$. For example, if $B=\{a,b\}$,
$\theta(a)=m$ and $\theta(b)=p$ and $(a,b)=\alpha$, then the
vector graph is
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\begin{tiny}
\put(2,5){$m$} \put(20,5){$p$} \put(2,10){$a$} \put(21,10){$b$}
\put(12,2){$\alpha$}
\end{tiny}
\end{picture}
\end{center}
We adopt the following conventions: if $m=2$ the number $2$ is
omitted, if $\alpha \in {\mathbb {R}}$ the arrow is omitted and if
$p=m$ and $\alpha=-1/2$, the value $-1/2$ is omitted.

(ii) Let $\pi =(B,\theta)$ be a vector graph. Then, we define dim
$\pi$ to be the dimension of the vector space spanned by $B$, and
$W(\pi)$ to be the group generated by all the reflections
$s_{a,\theta(a)}$ for $a \in B$. The vector graph $\pi$ is called
a {\it root graph} if dim $\pi=|B|$ and $W(\pi)$ is a finite
reflection group. Root graphs $\pi = (B, \theta)$ and
$\pi^{\prime} = (B^{\prime},\theta^{\prime})$ are {\it equivalent}
if the groups $W(\pi)$ and $W(\pi^{\prime})$ are conjugate.

(iii) For any root graph $\pi =(B,\theta)$ and for any $w \in W(\pi)$, let
$w\pi = (B_{w},\theta_{w})$, where $B_{w}=wB$ and $\theta_{w}(w(a))=\theta(a)$
with $a \in B$, then $w\pi$ is also a root graph which is equivalent to $\pi$ since
$s_{w(a),\theta_{w}(w(a))} = ws_{a,\theta(a)}w^{-1}$ for all $a \in B$ it follows
that $W(w\pi) = wW(\pi)w^{-1} = W(\pi)$.

(iv) We say that $\pi$ is {\it irreducible} if $W(\pi)$ is irreducible
(or that $\pi$ is connected). The vector graph $\pi$ is said
to be {\it congruent} to the vector graph ${\pi}^{\prime}=(B^{\prime},{\theta}^{\prime})$
if there is a $t\in {\bf Gl}({\mathbb {C}}^{\infty})$ such that ${\theta}^{\prime}(ta)={\theta}
(a)$ for $a\in B$ and the elements of $B$ are eigenvectors of $t$.


(v) A pair $(R,f)$ is called a {\it pre-root system} if $R$ is a
subset of non-zero elements of ${\mathbb {C}}^{\infty}$ and
$f:R\rightarrow{\mathbb {N}}\setminus\{1\}$ such that for all
${a,b \in R}, s_{a,f(a)}R=R$ and $f(s_{a,f(a)}a)=f(a)$ . To
$\Sigma=(R,f)$ is associated the reflection group $W(\Sigma)$
defined by $W(\Sigma)=$ \newline $<s_{a,f(a)}|a \in R>$.

(vi) A pre-root system $\Sigma$ is called a {\it root system} if
in addition $\alpha a \in R$ if and only if $\alpha a \in
{W(\Sigma)a}$ for all ${a \in R}, {\alpha \in \mathbb {C}}.$

We make the following remarks which have been proved in \cite{cohen}.

\newtheorem{remark}{Remark}[section]
\begin{remark}

Every root graph defines a pre-root system, for if $\pi = (B,\theta)$ is a root graph,
then $\Sigma = (R,f)$, where $R = W(\pi)B, \enspace f:R\rightarrow{\mathbb {N}}\setminus\{1\}$
is induced by the order function $o_{W(\pi)}$ defines a pre-root system with $W(\pi) = W(\Sigma)$.
\end{remark}

\begin{remark}
Every finite reflection group yields a pre-root system and every pre-root system contains
a root system, that is, if $\Sigma = (R,f)$ is a pre-root system, then there is a root system
$\Phi = (S,g)$ with $W(\Phi) = W(\Sigma), S \subset R$ and $g = f|_{S}$.
\end{remark}

\begin{remark}
Every reflection group has a root system, but not every root
system is obtained in the above way from a root graph.
\end{remark}

\begin{remark}
If a root system $\Phi$ is the pre-root system obtained from a
root graph $\pi$ as described in Remark 2.1, then $\pi$ is called
a {\it simple system} in $\Phi$. Can \cite{can} has shown that if
$w \in W$ then $w\pi = (wB,\theta_{w})$, where $\theta_{w}(w(a)) =
\theta(a), a \in B$ is a root graph which is equivalent to $\pi$
which yields the same root system $\Phi$ and so $w\pi$ is another
simple system in $\Phi$. Hence, the number of simple systems in
$\Phi$ is equal to the order of $W(\pi)$. If $\Phi$ is a root
system with simple system $\pi$, then the graph associated to
$\pi$ is called a {\it Cohen-Dynkin diagram} of $\Phi$.
\end{remark}

A group $G$ of unitary automorphisms of $V$ is said to be {\it
imprimitive} if $V$ is a direct sum $V= V_{1} \oplus \cdots \oplus
V_{k}$ of non-trivial proper subspaces $V_{i}(1\leq i\leq k)$ of
$V$ such that $\{V_{i}\;|\; 1 \le i \le k \}$ is invariant under
$G$. If such a direct splitting of $V$ does not exist, $G$ is said
to be {\it primitive}.

Let ${\mathcal S}_{n}$ be the (symmetric) group of all $n\times n$
permutation matrices and let $A(m,p,n)$ be the set of all diagonal
$n\times n$ matrices with $\zeta^{{\rho}_{i}},\rho_{i} {\in
{\mathbb {Z}}}$ in the $(i,i)$ position, where $\zeta$ is a
primitive $m$th root of unity and $\sum_{i=1}^{n}\rho_{i} \equiv 0
\pmod{p}$. Define $G(m,p,n)=A(m,p,n)\rtimes{\mathcal S}_{n}$, then
the imprimitive reflection groups in $V$ are of the form
$G(m,p,n)$, where $p|m$.


\begin{remark}
(i) $G(m,m,2)$ is conjugate to $W(I_{2}(m))$ (notation of \cite{bour}), the dihedral group
of order $2m$.

(ii) $G(1,1,n)=W(A_{n-1})\cong{\mathcal S}_{n}$, the Weyl group of type $A_{n-1}$.

(iii) $G(2,1,n)=W(B_{n})$, the Weyl group of type $B_{n}$.

(iv) $G(2,2,n)=W(D_{n})$, the Weyl group of type $D_{n}$.

(v) If $p\neq 1,m$, then $G(m,p,n)$ can be defined with $n+1$
generating reflections, but for $p=1,m$, then $n$ generating
reflections are sufficient.
\end{remark}

A root system for $G(m,p,n)$ may be defined as follows. Let $\zeta$ be a primitive
$m$-th root of unity and let $\{\epsilon_{1}, \ldots , \epsilon_{n}\}$ be the standard
basis for the complex vector space ${\mathbb C}^{n}$. If $d \in {\mathbb N}$, let
$\mu_{d}$ be the group of $d$-th roots of unity. Let

$$ \Omega_{l}(m,n) = \{\epsilon_{i}-\zeta^{a}\epsilon_{j},1\leq i,j \leq n,
\enspace i \neq j, \enspace a \in {\mathbb Z}/m{\mathbb Z}\}.$$
\noindent
and let

\begin{eqnarray*}
R_{l}(m,m,n) = \left\{ \begin{array}{ll}
                        \mu_{m}\Omega_{l}(m,n) & \mbox{if $m$ is even}, \\
                        \mu_{2m}\Omega_{l}(m,n) & \mbox{if $m$ is odd}.
                         \end{array}
                        \right.
\end{eqnarray*}
\noindent
Let $f_{l}(r) = 2$ for all $r \in R_{l}(m,m,n)$. Then $\Phi_{l}(m,m,n) = (R_{l}(m,m,n),f_{l})$
is a root system for $G(m,m,n)$.
\noindent
Let

$$ \Omega_{s}(n) = \{\epsilon_{i}, \enspace 1 \leq i \leq n \}$$
\noindent
and let $R_{s}(m/p,n) = \mu_{m/p}\Omega_{s}(n)$. Let $f_{s}(r) = m$ for all
$r \in R_{s}(m/p,n)$. Let $\Phi_{s}(m/p,n) = (R_{s}(m/p,n),f_{s})$. Then

$$\Phi(m,p,n) = \Phi_{l}(m,m,n) \bigcup \Phi_{s}(m/p,n)$$
\noindent
is a root system for $G(m,p,n)$ for $p \neq m$.

