Séminaire Lotharingien de Combinatoire 52 (2004), Article B52g

MACDONALD REPRESENTATIONS OF COMPLEX REFLECTION GROUPS

ALUN O MORRIS AND PATRICK MWAMBA

To Alain Lascoux to celebrate his sixtieth birthday and his massive contribution to algebraic combinatorics and representation theory

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.

1. Introduction.

In a now classic paper, I G Macdonald [18] 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 [17] 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 [24] 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 ≥ 2 and p = 1 if n = 1 and 34 exceptional cases which we denote ST_{i}, 4 ≤ i ≤ 37.

In the case of Weyl groups or finite Coxeter groups the theory is well developed and documented, see for example, [2]. 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 [3], [4], [23] and [25]. In addition, the most significant work on complex reflection groups in recent years has originated in the work of M. Broué, G. Malle et al, see for example, [5], [6], 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 [10] 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 [13], [14] and H Can [8], 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 [10]. 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 [26] 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.

2. Preliminaries.

In this section, the basic definitions and notation required later are given following the approach in [10], [13].

Let V be a complex vector space of dimension n. A reflection in V is a linear transformation of V of finite order with exactly (n − 1) eigenvalues equal to 1. A 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 ⊂ 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) 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) 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 ∈ V, a ≠ 0 and a primitive m-th root of unity ζ such that s_{a,m}x = x − (1 − ζ)(x, a)(a, a)^{−1}a for all x ∈ 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 θ_{G} : V → ℕ by θ_{G}(a) = |G_{W} |, where W =< a >^{⊥}, a ∈ V. The number θ_{G}(a) is called the order of a (with respect to G).

Each reflection s_{a,m} fixes a unique reflecting hyperplane

H_{s_{a,m}} = {x ∈ V | (x, a) = 0} = {x ∈ V | s_{a,m}x = x}.

Clearly H_{s_{ζ^{i}a,m}} = H_{s_{a,m}}^{i} = H_{s_{a,m}}, 0 ≤ i ≤ m − 1. For each of the reflecting hyperplanes H, choose a functional α_{H} ∈ V^{∗}, the dual space of V, such that ker(α_{H}) = H, that is, H = {x ∈ V | α_{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 A on which the group G acts in a natural way. Let C denote the set of G-orbits under this action, thus A = ∪_{C∈C}C. The order e_{H} depends on the orbit C = G.H ∈ C; thus whenever it is convenient we write e_{C} in place of e_{H} and ζ_{H} = ζ_{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 [10]. (i) A vector graph is a pair (B, θ), where B is a non-empty finite subset of ℂ^{∞}, the vector space with standard unitary inner product ( , ), such that for all a, b ∈ B, |(a, b)| = 1 if and only if a = b and θ is a map from B to ℕ \ {1}. We say that B is the set of vertices and θ(a), for a ∈ B, is the order of a. A vector graph (B, w) is represented by a directed value graph by assigning to each element a ∈ B a node a with weight θ(a) and if (a, b) ≠ 0, 1 a directed edge from a to b with weight (a, b). For example, if B = {a, b}, θ(a) = m and θ(b) = p and (a, b) = α, then the vector graph is

[figure: m; p; a; b; α]

We adopt the following conventions: if m = 2 the number 2 is omitted, if α ∈ ℝ the arrow is omitted and if p = m and α = −1/2, the value −1/2 is omitted.

(ii) Let π = (B, θ) be a vector graph. Then, we define dim π to be the dimension of the vector space spanned by B, and W (π) to be the group generated by all the reflections s_{a,θ(a)} for a ∈ B. The vector graph π is called a root graph if dim π = |B| and W (π) is a finite reflection group. Root graphs π = (B, θ) and π′ = (B′, θ′) are equivalent if the groups W (π) and W (π′) are conjugate.

(iii) For any root graph π = (B, θ) and for any w ∈ W (π), let wπ = (B_{w}, θ_{w}), where B_{w} = wB and θ_{w}(w(a)) = θ(a) with a ∈ B, then wπ is also a root graph which is equivalent to π since s_{w(a),θ_{w}(w(a))} = ws_{a,θ(a)}w^{−1} for all a ∈ B it follows that W (wπ) = wW (π)w^{−1} = W (π).

(iv) We say that π is irreducible if W (π) is irreducible (or that π is connected). The vector graph π is said to be congruent to the vector graph π′ = (B′, θ′) if there is a t ∈ Gl(ℂ^{∞}) such that θ′(ta) = θ(a) for a ∈ B and the elements of B are eigenvectors of t.

(v) A pair (R, f) is called a pre-root system if R is a subset of non-zero elements of ℂ^{∞} and f : R → ℕ \ {1} such that for all a, b ∈ R, s_{a,f(a)}R = R and f(s_{a,f(a)}a) = f(a) . To Σ = (R, f) is associated the reflection group W (Σ) defined by W (Σ) = < s_{a,f(a)}|a ∈ R >.

(vi) A pre-root system Σ is called a root system if in addition αa ∈ R if and only if αa ∈ W (Σ)a for all a ∈ R, α ∈ ℂ.

