\documentclass[12pt]{amsart} \usepackage{graphicx} \usepackage{amssymb,amsmath} \textheight 23cm \textwidth 16cm \oddsidemargin = 0cm \evensidemargin = 0cm \topmargin = 0cm % ---------------------------------------------------------------- \vfuzz2pt %Don't report over-full v-boxes if over-edge is small \hfuzz2pt %Don't report over-full h-boxes if over-edge is small \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % MATH ----------------------------------------------------------- \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Real}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}} % ---------------------------------------------------------------- \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{}% % ---------------------------------------------------------------- % ---------------------------------------------------------------- \setlength{\unitlength}{0.70mm} \renewcommand{\theequation}{\thesection.\arabic{equation}} \renewcommand{\arraystretch}{1.3} \setlength{\tabcolsep}{1mm} \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 %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--first part \thispagestyle{myheadings} \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \markright{\its S\'eminaire Lotharingien de Combinatoire \bfs 52 \rms (2004), Article~B52g\hfill} \def\thepage{} % % \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. %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--restoring the headers and pagenumbering \pagenumbering{arabic} \addtocounter{page}{1} \markboth{\SMALL ALUN O MORRIS AND PATRICK MWAMBA}{\SMALL MACDONALD REPRESENTATIONS OF COMPLEX REFLECTION GROUPS} % % \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 $W=^{\bot}, a \in V$. 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 \begin{center} \begin{picture}(25,10) \put(3,6){\circle{4}} \put(21,6){\circle{4}} \put(5,6){\vector(1,0){8}} \put(13,6){\line(1,0){6}} \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 $$. (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} \begin{center} \begin{picture}(120,20) \put(2,4){\circle{4}} \put(18,4){\circle{4}} \put(44,4){\circle{4}} \put(60,4){\circle{4}} \put(24,4){\circle*{.5}} \put(31,4){\circle*{.5}} \put(38,4){\circle*{.5}} \put(4,4){\line(1,0){12}} \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} \begin{center} \begin{picture}(120,6) \put(2,4){\circle{4}} \put(18,4){\circle{4}} \put(34,4){\circle{4}} \put(62,4){\circle{4}} \put(76,10){\circle{4}} \put(76,-2){\circle{4}} \put(41,4){\circle*{.5}} \put(55,4){\circle*{.5}} \put(48,4){\circle*{.5}} \put(4,4){\line(1,0){12}} \put(20,4){\line(1,0){12}} \put(64,5){\line(2,1){10}} \put(64,3){\line(2,-1){10}} \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}} \put(64,4){\circle*{.5}} \put(4,4){\line(1,0){12}} \put(20,4){\line(1,0){12}} \put(36,4){\line(1,0){12}} \put(80,5){\line(2,1){10}} \put(80,3){\line(2,-1){10}} \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 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$, 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. \begin{thebibliography}{99} \bibitem{ber} M.Benard, {\it Schur 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