\noindent
Let

$$ \pi(m,1,n) = \{\alpha_{1}=\epsilon_{1} - \epsilon_{2}, \ldots ,
\alpha_{n-1}=\epsilon_{n-1} - \epsilon_{n},\alpha_{n}=\epsilon_{n}
\} $$ \noindent where, $\theta(\alpha_{i}) =2, \enspace 1 \leq i
\leq n-1, \enspace \theta(\alpha_{n})=m$ and

$$ \pi(m,m,n) = \{\epsilon_{1} - \epsilon_{2}, \ldots ,\epsilon_{n-1} - \epsilon_{n},
\epsilon_{n-1} - \zeta\epsilon_{n} \}$$ \noindent where,
$\theta(\alpha_{i}) =2, \enspace 1 \leq i \leq n$. Then
$\pi(m,1,n)$ and $\pi(m,m,n)$ are vector graphs and so, simple
systems for $G(m,1,n)$ and $G(m,m,n)$ respectively.

\noindent
The corresponding Cohen-Dynkin diagrams for $G(m,1,n)$ and $G(m,m,n)$ are

\vspace{-0.5cm}
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\put(46,4){\line(1,0){12}}
\begin{tiny}
\put(0,8){$\alpha_{n}$} \put(14,8){$\alpha_{2}$}
\put(40,8){$\alpha_{n-1}$} \put(57,8){$\alpha_{n}$}
\put(58,3){$m$}
\end{tiny}
\begin{tiny}
\put(49,-1){$\frac{-1}{\sqrt{2}}$}
\end{tiny}
\end{picture}
\end{center}

\noindent
and

\vspace{1cm}

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\put(64,5){\line(2,1){10}}
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\put(76,0){\vector(0,1){4}}
\put(76,4){\line(0,1){4}}
\begin{tiny}
\put(0,8){$\alpha_{n}$} \put(16,8){$\alpha_{2}$}
\put(32,8){$\alpha_{3}$} \put(60,8){$\alpha_{n-2}$}
\put(74,14){$\alpha_{n}$} \put(74,-7){$\alpha_{n-1}$}
\put(78,4){$e^{\pi i/m}cos(\pi/m)$}
\end{tiny}
\end{picture}
\end{center}

\vspace{1cm}

\noindent
respectively. However, as pointed out by Can \cite{can}, based on the work of Popov
\cite{popov}, the vector graph $\pi(m,p,n)$ represented by

\vspace{1cm}

\begin{center}
\begin{picture}(120,6)
\put(2,4){\circle{4}}
\put(18,4){\circle{4}}
\put(34,4){\circle{4}}
\put(50,4){\circle{4}}
\put(78,4){\circle{4}}
\put(92,10){\circle{4}}
\put(92,-2){\circle{4}}
\put(57,4){\circle*{.5}}
\put(71,4){\circle*{.5}}
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\put(92,0){\vector(0,1){4}}
\put(92,4){\line(0,1){4}}
\begin{tiny}
\put(1,4){$q$} \put(0,8){$\alpha_{1}$} \put(16,8){$\alpha_{2}$}
\put(32,8){$\alpha_{3}$} \put(48,8){$\alpha_{4}$}
\put(76,8){$\alpha_{n-1}$} \put(90,14){$\alpha_{n+1}$}
\put(90,-7){$\alpha_{n}$} \put(94,4){$e^{\pi i/m}cos(\pi/m)$}
\end{tiny}
\begin{tiny}
\put(7,-1){$\frac{-1}{\sqrt{2}}$}
\end{tiny}
\end{picture}
\end{center}
is a vector graph for $G(m,p,n)$, where if $q=m/p$

$$ \pi(m,p,n) = \{\alpha_{1}=-\epsilon_{1},\alpha_{2}=\epsilon_{1} - \epsilon_{2}, \ldots ,
\alpha_{n}=\epsilon_{n-1} - \epsilon_{n},
\alpha_{n+1}=\epsilon_{n-1} - exp(2\pi i/q)\epsilon_{n} \}$$

\noindent
However, it is not a simple system for the corresponding root system $\Phi(m,p,n)$
as this set is clearly not linearly independent over ${\mathbb C}$.

\section{\bf Subsystems and primitive systems.}

Let $\Phi = (R,f)$ be a root system corresponding to the
reflection group $W(\Phi)$.If $S \subseteq R$ and $g = f|_{S}$,
then the pair $\Psi = (S,g)$ is called a {\it subsystem} of $\Phi$
if $\Psi$ is itself a root system. The corresponding {\it
reflection subgroup} $W(\Psi)$ is the subgroup of $W(\Phi)$
generated by the $s_{a,g(a)}$ with $a \in S$. Subsystems $\Psi_{1}
= (S_{1},g_{1})$ and $\Psi_{2} = (S_{2},g_{2})$ of $\Phi$ are {\it
conjugate under $W(\Phi)$} if $S_{2} = wS_{1}$ and $g_{2}(w(a)) =
g_{1}(a)$ for some $w \in W(\Phi)$ and for all $a \in S_{1}$; in
which case $W(w\Psi_{1}) = wW(\Psi_{1})w^{-1}$, that is,
$W(\Psi_{1})$ and $W(\Psi_{2})$ are conjugate subgroups in
$W(\Phi)$.

Now, as in Hughes \cite{hughes1} and Can \cite{can}, {\it primary systems} for root systems
$\Phi$ with simple system $\pi = (B,\theta)$ are defined. These play the role of positive
systems for real reflection groups. Let $B = \{a_{1}, \ldots , a_{n}\}$, and put
$r_{i} = s_{a_{i},\theta(a_{i})}, 1 \leq i \leq n$, then the corresponding
primary system is defined inductively as follows:

(i) Let $\Omega_{1} = B$.

(ii) Let $\Omega_{2} = \{r_{i}(a_{j}) \enspace | \enspace 1 \leq i,j \leq n, i \neq j, a_{j} \in
\Omega_{1}, r_{i}(a_{j}) \not\in \Omega_{1}\}$.

(iii) For $k \geq 3$, let
$$\Omega_{k} = \{r_{i}(a) \enspace | \enspace 1 \leq i,j \leq n, i \neq j,
a \in
\Omega_{k-1}, r_{i}(a) \neq \mu b \enspace \mbox{for any} \enspace b\in \Omega_{l}, l <k\},$$
where $\mu$ is a root of unity.

Then $\Omega = \bigcup_{k \geq 1} \Omega_{k}$ is a {\it primary
system } for the simple system $\pi$ of the root system $\Phi$.
The elements of $\Omega_{k}$ are called {\it primary roots of
height $k$}. The elements of $\Omega_{k}$ for the largest $k$ for
which $\Omega_{k} \neq \emptyset$ are called {\it highest roots}.

\begin{remark}
The primary system is not unique in that there is an element of
choice at each step. However, having fixed a primary system
$\Omega$ for the simple system $\pi$, if the simple system is
replaced by $w\pi$, then \cite{can} the corresponding primary
system obtained by making the same choices in the above algorithm
is the conjugate $w\Omega$ of $\Omega$. Thus, the choice of
primary system is of no consequence. In fact, different choices in
the above algorithm would result in conjugate primary systems.
\end{remark}
\begin{remark}
The roots of maximal height are not in general unique for complex reflection groups. For example,
\cite{hughes2}, $J_{3}(4)$ has a unique highest root of height $6$, but $L_{4}$ has two
highest roots of height $9$.
\end{remark}
\begin{remark}
In the case of real reflection groups, the primary systems are positive systems and the
highest roots are the longest roots.
\end{remark}

Hughes \cite{hughes1} has shown how subsystems of root systems may
be constructed. He has done this by extending the corresponding
method for real reflection groups to the complex case. Namely, an
{\it extended Cohen-Dynkin diagram} of a root system $\Phi$ is
formed by attaching the negative of a highest root to a
Cohen-Dynkin diagram for $\Phi$ and removing one or more nodes in
all possible ways and repeating this process on all the resulting
diagrams. At all stages, including the initial Cohen-Dynkin
diagram, all equivalent diagrams must also be considered. As there
could be more than one highest root in the complex case and since
a number of equivalent diagrams must be considered, the algorithm
is more difficult to apply in the complex case in comparison with
the real case. In \cite{hughes2}, a complete list of subsystems is
given for spaces of dimension $\geq 3$. Can \cite{can} has given
an alternative algorithm for obtaining a complete set of
subsystems. In particular, he has in \cite{can2} obtained
subsystems for the groups $G(m,1,n)$ and $G(m,m,n)$. He proved
that the subsystems (up to conjugacy) for the groups $G(m,1,n)$
and $G(m,m,n)$ are \begin{eqnarray}
\sum_{i=1}^{m}\sum_{j=1}^{s_{i}}B_{\lambda_{j}^{(i)}}^{m_{i}} +
\sum_{j=1}^{s}D_{\mu_{j}}^{m}, & &
\sum_{j=1}^{s_{1}}(\lambda_{j}^{(1)}+1)
+\sum_{i=2}^{m}\sum_{j=1}^{s_{1}}\lambda_{j}^{(i)} +
\sum_{j=1}^{s}\mu_{j}=n
\end{eqnarray}
and \begin{eqnarray}
\sum_{i=1}^{m}\sum_{j=1}^{s_{i}}D_{\lambda_{j}^{(i)}}^{m_{i}} , &
& \sum_{j=1}^{s_{1}}(\lambda_{j}^{(1)}+1)
+\sum_{i=2}^{m}\sum_{j=1}^{s_{1}}\lambda_{j}^{(i)}=n
\end{eqnarray}
respectively, where $m_{1}=1$ and $m_{i}=m,\;(i=2,\ldots, m)$.