We make the following remarks which have been proved in [10].

Remark 2.1. Every root graph defines a pre-root system, for if π = (B, θ) is a root graph, then Σ = (R, f), where R = W (π)B, f : R → ℕ \ {1} is induced by the order function o_{W (π)} defines a pre-root system with W (π) = W (Σ).

Remark 2.2. Every finite reflection group yields a pre-root system and every pre-root system contains a root system, that is, if Σ = (R, f) is a pre-root system, then there is a root system Φ = (S, g) with W (Φ) = W (Σ), S ⊂ R and g = f|_{S}.

Remark 2.3. Every reflection group has a root system, but not every root system is obtained in the above way from a root graph.

Remark 2.4. If a root system Φ is the pre-root system obtained from a root graph π as described in Remark 2.1, then π is called a simple system in Φ. Can [8] has shown that if w ∈ W then wπ = (wB, θ_{w}), where θ_{w}(w(a)) = θ(a), a ∈ B is a root graph which is equivalent to π which yields the same root system Φ and so wπ is another simple system in Φ. Hence, the number of simple systems in Φ is equal to the order of W (π). If Φ is a root system with simple system π, then the graph associated to π is called a Cohen-Dynkin diagram of Φ.

A group G of unitary automorphisms of V is said to be imprimitive if V is a direct sum V = V_{1} ⊕ ⋯ ⊕ V_{k} of non-trivial proper subspaces V_{i}(1 ≤ i ≤ k) of V such that {V_{i} | 1 ≤ i ≤ k} is invariant under G. If such a direct splitting of V does not exist, G is said to be primitive.

Let S_{n} be the (symmetric) group of all n × n permutation matrices and let A(m, p, n) be the set of all diagonal n × n matrices with ζ^{ρ_{i}}, ρ_{i}∈ ℤ in the (i, i) position, where ζ is a primitive mth root of unity and ∑_{i=1}^{n} ρ_{i} ≡ 0 (mod p). Define G(m, p, n) = A(m, p, n)⋊S_{n}, then the imprimitive reflection groups in V are of the form G(m, p, n), where p|m.

Remark 2.5. (i) G(m, m, 2) is conjugate to W (I_{2}(m)) (notation of [2]), the dihedral group of order 2m.

(ii) G(1, 1, n) = W (A_{n−1}) ≅ 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 ≠ 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.

A root system for G(m, p, n) may be defined as follows. Let ζ be a primitive m-th root of unity and let {ϵ_{1}, …, ϵ_{n}} be the standard basis for the complex vector space ℂ^{n}. If d ∈ ℕ, let μ_{d} be the group of d-th roots of unity. Let

Ω_{l}(m, n) = {ϵ_{i} − ζ^{a}ϵ_{j}, 1 ≤ i, j ≤ n, i ≠ j, a ∈ ℤ/mℤ}.

and let

R_{l}(m, m, n) = \{ \begin{matrix}μ_{m}Ω_{l}(m, n) & if m is even, \\ μ_{2m}Ω_{l}(m, n) & if m is odd. \end{matrix}

Let f_{l}(r) = 2 for all r ∈ R_{l}(m, m, n). Then Φ_{l}(m, m, n) = (R_{l}(m, m, n), f_{l}) is a root system for G(m, m, n). Let

Ω_{s}(n) = {ϵ_{i}, 1 ≤ i ≤ n}

and let R_{s}(m/p, n) = μ_{m/p}Ω_{s}(n). Let f_{s}(r) = m for all r ∈ R_{s}(m/p, n). Let Φ_{s}(m/p, n) = (R_{s}(m/p, n), f_{s}). Then

Φ(m, p, n) = Φ_{l}(m, m, n) ⋃ Φ_{s}(m/p, n)

is a root system for G(m, p, n) for p ≠ m. Let

π(m, 1, n) = {α_{1} = ϵ_{1} − ϵ_{2}, …, α_{n−1} = ϵ_{n−1} − ϵ_{n}, α_{n} = ϵ_{n}}

where, θ(α_{i}) = 2, 1 ≤ i ≤ n − 1, θ(α_{n}) = m and

π(m, m, n) = {ϵ_{1} − ϵ_{2}, …, ϵ_{n−1} − ϵ_{n}, ϵ_{n−1} − ζϵ_{n}}

where, θ(α_{i}) = 2, 1 ≤ i ≤ n. Then π(m, 1, n) and π(m, m, n) are vector graphs and so, simple systems for G(m, 1, n) and G(m, m, n) respectively. The corresponding Cohen-Dynkin diagrams for G(m, 1, n) and G(m, m, n) are

[figure: α_{n}; α_{2}; α_{n−1}; α_{n}; m; \frac{−1}{\sqrt{2}}]

and

[figure: α_{n}; α_{2}; α_{3}; α_{n−2}; α_{n}; α_{n−1}; e^{πi/m}cos(π/m)]

respectively. However, as pointed out by Can [8], based on the work of Popov [22], the vector graph π(m, p, n) represented by

[figure: q; α_{1}; α_{2}; α_{3}; α_{4}; α_{n−1}; α_{n+1}; α_{n}; e^{πi/m}cos(π/m); \frac{−1}{\sqrt{2}}]

is a vector graph for G(m, p, n), where if q = m/p

π(m, p, n) = {α_{1} = −ϵ_{1}, α_{2} = ϵ_{1} − ϵ_{2}, …, α_{n} = ϵ_{n−1} − ϵ_{n}, α_{n+1} = ϵ_{n−1} − exp(2πi/q)ϵ_{n}}

However, it is not a simple system for the corresponding root system Φ(m, p, n) as this set is clearly not linearly independent over ℂ.