In Tables 1 and 2, the results of applying these algorithms to all
the exceptional groups $ST_{4} - ST_{37}$ are given (excluding the
real reflection groups). We note that these results are in line
with the relationship between the extended Cohen-Dynkin diagram
and the diagrams and presentation for the corresponding
irreducible infinite discrete complex reflection groups given by G
Malle \cite{ma}.

From now on, if $\Phi = (R,f)$ is a root system, we will simply
write $\pi$ and $\Omega$ for the corresponding simple system and
primary system with the corresponding map $f$ being restricted to
these sets.


\begin{table}
\begin{center}
\begin{tabular}{ccccc}\hline
\begin{tiny}
{\bf Shephard}\end{tiny}&\begin{tiny}{\bf Cohen-Dynkin}\end{tiny}& & &\begin{tiny}{\bf Extended}
\end{tiny}\\[-0.4cm]
\begin{tiny}{\bf Todd type} \end{tiny} &\begin{tiny}{\bf diagram}\end{tiny}
&\begin{tiny}${i\alpha}$\end{tiny}&\begin{tiny}${i\alpha}^{\prime}$\end{tiny} & \begin{tiny}{\bf C-D diagram}\end{tiny}
\\ \hline
$ST_{4}$&\begin{picture}(25,20) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}&
$\frac{1}{\sqrt{3}}$ & $\frac{\sqrt{2}}{\sqrt{3}}$
&$\begin{picture}(60,20) \put(3,-5){\circle{4}}
\put(21,-5){\circle{4}} \put(12,9){\circle{4}}
\put(5,-5){\vector(1,0){8}} \put(13,-5){\line(1,0){6}}
\put(4,-3){\vector(2,3){4}} \put(4,-3){\line(2,3){7}}
\put(20,-3){\line(-2,3){7}} \put(20,-3){\vector(-2,3){4}}
\begin{tiny}
\put(2,1){$\alpha$}
\put(19,1){$\alpha$}
\put(10,-9){$\alpha$}
\put(2,-6){$3$}
\put(20,-6){$3$}
\put(11,8){$3$}
\end{tiny}
\end{picture}$\\[-0.2cm]
$ST_{5}$&\begin{picture}(25,20)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$4$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&$\frac{\sqrt{2}}{\sqrt{3}}$
&$\frac{1}{\sqrt 3}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$3$}
\put(20,2){$3$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{6}$&\begin{picture}(25,15) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$6$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\frac{1}{\sqrt 3}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{1}{\sqrt 3}))}^{\frac{1}{2}}$&
$\begin{picture}(60,20) \put(3,-5){\circle{4}}
\put(21,-5){\circle{4}} \put(12,9){\circle{4}}
\put(5,-5){\vector(1,0){8}} \put(13,-5){\line(1,0){6}}
\put(4,-3){\vector(2,3){4}} \put(4,-3){\line(2,3){7}}
\put(20,-3){\line(-2,3){7}} \put(20,-3){\vector(-2,3){4}}
\begin{tiny}
\put(-1,2){$\frac{-i}{\sqrt{3}}$}
\put(19,1){$\alpha^{\prime}$}
\put(10,-9){$\alpha$}
\put(2,-6){$3$}
\put(11,8){$3$}
\end{tiny}
\end{picture}$\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$} \put(30,-1){$\alpha$}
\put(20,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{8}$&\begin{picture}(25,12) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$4$}
\put(20,2){$4$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&$\frac{1}{\sqrt 2}$&$\frac{1}{\sqrt 2}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$4$}
\put(20,2){$4$}
\put(38,2){$4$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{9}$&\begin{picture}(25,18) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$4$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$6$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\frac{1}{\sqrt 2}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{1}{\sqrt 2}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$4$}
\put(38,2){$4$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(20,2){$4$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{10}$&\begin{picture}(25,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$4$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$4$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\frac{1}{\sqrt 3}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{1}{\sqrt 3}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$4$}
\put(20,2){$3$}
\put(38,2){$4$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(2,2){$3$}
\put(20,2){$4$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{14}$&\begin{picture}(25,12) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$8$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+ \frac{\sqrt{2}}{\sqrt{3}}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{\sqrt{2}}{\sqrt{3}}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$3$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(20,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{16}$&\begin{picture}(25,12) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$5$}
\put(20,2){$5$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\frac{1}{\sqrt 5}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{1}{\sqrt 5}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$5$}
\put(20,2){$5$}
\put(38,2){$5$}
\end{tiny}
\end{picture}\\[-0.2cm]
$ST_{17}$&\begin{picture}(25,12) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$5$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$6$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\alpha_{16}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\alpha_{16}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$5$}
\put(38,2){$5$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(20,2){$5$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{18}$&\begin{picture}(25,15) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$5$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$4$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\frac{1}{\sqrt 3}cot\frac{\pi}{5}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\frac{1}{\sqrt 3}cot\frac{\pi}{5}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$5$}
\put(20,2){$3$}
\put(38,2){$5$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(2,2){$3$}
\put(20,2){$5$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{20}$&\begin{picture}(25,18) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$5$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&$\frac{1}{2}(\frac{1 + \sqrt 5}{\sqrt 3})$
&$\frac{1}{2}(\frac{1 - \sqrt 5}{\sqrt 3})$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$3$}
\put(20,2){$3$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.3cm]
$ST_{21}$&\begin{picture}(25,12) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$10$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
&${(\frac{1}{2}(1+\alpha_{20}))}^{\frac{1}{2}}$
&${(\frac{1}{2}(1-\alpha_{20}))}^{\frac{1}{2}}$
&\begin{picture}(60,18)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(39,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(37,3){\vector(-1,0){8}}
\put(29,3){\line(-1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha$}
\put(30,-1){$\alpha^{\prime}$}
\put(2,2){$3$}
\put(38,2){$3$}
\end{tiny}
\end{picture}\\[-0.2cm]
&&&&\begin{picture}(60,14) \put(3,3){\circle{4}}
\put(21,3){\circle{4}} \put(39,3){\circle{4}}
\put(19,3){\vector(-1,0){8}} \put(11,3){\line(-1,0){6}}
\put(23,3){\vector(1,0){8}} \put(31,3){\line(1,0){6}}
\begin{tiny}
\put(11,-1){$\alpha^{\prime}$}
\put(30,-1){$\alpha$}
\put(20,2){$3$}
\end{tiny}
\end{picture}\\[-0.1cm]
\end{tabular}
\caption{Extended Cohen-Dynkin diagrams in dimension 2}
\end{center}
\end{table}


\begin{table}
\begin{center}
\begin{tabular}{cccc}\hline
\setlength{\unitlength}{0.40mm}
{\bf ST $\mid$ Cohen}&{\bf Cohen-Dynkin}& &{\bf Extended}\\
{\bf type}  &{\bf diagram}&&{\bf C-D diagram}\\ \hline
$ST_{24} \mid J_{3}(4)$ &
\begin{picture}(60,12)
\put(3,-5){\circle{4}} \put(21,-5){\circle{4}}
\put(12,9){\circle{4}} \put(5,-5){\line(1,0){14}}
\put(4,-3){\vector(2,3){4}} \put(4,-3){\line(2,3){7}}
\put(20,-3){\line(-2,3){7}}
\begin{tiny}
\put(2,1){$\alpha$}
\end{tiny}
\end{picture}&
$\mbox{\begin{tiny}$\alpha = \frac{(1-\sqrt{7}i)}{4}$\end{tiny}}$
&\begin{picture}(60,12) \put(3,-5){\circle{4}}
\put(21,-5){\circle{4}} \put(12,9){\circle{4}}
\put(39,-5){\circle{4}} \put(5,-5){\line(1,0){14}}
\put(23,-5){\line(1,0){14}} \put(4,-3){\vector(2,3){4}}
\put(4,-3){\line(2,3){7}} \put(20,-3){\line(-2,3){7}}
\begin{tiny}
\put(2,1){$\alpha$}
\end{tiny}
\end{picture}\\
$ST_{25} \mid L_{3}$ &
\begin{picture}(60,20)
\put(3,0){\circle{4}}
\put(21,0){\circle{4}}
\put(39,0){\circle{4}}
\put(19,0){\vector(-1,0){8}}
\put(11,0){\line(-1,0){6}}
\put(23,0){\vector(1,0){8}}
\put(31,0){\line(1,0){6}}
\begin{tiny}
\put(11,-4){$\alpha$}
\put(31,-4){$\alpha$}
\put(2,-1){$3$}
\put(20,-1){$3$}
\put(38,-1){$3$}
\end{tiny}
\end{picture}
&\begin{tiny}$\alpha = \frac{-i}{\sqrt{3}}$\end{tiny}&
\begin{picture}(60,28)
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\put(21,0){\circle{4}}
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\put(20,-1){$3$}
\put(38,-1){$3$}
\put(20,15){$3$}
\end{tiny}
\end{picture}\\
$ST_{26} \mid M_{3}$ &
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\end{tiny}
\end{picture}
&\begin{tiny}{$\alpha = \frac{-i}{\sqrt{3}}$}\end{tiny}&
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\put(2,-1){$3$}
\put(20,-1){$3$}
\put(38,-1){$3$}
\end{tiny}
\end{picture}\\
& &
$\beta =\frac{-1}{\sqrt{2}}$ &
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\put(20,-1){$3$}
\end{tiny}
\end{picture}\\
$ST_{27} \mid J_{3}(5)$ &
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\put(20,-3){\line(-2,3){7}}
\begin{tiny}
\put(2,1){$\alpha$}
\end{tiny}
\end{picture}
&\begin{tiny}
$\alpha=\frac{1}{2}+\omega cos\frac{\pi}{5}$
\end{tiny}
&
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\begin{tiny}
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\end{tiny}
\end{picture}\\
$ST_{29} \mid N_{4}$ &
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\put(20,-3){\line(-2,3){7}}
\begin{tiny}
\put(11,-8){$\alpha$}
\end{tiny}
\end{picture}
&$\alpha=\frac{1-i}{2}$&
\begin{picture}(60,12)
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\put(38,-3){\line(-2,3){7}}
\begin{tiny}
\put(20,1){$\alpha$}
\end{tiny}
\end{picture}\\
$ST_{31} \mid EN_{4}$ &
&$\alpha=\frac{1-i}{2}$&
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\put(4,-3){\line(2,3){7}}
\put(20,-3){\line(-2,3){7}}
\begin{tiny}
\put(11,-8){$\alpha$}
\end{tiny}
\end{picture}\\
$ST_{32} \mid L_{4}$ &
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\put(31,0){\line(1,0){6}}
\begin{tiny}
\put(11,-4){$\alpha$}
\put(29,-4){$\alpha$}
\put(49,-4){$\alpha$}
\put(2,-1){$3$}
\put(20,-1){$3$}
\put(38,-1){$3$}
\put(56,-1){$3$}
\end{tiny}
\end{picture}
&\begin{tiny}$\alpha =\frac{-i}{\sqrt{3}}$\end{tiny}&
\begin{picture}(60,20)
\put(3,0){\circle{4}}
\put(21,0){\circle{4}}
\put(39,0){\circle{4}}
\put(57,0){\circle{4}}
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\put(19,0){\vector(-1,0){8}}
\put(11,0){\line(-1,0){6}}
\put(55,0){\vector(-1,0){8}}
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\put(31,0){\line(1,0){6}}
\put(59,0){\vector(1,0){8}}
\put(67,0){\line(1,0){6}}
\begin{tiny}
\put(11,-4){$\alpha$}
\put(29,-4){$\alpha$}
\put(49,-4){$\alpha$}
\put(67,-4){$\alpha$}
\put(2,-1){$3$}
\put(20,-1){$3$}
\put(38,-1){$3$}
\put(56,-1){$3$}
\put(74,-1){$3$}
\end{tiny}
\end{picture}\\
&&&\begin{picture}(60,20)
\put(3,0){\circle{4}}
\put(21,0){\circle{4}}
\put(39,0){\circle{4}}
\put(57,0){\circle{4}}
\put(75,0){\circle{4}}
\put(5,0){\vector(1,0){8}}
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\put(73,0){\vector(-1,0){8}}
\put(65,0){\line(-1,0){6}}
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\put(11,-4){$\alpha$}
\put(29,-4){$\alpha$}
\put(49,-4){$\alpha$}
\put(67,-4){$\alpha$}
\put(2,-1){$3$}
\put(20,-1){$3$}
\put(38,-1){$3$}
\put(56,-1){$3$}
\put(74,-1){$3$}
\end{tiny}
\end{picture}\\
$ST_{33} \mid K_{5}$ &
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\put(57,-5){\circle{4}} \put(5,-5){\line(1,0){14}}
\put(23,-5){\line(1,0){14}} \put(41,-5){\line(1,0){14}}
\put(22,-3){\line(2,3){7}} \put(38,-3){\line(-2,3){7}}
\put(37,-5){\vector(-1,0){8}} \put(29,-5){\line(-1,0){6}}
\begin{tiny}
\put(28,-8){$\alpha$}
\end{tiny}
\end{picture}
&\begin{tiny}$\alpha=\frac{-\omega}{2}$\end{tiny}&\\
&\begin{picture}(60,22)
\put(3,-5){\circle{4}}
\put(21,-5){\circle{4}}
\put(21,11){\circle{4}}
\put(39,-5){\circle{4}}
\put(39,11){\circle{4}}
\put(5,-5){\line(1,0){14}}
\put(21,-3){\line(0,1){12}}
\put(23,11){\line(1,0){14}}
\put(39,-3){\line(0,1){12}}
\put(37,-5){\vector(-1,0){8}}
\put(29,-5){\line(-1,0){6}}
\begin{tiny}
\put(28,-8){$\alpha$}
\end{tiny}
\end{picture}&&
\begin{picture}(60,22)
\put(3,-5){\circle{4}}
\put(21,-5){\circle{4}}
\put(21,11){\circle{4}}
\put(57,11){\circle{4}}
\put(39,-5){\circle{4}}
\put(39,11){\circle{4}}
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\put(29,-5){\line(-1,0){6}}
\put(41,11){\line(1,0){14}}
\begin{tiny}
\put(28,-8){$\alpha$}
\end{tiny}
\end{picture}\\
$ST_{34} \mid K_{6}$ &
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\put(41,-5){\line(1,0){14}} \put(22,-3){\line(2,3){7}}
\put(38,-3){\line(-2,3){7}} \put(37,-5){\vector(-1,0){8}}
\put(29,-5){\line(-1,0){6}} \put(59,-5){\line(1,0){14}}
\begin{tiny}
\put(28,-8){$\alpha$}
\end{tiny}
\end{picture}
&\begin{tiny}$\alpha=\frac{-\omega}{2}$\end{tiny}&\\
&\begin{picture}(60,22)
\put(3,-5){\circle{4}}
\put(21,-5){\circle{4}}
\put(39,-5){\circle{4}}
\put(39,11){\circle{4}}
\put(57,-5){\circle{4}}
\put(57,11){\circle{4}}
\put(5,-5){\line(1,0){14}}
\put(23,-5){\line(1,0){14}}
\put(39,-3){\line(0,1){12}}
\put(41,11){\line(1,0){14}}
\put(57,-3){\line(0,1){12}}
\put(55,-5){\vector(-1,0){8}}
\put(47,-5){\line(-1,0){6}}
\begin{tiny}
\put(46,-8){$\alpha$}
\end{tiny}
\end{picture}&&
\begin{picture}(60,22)
\put(3,-5){\circle{4}}
\put(21,-5){\circle{4}}
\put(39,-5){\circle{4}}
\put(57,-5){\circle{4}}
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\begin{tiny}
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\end{picture}\\
& & &
\end{tabular}
\caption{Extended Cohen-Dynkin diagrams in dimension $>2$}
\end{center}
\end{table}



\section{\bf Macdonald representations.}

Let $R$ be the ring of polynomial functions on $V$ which can be
identified with the symmetric algebra of $V$. The reflection group
$G$ acts on $R$ as follows $$ (gf)(x) = f(g^{-1}x)$$ for all $ f
\in R, g \in G, x \in V$. In particular, $$ (g \alpha_{H})(x) =
\chi_{H}^{-1}\alpha_{H}(x)$$ for all $g \in G_{H}, x \in V$, where
$\tilde{G}_{H} = \{\chi_{H}^{-i}\; |\; 0 \leq i \leq e_{H}-1 \}$
is the character group of $G_{H}$. Define $\pi_{C} = \prod_{H \in
C}\alpha_{H}$ for $C \in \mathcal{C}$ and $\pi_{G} = \prod_{C \in
\mathcal{C}}\pi_{C}.$

When reflection subgroups are determined by subsystems of root
systems, the functions $\pi_{G}$ defined earlier will be denoted
$\pi_{\Phi}$.