3. Subsystems and primitive systems.

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

Now, as in Hughes [13] and Can [8], primary systems for root systems Φ with simple system π = (B, θ) are defined. These play the role of positive systems for real reflection groups. Let B = {a_{1}, …, a_{n}}, and put r_{i} = s_{a_{i},θ(a_{i})}, 1 ≤ i ≤ n, then the corresponding primary system is defined inductively as follows:

(i) Let Ω_{1} = B.

(ii) Let Ω_{2} = {r_{i}(a_{j}) | 1 ≤ i, j ≤ n, i ≠ j, a_{j} ∈ Ω_{1}, r_{i}(a_{j}) ∉ Ω_{1}}.

(iii) For k ≥ 3, let

Ω_{k} = {r_{i}(a) | 1 ≤ i, j ≤ n, i ≠ j, a ∈ Ω_{k−1}, r_{i}(a) ≠ μb for any b ∈ Ω_{l}, l < k}, where μ is a root of unity.

Then Ω = ⋃_{k≥1} Ω_{k} is a primary system for the simple system π of the root system Φ. The elements of Ω_{k} are called primary roots of height k. The elements of Ω_{k} for the largest k for which Ω_{k} ≠ ∅ are called highest roots.

Remark 3.1. The primary system is not unique in that there is an element of choice at each step. However, having fixed a primary system Ω for the simple system π, if the simple system is replaced by wπ, then [8] the corresponding primary system obtained by making the same choices in the above algorithm is the conjugate wΩ of Ω. Thus, the choice of primary system is of no consequence. In fact, different choices in the above algorithm would result in conjugate primary systems.

Remark 3.2. The roots of maximal height are not in general unique for complex reflection groups. For example, [14], J_{3}(4) has a unique highest root of height 6, but L_{4} has two highest roots of height 9.

Remark 3.3. In the case of real reflection groups, the primary systems are positive systems and the highest roots are the longest roots.

Hughes [13] 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 extended Cohen-Dynkin diagram of a root system Φ is formed by attaching the negative of a highest root to a Cohen-Dynkin diagram for Φ 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 [14], a complete list of subsystems is given for spaces of dimension ≥ 3. Can [8] has given an alternative algorithm for obtaining a complete set of subsystems. In particular, he has in [9] 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

(3.1)  ∑_{i=1}^{m} ∑_{j=1}^{s_{i}} B_{λ_{j}^{(i)}}^{m_{i}} + ∑_{j=1}^{s} D_{μ_{j}}^{m}, ∑_{j=1}^{s_{1}}(λ_{j}^{(1)} + 1) + ∑_{i=2}^{m} ∑_{j=1}^{s_{1}} λ_{j}^{(i)} + ∑_{j=1}^{s} μ_{j} = n

and

(3.2)  ∑_{i=1}^{m} ∑_{j=1}^{s_{i}} D_{λ_{j}^{(i)}}^{m_{i}}, ∑_{j=1}^{s_{1}}(λ_{j}^{(1)} + 1) + ∑_{i=2}^{m} ∑_{j=1}^{s_{1}} λ_{j}^{(i)} = n

respectively, where m_{1} = 1 and m_{i} = m, (i = 2, …, 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 [19].

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

Shephard Cohen-Dynkin  Extended
Todd type  diagram  iα  iα′  C-D diagram

[figure: ST_{4}; 3; 3; s; t; 3; α; \frac{1}{\sqrt{3}}; \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}; α; α; α; 3; 3; 3]

[figure: ST_{5}; 3; 3; s; t; 4; α; \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}; \frac{1}{\sqrt{3}}; α; α′; 3; 3; 3]

[figure: ST_{6}; 3; s; t; 6; α; (\frac{1}{2}(1 + \frac{1}{\sqrt{3}}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \frac{1}{\sqrt{3}}))^{\frac{1}{2}}; \frac{−i}{\sqrt{3}}; α′; α; 3; 3]
[figure: α′; α; 3]
[figure: ST_{8}; 4; 4; s; t; 3; α; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}}; α; α′; 4; 4; 4]
[figure: ST_{9}; 4; s; t; 6; α; (\frac{1}{2}(1 + \frac{1}{\sqrt{2}}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \frac{1}{\sqrt{2}}))^{\frac{1}{2}}; α; α′; 4; 4]
[figure: α′; α; 4]
[figure: ST_{10}; 4; 3; s; t; 4; α; (\frac{1}{2}(1 + \frac{1}{\sqrt{3}}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \frac{1}{\sqrt{3}}))^{\frac{1}{2}}; α; α′; 4; 3; 4]
[figure: α′; α; 3; 4; 3]
[figure: ST_{14}; 3; s; t; 8; α; (\frac{1}{2}(1 + \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \begin{matrix}√ & \overline{2} \\ √ & \overline{3} \end{matrix}))^{\frac{1}{2}}; α; α′; 3; 3]
[figure: α′; α; 3]
[figure: ST_{16}; 5; 5; s; t; 3; α; (\frac{1}{2}(1 + \frac{1}{\sqrt{5}}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \frac{1}{\sqrt{5}}))^{\frac{1}{2}}; α; α′; 5; 5; 5]