We give three examples

\newtheorem{example}[remark]{Example}
\begin{example}
For the real reflection group of type $A_{n-1}$ (symmetric groups
of order $n!$), the root system is $\{\epsilon_{i} - \epsilon_{j},
1 \leq i,j \leq n\}$ and a simple system is $\{\epsilon_{i} -
\epsilon_{i+1}, 1 \leq i \leq n-1 \}$ and corresponding primary
(positive) system $\{\epsilon_{i}-\epsilon_{j}, 1 \leq i < j \leq
n\}$. Then
$$\pi_{A_{n-1}}(x) = \prod_{1 \leq i < j \leq n} (x_{i} - x_{j})$$
which is the Vandermonde determinant.
\end{example}
\begin{example}
For the complex reflection group of type $G(m,m,n)$ and $G(m,1,n)$
using the primary root systems given above, we obtain
$$\pi_{G(m,m,n)}(x) = \prod_{1 \leq i<j \leq n}\prod_{a=1}^{m}(x_{i} - \zeta^{a}x_{j})
=\prod_{1 \leq i<j \leq n}(x_{i}^{m} - x_{j}^{m}) $$

\noindent and

$$\pi_{G(m,1,n)}(x) = \prod_{1 \leq i<j \leq n}\prod_{a=1}^{m}(x_{i} - \zeta^{a}x_{j})
\prod_{i=1}^{n}x_{i} = \prod_{1 \leq i<j \leq n}(x_{i}^{m} -
x_{j}^{m})\prod_{i=1}^{n}x_{i} .$$
\end{example}
\begin{example}
For the complex reflection group of type $ST_{4}$, a simple system
is \cite{hm} $\{\epsilon_{1}, -i/\sqrt{3}(\epsilon_{1} +
\sqrt{2}\epsilon_{2})\}$ with primary system $\Omega_{ST4} =
\{-i/\sqrt{3}(\epsilon_{1} + \omega^{k} \sqrt{2}\epsilon_{2}), 0
\leq k \leq 2 \}$ and root system $\mu_{6}\Omega_{ST4}$. Then
$$ \pi_{ST4}(x) = x_{1}(x_{1}- \sqrt{2}x_{2})
(x_{1}- \omega\sqrt{2}x_{2})(x_{1}- \omega^{2}\sqrt{2}x_{2}) =
x_{1}(x_{1}^{3} + 2\sqrt{2}x_{2}^{3})/3\sqrt{3}$$
\end{example}

The following lemma is the crucial result which extends the well
known result in the real case that the positive roots are permuted
by a reflection $s_{a}$ except that $s_{a}(a)=-a$, where $a$ is
any positive root.
\begin{lem}
\begin{eqnarray*}
s_{H}\pi_{C} & = &  \left\{\begin{array}{ll} \pi_{C} & \mbox{if}
\;H \not\in C \\(\chi_{H})^{-1}\pi_{C} & \mbox{if} \; H \in C.
\end{array}\right.
\end{eqnarray*}
\end{lem}

Proof. If $H^{\prime} \not\in C$, let $v_{0} \in H^{\prime}$ be a
point, thus $\alpha_{H}(v_{0}) \neq 0$ for all $H \in C$ which
implies that $\pi_{C}(v_{o}) = \prod_{H \in C}\alpha_{H}(v_{0})
\neq 0$. Hence
$$(s_{H^{\prime}}\pi_{C})(v_{0}) =
\pi_{C}(s_{H^{\prime}}^{-1}v_{0}) = \pi_{C}(v_{0}).$$ Furthermore,
if $H \in C$, then
$$s_{H}\pi_{C} =
\prod_{H \in C}s_{H}\alpha_{H} =
\left(\frac{s_{H}\alpha_{H}}{\alpha_{H}}\right )\pi_{C} =
\chi_{H}^{-1}\pi_{C}.$$ Let $\overline{\pi}_{C} =
\prod_{H^{\prime} \in C, H^{\prime} \neq H}\alpha_{H^{\prime}}$
and let $v_{0} \in H$ be a point such that
$\alpha_{H^{\prime}}(v_{0}) \neq 0$ for all $H^{\prime} \neq H$,
then $\overline{\pi}_{C}(v_{0}) \neq 0$. Hence
$$(s_{H}\overline{\pi}_{C})(v_{0}) =
\overline{\pi}_{C}(s_{H}^{-1}v_{0}) = \overline{\pi}_{C}(v_{0}),$$
which proves the lemma.

Let $R^{G}$ be the subring of $G$-invariant elements in $R$. Then,
it is well known \cite{bour} that $R^{G}$ is generated by $n$
algebraically independent homogeneous elements $p_{1}, \ldots,
p_{n}$. If $d_{i} = deg(p_{i})$, then \begin{eqnarray}  |G|
& = & \prod_{i=1}^{n}d_{i}\\
  |\mbox{complex reflections in} \; G| & = & \sum_{i=1}^{n}(d_{i}-1) \; = \; \sum_{H \in \mathcal{A}}(e_{H}-1).
\end{eqnarray}
(In fact, \cite{os} if $m_{i} = d_{i}-1$ are the exponents and
$n_{i}$ are the coexponents of $G$, then
\begin{eqnarray}
|\mbox{complex reflections in} \; G| & = & \sum_{i=1}^{n}m_{i}  \\
|\mbox{reflecting hyperplanes}|  & = & \sum_{i=1}^{n}n_{i}.)
\end{eqnarray}


If $\chi$ is a linear character of $G$, let $R_{\chi}^{G}$ be the
$R^{G}$-module of relative invariants of $G$ introduced by R.
Stanley \cite{st}, that is,
$$ R_{\chi}^{G} = \{f \in R \;|\;gf=\chi(f)f \;\mbox{for all}\; g
\in G. \}$$ Then Stanley has proved, amongst other things, that if
$f \in R_{\chi}^{G}$, then $f$ is divisible by $f_{\chi} =
\prod_{C \in \mathcal{C}}\pi_{C}^{l_{C}},$ where $0 \leq l_{C}
\leq e_{C}-1$ and $l_{C}$ is the least positive integer such that
$\chi(s_{C}) = \zeta_{C}^{l_{C}}$, for a fixed generator $s_{C}
\in G_{H}$ for some $H \in C$. Furthermore, $R_{\chi}^{G} =
f_{\chi}R^{G}$ and $f_{\chi} \in R_{\chi}^{G}$, that is,
$$gf_{\chi} = \chi(g)f_{\chi}\;\mbox{for all}\; g \in G.$$

Now let $G^{\prime}$ be a reflection subgroup of $G$ and let
$\chi$ also denote the restriction of the linear character $\chi$
to $G^{\prime}$. Let $f_{\chi}^{\prime}$ be defined as above for
the subgroup $G^{\prime}$. Let $P_{G^{\prime}}^{\chi}$ be the
subspace of $R$ generated by the polynomial functions
$gf_{\chi}^{\prime}$ for all $g \in G$, then the vector space
$P_{G^{\prime}}^{\chi}$ is a $\mathbb{C}G$-module. The special
case $\chi =1$ will be denoted by $P_{G^{\prime}}$. Then we have
the following theorem which generalizes the well known result of I
G Macdonald \cite{mcd} to complex reflection groups - in fact, his
proof carries over almost verbatim. It can be shown by using some
of the above results that (i) below is a direct consequence of a
theorem in \cite{lusspa}.
\begin{thm}
(i) The module $P_{G^{\prime}}^{\chi}$ is an absolutely
irreducible $\mathbb{C}G$-module.

(ii) If $G^{\prime}$ and $G^{\prime\prime}$ are two non-isomorphic
reflection subgroups of $G$ with
$$|\mbox{reflecting hyperplanes
for}\;G^{\prime}| \neq |\mbox{reflecting hyperplanes for}
\;G^{\prime\prime}|,$$
then the modules $P_{G^{\prime}}^{\chi}$
and $P_{G^{\prime\prime}}^{\chi}$ are not isomorphic.

(iii) The modules $P_{G^{\prime}}^{\chi}$ and $\chi \otimes
P_{G^{\prime}}$ are isomorphic.
\end{thm}

Proof. (i) Let $\phi:P_{G^{\prime}}^{\chi}\rightarrow
P_{G^{\prime}}^{\chi}$ be a $\mathbb{C}G$-homomorphism, then
$$g\phi(f_{\chi}^{\prime}) = \phi(gf_{\chi}^{\prime}) =
\phi(\chi(g)f_{\chi}^{\prime}) = \chi(g)f_{\chi}^{\prime}$$ for
all $g \in G$ and so, by the above result
$\phi(f_{\chi}^{\prime})$ is divisible by $f_{\chi}^{\prime}$.
Since $\phi(f_{\chi}^{\prime})$ and $f_{\chi}^{\prime}$ are of the
same degree and so $\phi(f_{\chi}^{\prime})$ is a scalar multiple
of $f_{\chi}^{\prime}$ and (i) follows.