[figure: ST_{17}; 5; s; t; 6; α; (\frac{1}{2}(1 + α_{16}))^{\frac{1}{2}}; (\frac{1}{2}(1 − α_{16}))^{\frac{1}{2}}; α; α′; 5; 5]
[figure: α′; α; 5]
[figure: ST_{18}; 5; 3; s; t; 4; α; (\frac{1}{2}(1 + \frac{1}{\sqrt{3}}cot\frac{π}{5}))^{\frac{1}{2}}; (\frac{1}{2}(1 − \frac{1}{\sqrt{3}}cot\frac{π}{5}))^{\frac{1}{2}}; α; α′; 5; 3; 5]
[figure: α′; α; 3; 5; 3]
[figure: ST_{20}; 3; 3; s; t; 5; α; \frac{1}{2}(\frac{1+\sqrt{5}}{\sqrt{3}}); \frac{1}{2}(\frac{1−\sqrt{5}}{\sqrt{3}}); α; α′; 3; 3; 3]
[figure: ST_{21}; 3; s; t; 10; α; (\frac{1}{2}(1 + α_{20}))^{\frac{1}{2}}; (\frac{1}{2}(1 − α_{20}))^{\frac{1}{2}}; α; α′; 3; 3]
[figure: α′; α; 3]

Table 1. Extended Cohen-Dynkin diagrams in dimension 2

ST | Cohen Cohen-Dynkin  Extended
type  diagram  C-D diagram
[figure: ST_{24} | J_{3}(4); α; α = ^{\frac{(1−\sqrt{7}i)}{4}}; α]

[figure: ST_{25} | L_{3}; α; α; 3; 3; 3; α = \frac{−i}{\sqrt{3}}; α; α; α; 3; 3; 3; 3]

[figure: ST_{26} | M_{3}; α; β; 3; 3; α = \frac{−i}{\sqrt{3}}; α; α; β; 3; 3; 3]

β = \frac{−1}{\sqrt{2}}  [figure: α; β; 3; 3]

[figure: ST_{27} | J_{3}(5); α; α = \frac{1}{2} + ωcos \frac{π}{5}; α]

[figure: ST_{29} | N_{4}; α; α = \frac{1−i}{2}; α]

ST_{31} | EN_{4}  α = \frac{1−i}{2}  [figure: α]

[figure: ST_{32} | L_{4}; α; α; α; 3; 3; 3; 3; α = \frac{−i}{\sqrt{3}}; α; α; α; α; 3; 3; 3; 3; 3]

[figure: α; α; α; α; 3; 3; 3; 3; 3]
ST_{33} | K_{5}  [figure: α]  α = \frac{−ω}{2}

[figure: α; α]

ST_{34} | K_{6}  [figure: α] α = \frac{−ω}{2}

[figure: α; α]

Table 2. Extended Cohen-Dynkin diagrams in dimension > 2

4. 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 ∈ R, g ∈ G, x ∈ V. In particular,

(gα_{H})(x) = χ_{H}^{−1}α_{H}(x)

for all g ∈ G_{H}, x ∈ V, where G̃_{H} = {χ_{H}^{−i} | 0 ≤ i ≤ e_{H} − 1} is the character group of G_{H}. Define π_{C} = ∏_{H∈C} α_{H} for C ∈ C and π_{G} = ∏_{C∈C} π_{C}.

When reflection subgroups are determined by subsystems of root systems, the functions π_{G} defined earlier will be denoted π_{Φ}.

We give three examples

Example 4.1. For the real reflection group of type A_{n−1} (symmetric groups of order n!), the root system is {ϵ_{i} − ϵ_{j}, 1 ≤ i, j ≤ n} and a simple system is {ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ n − 1} and corresponding primary (positive) system {ϵ_{i} − ϵ_{j}, 1 ≤ i < j ≤ n}. Then

π_{A_{n−1}}(x) = ∏_{1≤i<j≤n}(x_{i} − x_{j})

which is the Vandermonde determinant.

Example 4.2. 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

π_{G(m,m,n)}(x) = ∏_{1≤i<j≤n} ∏_{a=1}^{m}(x_{i} − ζ^{a}x_{j}) = ∏_{1≤i<j≤n}(x_{i}^{m} − x_{j}^{m})

and

π_{G(m,1,n)}(x) = ∏_{1≤i<j≤n} ∏_{a=1}^{m}(x_{i} − ζ^{a}x_{j}) ∏_{i=1}^{n} x_{i} = ∏_{1≤i<j≤n}(x_{i}^{m} − x_{j}^{m}) ∏_{i=1}^{n} x_{i}.