(ii) The proof is similar to that of (i), but resulting this time
in a zero map.

(iii) The map $\phi:P_{G^{\prime}}^{\chi}\rightarrow \chi \otimes
P_{G^{\prime}}$ is $\phi(f_{\chi}) = \chi \otimes f_{1}$, which is
easily seen to be the required isomorphism.

\vspace{0.2cm}

The last result (iii) shows that basically it is only necessary to
construct the modules(representations) $P_{G^{\prime}}^{\chi}$.

Some examples are now given to illustrate the usefulness of the
above approach.
\begin{example}

In the case $G(m,1,n)$, this construction gives all the
irreducible Macdonald modules. It is well known in this case that
the irreducible representations are in one-to-one correspondence
with the set of $m$-partitions
$(\lambda^{(1)},\ldots,\lambda^{(m)})$ of $n$; thus we need only
take the subsystems of type
$\sum_{i=1}^{m}\sum_{j=1}^{s_{i}}B_{\lambda_{j}^{(i)}}^{m_{i}}$
listed in (3.1) - here $\lambda^{(i)}$ is the partition
$(\lambda_{1}^{(i)},\ldots,\lambda_{s_{i}}^{(i)})$. The
representations of the generalized symmetric group $G(m,1,n)$ have
been considered by Can \cite{can0} and Hughes \cite{hughes} from a
different point of view (generalizing the concepts of tabloids and
polytabloids or the symmetric groups), it is clear that this can
be modified to adopt the Macdonald module approach.
\end{example}
\begin{example}
We consider the group $ST_{4}$. As was seen in Table 1, this group
has Cohen-Dynkin diagram

\vspace{-0.2cm}

\setlength{\unitlength}{0.70mm}

\begin{center}
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\put(21,3){\circle{4}}
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\begin{tiny}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
\end{center}

\vspace{-0.2cm}

\noindent
and extended Cohen-Dynkin diagram

\vspace{-0.5cm}

\begin{center}
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%\put(6,0){\line(2,3){3}}
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%\put(22,0){\line(-2,3){3}}
\begin{tiny}
\put(2,1){$\alpha$}
\put(19,1){$\alpha$}
\put(10,-9){$\alpha$}
\put(2,-6){$3$}
\put(20,-6){$3$}
\put(11,8){$3$}
\end{tiny}
\end{picture},
\end{center}
where $\alpha = -i/\sqrt{3}$.

\noindent
Thus, in this case, there are only $3$ non-conjugate subsystems

\begin{picture}(45,10)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(20,2){$3$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture}
\begin{picture}(25,10)
\put(3,3){\circle{4}}
\begin{tiny}
\put(2,2){$3$}
\end{tiny}
\end{picture}
$\emptyset$.

\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|ccccccc|} \hline
$ST_{4}$ & $cl_{1}$ & $cl_{2}$ & $cl_{3}$ & $cl_{4}$ & $cl_{5}$ & $cl_{6}$ & $cl_{7}$ \\
order & 1 & 2 & 6 & 6 & 3 & 3 & 4 \\
class order & 1 & 1 & 4 & 4 & 4 & 4 & 6 \\ \hline
$\chi_{1}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
$\chi_{2}$ & 1 & 1 & $\omega$ & $\omega^{2}$ & $\omega$ & $\omega^{2}$ & 1 \\
$\chi_{3}$ & 1 & 1 & $\omega^{2}$ & $\omega$ & $\omega^{2}$ & $\omega$ & 1 \\
$\chi_{4}$ & 2 & -2 & 1 & 1 & -1 & -1 & 0 \\
$\chi_{5}$ & 2 & -2 & $\omega$ & $\omega^{2}$ & $-\omega$ & $-\omega^{2}$ & 0 \\
$\chi_{6}$ & 2 & -2 & $\omega^{2}$ & $\omega$ & $-\omega^{2}$ & $-\omega$ & 0 \\
$\chi_{7}$ & 3 & 3 & 0 & 0 & 0 & 0 & -1 \\ \hline
\end{tabular}
\vskip5pt
\end{tiny}
\caption{Character Table of $ST_{4}$}
\end{center}
\end{table}

From Table 3, we see that $ST_{4}$ has three linear characters $\chi_{1},\chi_{2}$
and $\chi_{3}$ and since $\chi_{5} = \chi_{4} \otimes \chi_{2}$ and
$\chi_{6} = \chi_{4} \otimes \chi_{3}$, thus the only modules required are those
corresponding to $\chi_{1}, \chi_{4}$ and $\chi_{7}$ of degrees 1,2 and 3 respectively.
The first two are obtained from the subsystems
\begin{picture}(25,8)
\begin{tiny}
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(2,2){$3$}
\put(20,2){$3$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{tiny}
\end{picture} and
\begin{picture}(8,8)
\begin{tiny}
\put(3,3){\circle{4}}
\put(2,2){$3$}
\end{tiny}
\end{picture}
of $ST_{4}$. The third is obtained by using the fact that $ST_{4}$ is a subgroup
of $ST_{6}$ with Cohen-Dynkin diagram
\begin{picture}(25,8)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(2,2){$3$}
\put(11,5){$6$}
\put(10,-1){$\beta$}
\end{tiny}
\end{picture},
where
\begin{tiny}
$\beta = -i(\frac{1}{2}(1+\frac{1}{\sqrt{3}}))^{\frac{1}{2}}$.
\end{tiny}
The representation of degree $3$ is the representation of $ST_{6}$
corresponding to the subsystem
\begin{picture}(25,8)
\begin{tiny}
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(2,2){$3$}
\put(20,2){$3$}
\end{tiny}
\end{picture}
which remains irreducible on restriction to $ST_{4}$, indeed this subsystem is also a subsystem
of $ST_{4}$.

The irreducible representations obtained from the three subsystems are given in Table 4.
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|}\hline
{\bf Subsystem} & {\bf Basis} & {\bf s} & {\bf t} \\ \hline
\begin{picture}(25,10)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\put(11,5){$3$}
\put(10,-1){$\alpha$}
\end{picture} &
$\{s\}$ & $(\omega)$ & $(\omega)$ \\
\begin{picture}(25,10)
\put(3,3){\circle{4}}
\put(2,2){$3$}
\put(2,-1){$s$}
\end{picture} &
$\{ s, \tau_{t}s \}$ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $ &
$ \left( \begin{array}{cc}
\omega & \omega \\
0 & 1
\end{array} \right) $\\
\begin{picture}(25,10)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(2,2){$3$}
\put(20,2){$3$}
\put(2,-1){$s$}
\put(20,-1){$t$}
\end{picture} &
$\{ s, \tau_{t}s \}$ &
$ \left( \begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
0 & 0 & -1 \\
-1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $\\
$\emptyset$ & & (1) & (1)\\ \hline
\end{tabular}
\end{tiny}
\vskip5pt
\caption{Irreducible Representations of $ST_{4}$}
\end{center}
\end{table}

\noindent
\end{example}
\begin{example}
Some more general results may be obtained for an arbitrary
two-dimensional group. For an arbitrary root system

\setlength{\unitlength}{0.70mm}

\begin{center}
\begin{picture}(60,10)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(1,2){$m$}
\put(20,2){$n$}
\put(11,5){$k$}
\put(10,-1){$\alpha$}
\put(2,-2){$u$}
\put(20,-2){$v$}
\end{tiny}
\end{picture}
\end{center}
let $\zeta$ and $\eta$ denote primitive $m$-th and $n$-th roots of unity respectively. For
the subsystem
\begin{eqnarray*}
 S & = &  \begin{picture}(10,10)
\put(3,3){\circle{4}}
\begin{tiny}
\put(1,2){$m$}
\put(2,-2){$u$}
\end{tiny}
\end{picture}
\end{eqnarray*}

\noindent if $s= \tau_{u}$ and $t= \tau_{v}$, then $\pi_{S} = u$
and $P_{S} = <u,tu=u+(1-\eta)\alpha v>$, and we see that

$s \longmapsto  \left( \begin{array}{cc}
\zeta & -(1-\zeta)(1+(1-\eta)\alpha^{2}) \\
0 & 1
\end{array} \right) $ and $t \longmapsto  \left( \begin{array}{cc}
0 & -\eta \\
1 & 1 + \eta
\end{array} \right). $

It can be shown that the subsystem
\begin{picture}(10,10)
\put(3,3){\circle{4}}
\begin{tiny}
\put(1,2){$m$}
\put(2,-2){$u$}
\end{tiny}
\end{picture}
gives an equivalent representation.