Example 4.3. For the complex reflection group of type ST_{4}, a simple system is [15] \{ϵ_{1}, −i/\sqrt{3}(ϵ_{1} + \sqrt{2}ϵ_{2})\} with primary system Ω_{ST 4} = \{−i/\sqrt{3}(ϵ_{1} + ω^{k}\sqrt{2}ϵ_{2}), 0 ≤ k ≤ 2\} and root system μ_{6}Ω_{ST 4}. Then

π_{ST 4}(x) = x_{1}(x_{1} − \sqrt{2}x_{2})(x_{1} − ω\sqrt{2}x_{2})(x_{1} − ω^{2}\sqrt{2}x_{2}) = x_{1}(x_{1}^{3} + 2\sqrt{2}x_{2}^{3})/3\sqrt{3}

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.

Lemma 4.1.

s_{H}π_{C} = \{ \begin{matrix}π_{C} & if H ∉ C \\ (χ_{H})^{−1}π_{C} & if H ∈ C. \end{matrix}

Proof. If H′ ∉ C, let v_{0} ∈ H′ be a point, thus α_{H}(v_{0}) ≠ 0 for all H ∈ C which implies that π_{C}(v_{o}) = ∏_{H∈C} α_{H}(v_{0}) ≠ 0. Hence

(s_{H′}π_{C})(v_{0}) = π_{C}(s_{H′}^{−1}v_{0}) = π_{C}(v_{0}).

Furthermore, if H ∈ C, then

s_{H}π_{C} = ∏_{H∈C} s_{H}α_{H} = (\frac{s_{H}α_{H}}{α_{H}} ) π_{C} = χ_{H}^{−1}π_{C}.

Let \overline{π}_{C} = ∏_{H′∈C,H′≠H} α_{H′} and let v_{0} ∈ H be a point such that α_{H′}(v_{0}) ≠ 0 for all H′ ≠ H, then \overline{π}_{C}(v_{0}) ≠ 0. Hence

(s_{H} \overline{π}_{C})(v_{0}) = \overline{π}_{C}(s_{H}^{−1}v_{0}) = \overline{π}_{C}(v_{0}),

which proves the lemma.

Let R^{G} be the subring of G-invariant elements in R. Then, it is well known [2] that R^{G} is generated by n algebraically independent homogeneous elements p_{1}, …, p_{n}. If d_{i} = deg(p_{i}), then

(4.1)  |G| = ∏_{i=1}^{n} d_{i}

(4.2)  |complex reflections in G| = ∑_{i=1}^{n}(d_{i} − 1) = ∑_{H∈A}(e_{H} − 1).

(In fact, [21] if m_{i} = d_{i} − 1 are the exponents and n_{i} are the coexponents of G, then

(4.3)  |complex reflections in G| = ∑_{i=1}^{n} m_{i}

(4.4)  |reflecting hyperplanes| = ∑_{i=1}^{n} n_{i}.)

If χ is a linear character of G, let R_{χ}^{G} be the R^{G}-module of relative invariants of G introduced by R. Stanley [26], that is,

R_{χ}^{G} = {f ∈ R | gf = χ(f)f for all g ∈ G.}

Then Stanley has proved, amongst other things, that if f ∈ R_{χ}^{G}, then f is divisible by f_{χ} = ∏_{C∈C} π_{C}^{l_{C}}, where 0 ≤ l_{C} ≤ e_{C} − 1 and l_{C} is the least positive integer such that χ(s_{C}) = ζ_{C}^{l_{C}}, for a fixed generator s_{C} ∈ G_{H} for some H ∈ C. Furthermore, R_{χ}^{G} = f_{χ}R^{G} and f_{χ} ∈ R_{χ}^{G}, that is,

gf_{χ} = χ(g)f_{χ} for all g ∈ G.

Now let G′ be a reflection subgroup of G and let χ also denote the restriction of the linear character χ to G′. Let f′_{χ} be defined as above for the subgroup G′. Let P_{G′}^{χ} be the subspace of R generated by the polynomial functions gf′_{χ} for all g ∈ G, then the vector space P_{G′}^{χ} is a ℂG-module. The special case χ = 1 will be denoted by P_{G′}. Then we have the following theorem which generalizes the well known result of I G Macdonald [18] 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 [17].

Theorem 4.2. (i) The module P_{G′}^{χ} is an absolutely irreducible ℂG-module. (ii) If G′ and G′′ are two non-isomorphic reflection subgroups of G with

|reflecting hyperplanes for G′| ≠ |reflecting hyperplanes for G′′|,

then the modules P_{G′}^{χ} and P_{G′′}^{χ} are not isomorphic.

(iii) The modules P_{G′}^{χ} and χ ⊗ P_{G′} are isomorphic.