The explicit results for the separate root systems are listed in Table 5;
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c||c|c|c|}\hline
{\bf ST-type}  & {\bf s} & {\bf t} & {\bf ST-type}  & {\bf s} & {\bf t}  \\ \hline
$ST_{4}$ &
$ \left( \begin{array}{cc}
\omega & -1 \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $ &
$ST_{14}$ &
$ \left( \begin{array}{cc}
\omega & \sqrt{2}i\omega^{2} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) $\\
$ST_{5}$ &
$ \left( \begin{array}{cc}
\omega & \omega^{2} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $ &
$ST_{16}$ &
$ \left( \begin{array}{cc}
\eta & -1 \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\eta \\
1 & 1 + \eta
\end{array} \right) $\\
$ST_{6}$ &
$ \left( \begin{array}{cc}
\omega & i \omega^{2} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) $ &
$ST_{17}$ &
$ \left( \begin{array}{cc}
\eta & (1- \eta)\alpha \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) $\\
 $ST_{8}$ &
$ \left( \begin{array}{cc}
i & -1 \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -i \\
1 & 1 + i
\end{array} \right) $ &
$ST_{18}$ &
$ \left( \begin{array}{cc}
\eta & \omega^{2} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $\\
$ST_{9}$ &
$ \left( \begin{array}{cc}
i & -\epsilon^{3} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) $ &
$ST_{20}$ &
$ \left( \begin{array}{cc}
\omega & \omega^{2}-\omega(\eta+\eta^{4}) \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $\\
$ST_{10}$ &
$ \left( \begin{array}{cc}
i & \omega^{2} \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $ &
$ST_{21}$ &
$ \left( \begin{array}{cc}
\omega & i\omega^{2}(1+\eta+\eta^{4}) \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) $\\ \hline
\end{tabular}
\end{tiny}
\vskip5pt
\caption{Irreducible Representations of degree 2}
\end{center}
\end{table}
thus we have the
'basic' representations of degree two for all the two-dimensional reflection groups. In Table 5, $\omega, i, \eta$ and $\epsilon$ are primitive cube, fourth, fifth and eigth
roots of unity respectively and \begin{tiny} $\alpha = -i\sqrt{\frac{1}{2}(1+\frac{1}{\sqrt{5}})}$.
\end{tiny}
\end{example}
\begin{example}

The irreducible representations of degree three are obtained by
taking the subsystems \setlength{\unitlength}{0.70mm}
\begin{picture}(25,8)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\begin{tiny}
\put(1,2){$m$}
\put(19,2){$m$}
\end{tiny}
\end{picture}
and
\begin{picture}(25,8)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\begin{tiny}
\put(2,2){$n$}
\put(20,2){$n$}
\end{tiny}
\end{picture}
of the root system
\begin{picture}(25,8)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\begin{tiny}
\put(1,2){$m$}
\put(20,2){$n$}
\end{tiny}
\end{picture}.
In Table 6, as examples, the irreducible representations of degree three of
$ST_{4},ST_{5},ST_{6},ST_{8}$ and $ST_{9}$ are given, the two representations of $ST_{9}$
are clearly not equivalent.

\begin{table}
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|}\hline
{\bf ST-type} & {\bf s} & {\bf t} \\ \hline
$ST_{4}$&
$ \left( \begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
0 & 0 & -1 \\
-1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $\\\hline
$ST_{5}$&
$ \left( \begin{array}{ccc}
\omega & 0 & -1 \\
0 & 0 & -\omega^{2} \\
0 & 1 & -\omega
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $\\\hline
$ST_{6}$&
$ \left( \begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $\\ \hline
$ST_{8}$&
$ \left( \begin{array}{ccc}
i & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & i
\end{array} \right) $\\ \hline
$ST_{9}$&
$ \left( \begin{array}{ccc}
i & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & -1
\end{array} \right) $\\
& $ \left( \begin{array}{ccc}
0 & 0 & -i \\
1 & 0 & 1 \\
0 & 1 & i
\end{array} \right) $ &
$ \left( \begin{array}{ccc}
-1 & -i & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $\\ \hline
\end{tabular}
\end{tiny}
\vskip5pt
\caption{Irreducible Representations of degree 3}
\end{center}
\end{table}
\end{example}

Further examples are given in the Appendix below.

\section{Acknowlegements}
The first author is grateful to Eric Opdam for a helpful
discussion concerning this work, in particular for drawing his
attention to the paper by Richard Stanley.


\appendix{\bf Appendix.}
The group $ST_{24}$ has irreducible representations of degrees
$$\underbrace{1,1},\underbrace{3,3},
\overline{\underbrace{3,3}},\underbrace{6,6},
\underbrace{7,7},\underbrace{8,8}.$$ Information concerning the
irreducible representations and characters of the complex
reflection groups may be found in M. Benard \cite{ber}.

\begin{table}
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|}\hline
{\bf Subsystem} & {\bf s} & {\bf t} & {\bf u} \\
\hline
\begin{picture}(25,10)
\put(3,3){\circle{4}}  \put(2,-2){$s$}
\end{picture}&
 $ \left( \begin{array}{ccc}
-1 & 0 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
0 & 1 & -\beta \\
1 & 0 & \beta \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
0 & -\bar{\alpha}& 1 \\
0 & 1 & 0 \\
1 &  \bar{\alpha}& 0
\end{array} \right) $\\
\begin{picture}(42,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}\put(39,3){\circle{4}}
\put(2,-2){$s$} \put(20,-2){$t$}
\put(38,-2){$\sigma$}\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\end{picture}&
 $ \left(%
\begin{array}{cccccccc}
  -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 \\
  0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
  0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
  0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\
  0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
  0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$ & $ \left(%
\begin{array}{cccccccc}
  -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
  0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\
  0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
\end{array}%
\right)$ & $ \left(%
\begin{array}{cccccccc}
  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & -1 & 0 & -1 & 0 & -1 & 0 \\
  0 & 0 & 0 & -1 & 0 & -1 & 0 & -1 \\
  0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\
  0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
  0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
  0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
\end{array}%
\right)$ \\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}  \put(2,-2){$s$}
\put(20,-2){$\sigma$}
\end{picture} &
 $\left(
\begin{array}{cccccc}
  -1 & 0 & -1 & 0 & 0 & 0 \\
  0 & 1 & 0 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0 \\
\end{array}
\right)$ &  $\left(%
\begin{array}{cccccc}
  0 & 1 & 0 & 1 & 0 & 1 \\
  1 & 0 & 0 & -1 & 0 & -1 \\
  0 & 0 & 0 & -1 & 1 & -1 \\
  0 & 0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 1 & 1 & 0 & 1 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$& $\left(%
\begin{array}{cccccc}
  0 & 0 & 1 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1 & -1 & 0 \\
  1 & 0 & 0 & 0 & -1 & 0 \\
  0 & 1 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$\\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}
\put(5,3){\line(1,0){14}}
 \put(2,-2){$s$} \put(20,-2){$u$}
\end{picture} & $\left(
\begin{array}{ccccccc}
  -1 & 0 & 0 & 0 & -2 & 0 & 0 \\
  0 & 0 & 1 & 0 & 1 & 0 & 0\\
  0 & 1 & 0 & 0 & -1 & 0 & 0\\
  0 & 0 & 0 & 0 & -1 & 1 & 1 \\
  0 & 0 & 0 & 0 & 1 & 0 & 0\\
  0 & 0 & 0 & 1 & 1 & 0 & -1\\
  0 & 0 & 0 & 0 & 0 & 0 & 1\\
\end{array}
\right)$ &  $\left(
\begin{array}{ccccccc}
  0 & 1 & 1 & 0 & 0 & -1 & 0 \\
  1 & 0 & -1 & 0 & 0 & 1 & 0\\
  0 & 0 & 1 & 0 & 0 & 0 & 0\\
  0 & 0 & 0 & -1 & 0 & -2 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 & 1\\
  0 & 0 & 0 & 0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0 & 1 & -1 & 0\\
\end{array}
\right)$ & $\left(
\begin{array}{ccccccc}
  -1 & 0 & 0 & 0 & 0 & -2 & 0 \\
  0 & 0 & 0 & 1 & 0 & 1 & -1\\
  0 & 0 & 0 & 0 & 1 & -1 & 0\\
  0 & 1 & 0 & 0 & 0 & -1 & 1 \\
  0 & 0 & 1 & 0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0 & 0 & 0 & 1\\
\end{array}
\right)$ \\ \hline
\end{tabular}
\end{tiny}
\caption{Irreducible Representations of $ST_{24}$}
\end{center}
\end{table}


Thus the representations given in Table 7 lead to all the
irreducible representations of $ST_{24}$.

\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|}\hline
{\bf Subsystem} & {\bf s} & {\bf t} & {\bf u} \\
\hline
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(2,2){$3$} \put(2,-2){$s$}
\end{picture}&
 $ \left( \begin{array}{ccc}
\omega & -1 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
0 & -\omega & 0 \\
1 & -\omega^{2} & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & -\omega \\
0 & 1 & -\omega^{2}
\end{array} \right) $\\
\begin{picture}(42,10)
\put(3,3){\circle{4}}
\put(21,3){\circle{4}}\put(39,3){\circle{4}}\put(2,2){$3$}
\put(20,2){$3$}\put(38,2){$3$} \put(2,-2){$s$} \put(20,-2){$u$}
\put(38,-2){$\sigma$}
\end{picture}&
 $ \left( \begin{array}{cc}
\omega & -1 \\
0 & 1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
0 & -\omega \\
1 & -\omega^{2}
\end{array} \right) $ & $ \left( \begin{array}{cc}
\omega & -1 \\
0 & 1
\end{array} \right) $ \\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}} \put(2,2){$3$}
\put(20,2){$3$} \put(2,-2){$s$} \put(20,-2){$u$}
\end{picture} &
 $\left(
\begin{array}{cccccc}
  \omega & 0 & 0 & -1 & 0 & \omega^{2} \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 1 & 0 & -\omega & 0 & -\omega \\
  0 & 0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 1 & \omega & 0 & \omega \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$ &  $\left(%
\begin{array}{cccccc}
  0 & 0 & 0 & 1 & -\omega^{2} & -\omega \\
  1 & 0 & 0 & 0 & \omega^{2} & \omega \\
  0 & 0 & \omega & 0 & \omega & -\omega \\
  0 & 1 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$& $\left(%
\begin{array}{cccccc}
  \omega &\omega & 0 & 0 & -\omega^{2} & 0 \\
  0 & 1 & 0 & 0 & 0 & 0 \\
  0 & -1 & 0 & 0 & -1 & \omega \\
  0 & \omega^{2} & \omega^{2} & 0 & \omega^{2} & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0 \\
\end{array}%
\right)$\\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}} \put(13,3){\line(1,0){6}}
\put(2,2){$3$} \put(20,2){$3$} \put(2,-2){$s$} \put(20,-2){$t$}
\put(11,5){$3$} \put(10,-2){$\alpha$}
\end{picture} & $\left(
\begin{array}{cccccc}
  \omega & 0 & 0 & 0 & -1 & -\omega^{2} \\
  0 & \omega & 0 & 0 & \omega^{2} & 0 \\
  0 & 0 & 0 & 0 & -\omega^{2} & -1 \\
  0 & 0 & 0 & \omega & \omega & \omega^{2} \\
  0 & 0 & 1 & 0 & -\omega & 1 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$ &  $\left(%
\begin{array}{cccccc}
  \omega & 0 & 0 & -1 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 1 & 0 & -\omega & 0& 0 \\
  0 & 0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & \omega & \omega & 0 \\
  0 & 0 & 1 & \omega & 0 & 0 \\
\end{array}%
\right)$& $\left(%
\begin{array}{cccccc}
  0 &0 & 0 & 1 & 0 & -\omega^{2} \\
  1 & 0 & 0 & 0 & 0 & \omega^{2} \\
  0 & 0 & \omega & 0 & 0 & \omega \\
  0 & 1 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & \omega & -\omega \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$\\ \hline
\end{tabular}
\end{tiny}
\caption{Irreducible Representations of $ST_{25}$}
\end{center}
\end{table}


The group $ST_{25}$ has irreducible representations of degrees
$$\underbrace{1,1,1},\underbrace{2,2,2},\underbrace{3,3,3},
\overline{\underbrace{3,3,3}},\underbrace{6,6,6},\overline{\underbrace{6,6,6}},
\underbrace{8,8,8},3,9,\overline{9}.$$

Thus the representations in Table 8 lead to all the irreducible
representations except the final ones of degrees 8, 3 and 9.

\begin{table}
\begin{center}
\begin{tiny}
\begin{tabular}{|c|c|c|c|}\hline
{\bf Subsystem} & {\bf s} & {\bf t} & {\bf u} \\
\hline
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(2,-2){$s$}
\end{picture}&
 $ \left( \begin{array}{ccc}
-1 & \omega^{2} & \omega^{2} \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
0 & -\omega & 0 \\
1 & -\omega^{2} & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & -\omega \\
0 & 1 & -\omega^{2}
\end{array} \right) $\\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(2,2){$3$} \put(2,-2){$t$}
\end{picture}&
 $ \left( \begin{array}{ccc}
0 & 1 & -1 \\
1 & 0 & 1 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
\omega & 0 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) $ & $ \left( \begin{array}{ccc}
0 & -1 & -\omega \\
0 & 1 & 0 \\
1 & 1 & -\omega^{2}
\end{array} \right) $\\
\begin{picture}(42,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}\put(39,3){\circle{4}}
\put(38,2){$3$} \put(2,-2){$\sigma_{1}$} \put(20,-2){$s$}
\put(38,-2){$t$} \put(23,3){\vector(1,0){8}}
\put(31,3){\line(1,0){6}}\put(5,3){\line(1,0){14}}
\end{picture}&
 $ \left( \begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array} \right) $ &
$ \left( \begin{array}{cc}
\omega & -1 \\
0 & 1
\end{array} \right) $ & $ \left( \begin{array}{cc}
0 & -\omega\\
1 & -\omega^{2}
\end{array} \right) $ \\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}} \put(13,3){\line(1,0){6}}
\put(2,2){$3$} \put(20,2){$3$} \put(2,-2){$t$} \put(20,-2){$u$}
\put(11,5){$4$} \put(10,-2){$\alpha$}
\end{picture} &
 $\left(
\begin{array}{cccccc}
  0 & 1 & 0 & 0 & -\omega^{2} & \omega \\
  1 & 0 & 0 & 0 & \omega^{2} & -\omega \\
  0 & 0 & 0 & 1 & \omega & \omega \\
  0 & 0 & 1 & 0 & -\omega & -\omega \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$ &  $\left(%
\begin{array}{cccccc}a \\
  \omega & 0 & 0 & 0 & 0 & \omega \\
  0 & 0 & 0 & 0 & 1 & -\omega \\
  0 & 1 & 0 & 0 & 0 & \omega \\
  0 & 0 & 0 & \omega & 0 & -\omega \\
  0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$& $\left(%
\begin{array}{cccccc}
  \omega & 0 & -\omega & 0 & 0 & 0 \\
  0 & \omega & \omega & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & -1 & 0 & 0 & \omega \\
  0 & 0 & \omega^{2} & \omega^{2} & 0 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
\end{array}%
\right)$\\
\begin{picture}(25,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}
\put(5,3){\vector(1,0){8}} \put(13,3){\line(1,0){6}}
 \put(20,2){$3$} \put(2,-2){$s$} \put(20,-2){$t$}
\put(11,5){$4$} \put(10,-2){$\alpha$}
\end{picture} & $\left(
\begin{array}{cccccc}
  -1 & 0 & -\omega & 0 & 0 & 0 \\
  0 & -1 & -1 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & -\omega^{2} & -1 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 & 1 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
\end{array}
\right)$ &  $\left(%
\begin{array}{cccccc}
  \omega & 0 & 0 & -1 & 0 & -1 \\
  0 & 0 & 0 & 0 & 1 & 1 \\
  0 & 1 & 0 & -\omega & 0& -1 \\
  0 & 0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 1 & \omega & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$& $\left(%
\begin{array}{cccccc}
  0 &0 & 0 & 1 & -\omega^{2} & 0 \\
  1 & 0 & 0 & 0 & \omega^{2} & 1 \\
  0 & 0 & \omega & 0 & \omega & \omega \\
  0 & 1 & 0 & 0 & 0 & -1 \\
  0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$\\
\begin{picture}(42,10)
\put(3,3){\circle{4}} \put(21,3){\circle{4}}\put(39,3){\circle{4}}
\put(2,-2){$s$} \put(20,-2){$t$}
\put(38,-2){$\sigma$}\put(5,3){\vector(1,0){8}}
\put(13,3){\line(1,0){6}}
\end{picture}&
 $ \left(%
\begin{array}{cccccccc}
  -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
  0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$ & $ \left(%
\begin{array}{cccccccc}
  0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
\end{array}%
\right)$ & $ \left(%
\begin{array}{cccccccc}
  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
  0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
  0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}%
\right)$ \\ \hline
\end{tabular}
\end{tiny}
\caption{Irreducible Representations of $ST_{26}$}
\end{center}
\end{table}

The group $ST_{26}$ has irreducible representations of degrees
$$\underbrace{1,1,1,1,1,1},\underbrace{2,2,2,2,2,2},\underbrace{3,3,3,3,3,3},
\overline{\underbrace{3,3,3,3,3,3}},\underbrace{6,6,6,6,6,6},\overline{\underbrace{6,6,6,6,6,6}},
\underbrace{8,8,8,8,8,8},$$
$$\underbrace{3,3},\underbrace{9,9},\overline{\underbrace{9,9}}.$$

Thus the representations given in Table 9 lead to all the
irreducible representations except the final ones of degrees 3 and
9.




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\end{document}