Proof. (i) Let ϕ : P_{G′}^{χ} → P_{G′}^{χ} be a ℂG-homomorphism, then

gϕ(f′_{χ}) = ϕ(gf′_{χ}) = ϕ(χ(g)f′_{χ}) = χ(g)f′_{χ}

for all g ∈ G and so, by the above result ϕ(f′_{χ}) is divisible by f′_{χ}. Since ϕ(f′_{χ}) and f′_{χ} are of the same degree and so ϕ(f′_{χ}) is a scalar multiple of f′_{χ} and (i) follows.

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

(iii) The map ϕ : P_{G′}^{χ} → χ ⊗ P_{G′} is ϕ(f_{χ}) = χ ⊗ f_{1}, which is easily seen to be the required isomorphism.

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

Some examples are now given to illustrate the usefulness of the above approach.

Example 4.4. 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 (λ^{(1)}, …, λ^{(m)}) of n; thus we need only take the subsystems of type ∑_{i=1}^{m} ∑_{j=1}^{s_{i}} B_{λ_{j}^{(i)}}^{m_{i}} listed in (3.1) - here λ^{(i)} is the partition

(λ_{1}^{(i)}, …, λ_{s_{i}}^{(i)}). The representations of the generalized symmetric group G(m, 1, n) have been considered by Can [7] and Hughes [11] 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.

Example 4.5. We consider the group ST_{4}. As was seen in Table 1, this group has Cohen-Dynkin diagram

[figure: 3; 3; s; t; 3; α]

and extended Cohen-Dynkin diagram

[figure: α; α; α; 3; 3; 3] ,

where α = −i/\sqrt{3}. Thus, in this case, there are only 3 non-conjugate subsystems

[figure: 3; 3; 3; α]  ^{_{3}}  ∅.

From Table 3, we see that ST_{4} has three linear characters χ_{1}, χ_{2} and χ_{3} and since χ_{5} = χ_{4} ⊗ χ_{2} and χ_{6} = χ_{4} ⊗ χ_{3}, thus the only modules required are those corresponding to χ_{1}, χ_{4} and χ_{7} of degrees 1,2 and 3 respectively. The first two are obtained from the

subsystems [figure: 3; 3; 3; α] and ^{_{3}} of ST_{4}. The third is obtained by using the fact that ST_{4} is

a subgroup of ST_{6} with Cohen-Dynkin diagram [figure: 3; 6; β], where β = −i( \frac{1}{2} (1 + \frac{1}{\sqrt{3}} )) \frac{1}{2} . The representation of degree 3 is the representation of ST_{6} corresponding to the subsystem

[figure: 3; 3] which remains irreducible on restriction to ST_{4}, indeed this subsystem is also a subsystem of ST_{4}.

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
χ_{1} 1 1 1 1 1 1 1
χ_{2}  1 1 ω ω^{2} ω ω^{2} 1
χ_{3}  1 1 ω^{2} ω ω^{2} ω 1
χ_{4}  2 -2 1 1 -1 -1 0
χ_{5}  2 -2 ω ω^{2} −ω −ω^{2} 0
χ_{6}  2 -2 ω^{2} ω −ω^{2} −ω 0
χ_{7}  3 3 0 0 0  0 -1

Table 3. Character Table of ST_{4}

The irreducible representations obtained from the three subsystems are given in Table 4.

Subsystem Basis  s  t
[figure: 3; 3; s; t; 3; α]  {s}  (ω)  (ω)
 ; 3 ; s  \{s, τ_{t}s\} ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )  ( \begin{matrix}ω & ω \\ 0 & 1 \end{matrix} )
[figure: 3; 3; s; t] \{s, τ_{t}s\} ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}0 & 0 & −1 \\ −1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )
∅  (1)  (1)

Table 4. Irreducible Representations of ST_{4}

Example 4.6. Some more general results may be obtained for an arbitrary two-dimensional group. For an arbitrary root system

[figure: m; n; k; α; u; v]

let ζ and η denote primitive m-th and n-th roots of unity respectively. For the subsystem

S =  ; m ; u

if s = τ_{u} and t = τ_{v}, then π_{S} = u and P_{S} =< u, tu = u + (1 − η)αv >, and we see that

s ⟼ ( \begin{matrix}ζ & −(1 − ζ)(1 + (1 − η)α^{2}) \\ 0 & 1 \end{matrix} ) and t ⟼ ( \begin{matrix}0 & −η \\ 1 & 1 + η \end{matrix} ) .

It can be shown that the subsystem ; m ; u gives an equivalent representation.

The explicit results for the separate root systems are listed in Table 5; thus we have the ’basic’ representations of degree two for all the two-dimensional reflection groups. In Table 5, ω, i, η and ϵ are primitive cube, fourth, fifth and eigth roots of unity respectively

and α = −i\sqrt{\frac{1}{2} (1 + \frac{1}{\sqrt{5}} )}.

ST-type  s  t  ST-type  s  t
ST_{4}  ( \begin{matrix}ω & −1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )  ST_{14}  ( \begin{matrix}ω & \sqrt{2}iω^{2} \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )
ST_{5}  ( \begin{matrix}ω & ω^{2} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )  ST_{16}  ( \begin{matrix}η & −1 \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −η \\ 1 & 1 + η \end{matrix} )
ST_{6}  ( \begin{matrix}ω & iω^{2} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )  ST_{17}  ( \begin{matrix}η & (1 − η)α \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )
ST_{8}  ( \begin{matrix}i & −1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −i \\ 1 & 1 + i \end{matrix} ) ST_{18}  ( \begin{matrix}η & ω^{2} \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )
ST_{9}  ( \begin{matrix}i & −ϵ^{3} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )  ST_{20}  ( \begin{matrix}ω & ω^{2} − ω(η + η^{4}) \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )
ST_{10}  ( \begin{matrix}i & ω^{2} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )  ST_{21}  ( \begin{matrix}ω & iω^{2}(1 + η + η^{4}) \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

Table 5. Irreducible Representations of degree 2

ST-type  s  t
ST_{4}  ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}0 & 0 & −1 \\ −1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )

ST_{5}  ( \begin{matrix}ω & 0 & −1 \\ 0 & 0 & −ω^{2} \\ 0 & 1 & −ω \end{matrix} ) ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )

ST_{6}  ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

ST_{8}  ( \begin{matrix}i & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )  ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & i \end{matrix} )

ST_{9}  ( \begin{matrix}i & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} )

( \begin{matrix}0 & 0 & −i \\ 1 & 0 & 1 \\ 0 & 1 & i \end{matrix} ) ( \begin{matrix}−1 & −i & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

Table 6. Irreducible Representations of degree 3

Example 4.7. The irreducible representations of degree three are obtained by taking the

subsystems [figure: m; m] and [figure: n; n] of the root system [figure: m; n]. 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.

Further examples are given in the Appendix below.

5. 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.

Subsystem  s  t  u
_{s}  ( \begin{matrix}−1 & 0 & −1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & 1 & −β \\ 1 & 0 & β \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ᾱ & 1 \\ 0 & 1 & 0 \\ 1 & ᾱ & 0 \end{matrix} )

[figure: s; t; σ] ( \begin{matrix}−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{matrix} ) ( \begin{matrix}−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{matrix} ) ( \begin{matrix}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{matrix} )

[figure: s; σ]  ( \begin{matrix}−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{matrix} )  ( \begin{matrix}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{matrix} )  ( \begin{matrix}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{matrix} )

[figure: s; u]  ( \begin{matrix}−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{matrix} ) ( \begin{matrix}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{matrix} )  ( \begin{matrix}−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{matrix} )

Table 7. Irreducible Representations of ST_{24}

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 [1].

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

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.

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.

Subsystem  s  t  u
 ; 3 ; s  ( \begin{matrix}ω & −1 & −1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ω & 0 \\ 1 & −ω^{2} & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & −ω \\ 0 & 1 & −ω^{2} \end{matrix} )

[figure: 3; 3; 3; s; u; σ]  ( \begin{matrix}ω & −1 \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )  ( \begin{matrix}ω & −1 \\ 0 & 1 \end{matrix} )

[figure: 3; 3; s; u]  ( \begin{matrix}ω & 0 & 0 & −1 & 0 & ω^{2} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & −ω & 0 & −ω \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & ω & 0 & ω \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 0 & 0 & 1 & −ω^{2} & −ω \\ 1 & 0 & 0 & 0 & ω^{2} & ω \\ 0 & 0 & ω & 0 & ω & −ω \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}ω & ω & 0 & 0 & −ω^{2} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & −1 & ω \\ 0 & ω^{2} & ω^{2} & 0 & ω^{2} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} )

[figure: 3; 3; s; t; 3; α]  ( \begin{matrix}ω & 0 & 0 & 0 & −1 & −ω^{2} \\ 0 & ω & 0 & 0 & ω^{2} & 0 \\ 0 & 0 & 0 & 0 & −ω^{2} & −1 \\ 0 & 0 & 0 & ω & ω & ω^{2} \\ 0 & 0 & 1 & 0 & −ω & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}ω & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & −ω & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & ω & ω & 0 \\ 0 & 0 & 1 & ω & 0 & 0 \end{matrix} )  ( \begin{matrix}0 & 0 & 0 & 1 & 0 & −ω^{2} \\ 1 & 0 & 0 & 0 & 0 & ω^{2} \\ 0 & 0 & ω & 0 & 0 & ω \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ω & −ω \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

Table 8. Irreducible Representations of ST_{25}

References

[1] M.Benard, Schur indices and splitting fields of the unitary reflection groups, J. of Algebra 38 (1976), 318–347.

[2] N.Bourbaki, Groupes et algèbres de Lie, Chap IV, V, VI (Hermann, Paris, 1968).

[3] K.Bremke and G.Malle, Reduced words and a length function for G(e, 1, n), Indag. Math. 8 1997, 453–469.

[4] K.Bremke and G.Malle, Root systems and length functions, Geometriae Dedicata 72 (1998), 83–97.

[5] M.Broué and G.Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189.

[6] M.Broué, G.Malle and R.Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. reine angew. Math. 500 (1998), 127–190.

[7] H.Can, Representations of the generalized symmetric groups, Beiträge zur Algebra und Geometrie 37 (1996), 289–307.

[8] H.Can, Some combinatorial results for complex reflection groups, Europ. J. Combinatorics 19 (1998), 901–909.

[9] H.Can, A new approach to the construction of sybsystems of complex root systems, (to appear).

[10] A.M.Cohen, Finite complex reflection groups, Ann. scient. Éc. Norm. Sup., 9 (1976), 379–436.

[11] M.C.Hughes, Representations of complex imprimitive reflection groups, Math. Proc. Camb. Phil. Soc., 94 (1983), 425–436.

[12] M.C.Hughes, The conjugacy classes of complex imprimitive reflection groups, Proc. Ned. Akad. v. Wet., A 88 (1985), 313–330.

[13] M.C.Hughes, Complex reflection groups, Comm. in Algebra 18 (1990), 3999–4029.

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[15] M.C.Hughes and A.O.Morris, Root systems for two dimensional complex reflection groups, Sèminaire Lotharingen de Combinatoire 45 (2001), Article B45e (18 pages).

[16] D.W.Koster, Complex reflection groups, Ph.D. thesis, University of Wisconsin, 1975.

[17] G.Lusztig and N.Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), 41–52.

[18] I.G.Macdonald, Some irreducible representations of Weyl groups, Bull. London Math. Soc. 4 (1972), 148–150.

Subsystem  s  t  u
_{s}  ( \begin{matrix}−1 & ω^{2} & ω^{2} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ω & 0 \\ 1 & −ω^{2} & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & −ω \\ 0 & 1 & −ω^{2} \end{matrix} )

 ; 3 ; t  ( \begin{matrix}0 & 1 & −1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}ω & 0 & −1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −1 & −ω \\ 0 & 1 & 0 \\ 1 & 1 & −ω^{2} \end{matrix} )

[figure: 3; σ_{1}; s; t]  ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} )  ( \begin{matrix}ω & −1 \\ 0 & 1 \end{matrix} )  ( \begin{matrix}0 & −ω \\ 1 & −ω^{2} \end{matrix} )

[figure: 3; 3; t; u; 4; α]  ( \begin{matrix}0 & 1 & 0 & 0 & −ω^{2} & ω \\ 1 & 0 & 0 & 0 & ω^{2} & −ω \\ 0 & 0 & 0 & 1 & ω & ω \\ 0 & 0 & 1 & 0 & −ω & −ω \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}a \\ ω & 0 & 0 & 0 & 0 & ω \\ 0 & 0 & 0 & 0 & 1 & −ω \\ 0 & 1 & 0 & 0 & 0 & ω \\ 0 & 0 & 0 & ω & 0 & −ω \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}ω & 0 & −ω & 0 & 0 & 0 \\ 0 & ω & ω & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & ω \\ 0 & 0 & ω^{2} & ω^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} )

[figure: 3; s; t; 4; α]  ( \begin{matrix}−1 & 0 & −ω & 0 & 0 & 0 \\ 0 & −1 & −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & −ω^{2} & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} )  ( \begin{matrix}ω & 0 & 0 & −1 & 0 & −1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & −ω & 0 & −1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & ω & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )  ( \begin{matrix}0 & 0 & 0 & 1 & −ω^{2} & 0 \\ 1 & 0 & 0 & 0 & ω^{2} & 1 \\ 0 & 0 & ω & 0 & ω & ω \\ 0 & 1 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

[figure: s; t; σ] ( \begin{matrix}−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{matrix} ) ( \begin{matrix}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{matrix} ) ( \begin{matrix}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{matrix} )

Table 9. Irreducible Representations of ST_{26}

[19] G. Malle, Presentations for crystallographic complex reflection groups, Transformation Groups 1 (1996), 259–277.

[20] E.M.Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, MSJ-Memoirs, 8 (2000), Mathematics Society of Japan, 1–90.

[21] P.Orlik and L.Solomon, Unitary reflection groups and cohomology, Inv. Math. 59 (1980), 77–94.

[22] V.L.Popov, Discrete complex reflection groups, Comm. of the Math. Institute, vol. 15, Rijksuniversiteit Utrecht, 1982.

[23] K.Rampetas and T.Shoji, Length functions and Demazure operators for G(e, 1, n) I, Indag. Math. 9 (1998), 563–580; II, ibid. 9 (1998), 581–594.

[24] G.C.Shephard and J.A.Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

[25] T.Shoji, Length functions for G(r, p, n), Advanced Studies in Pure Mathematics, 28, (2000) in Combinatorial Methods in Representation Theory, 327–342.

[26] R.P.Stanley, Relative invariants of finite groups generated by pseudoreflections J. of Algebra 49 (1977), 134–148.

Institute of Mathematical and Physical Sciences, University of Wales, Aberystwyth, Ceredigion SY23 3BZ, Wales, U.K.

MACDONALD REPRESENTATIONS OF COMPLEX REFLECTION GROUPS 17

Department of Mathematics, University of Zambia, Lusaka, Zambia

E-mail address: alun@morus25.fsnet.co.uk