Normalisers of parabolic subgroups in finite unitary reflection groups
Muraleedaran Krishnasamy D. E. Taylor
math.GR 20F55


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thmTheorem[section]
cor[thm]Corollary
lemma[thm]Lemma
hypHypothesis
definition
defn[thm]Definition
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rem[thm]Remark
example[thm]Example

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⟨2mu=⋊kmuraleedaran@sydney.edu.audonald.taylor@sydney.edu.auSchool of Mathematics and Statistics

The University of Sydney

NSW 2006, Australia
It is well known that the normaliser of a parabolic subgroup of a finite
Coxeter group is the semidirect product of the parabolic subgroup by the
stabiliser of a set of simple roots.  We show that a similar result holds
for all finite unitary reflection groups: namely, the normaliser of a
parabolic subgroup is a semidirect product and in many, but not all, cases
the complement can be obtained as the stabiliser of a set of roots. In
addition we record when the complement acts as a reflection group on the
space of fixed points of the parabolic subgroup.


unitary reflection group complex reflection group parabolic
subgroup normaliser
2010 Mathematics subject classification: 20F55



§ INTRODUCTION

Let V be a finite-dimensional vector space over the complex numbers 
and suppose that V is equipped with a positive definite hermitian form
(-,-). A reflection is a linear transformation of V of finite
order whose fixed point space is a hyperplane. A unitary reflection
group on V is a group of linear transformations of V generated by
reflections which preserve (-,-).

In general a reflection of order m has the form

    r_a,ζ(v) = v - (1-ζ)(v,a)/(a,a) a,

where a is a non-zero element of V called a root of r_a,ζ
and ζ∈ is a primitive mth root of unity.

A subgroup of G is a reflection subgroup if it is generated by its
reflections. A parabolic subgroup of G is the pointwise stabiliser
G(X) of a subset of X of V; by a theorem of Steinberg
<cit.>G(X) is a reflection subgroup. Given a parabolic
subgroup P = G(X), let U = _V(P) be the space of fixed points of P.
Then G(U) ⊆ G(X) = P⊆ G(U) whence P = G(U). A
straightforward calculation shows that the normaliser N_G(P) of P in G
is G_U, the setwise stabiliser of U. Similarly, N_G(P) = G_U^⊥,
where U^⊥ is the orthogonal complement of U in V.


Given a parabolic subgroup P of G, it is the purpose of this
paper—extending the work in <cit.>—to describe the structure
of the normaliser N_G(P). In particular, we show that in all cases there
exists a complement H to P in its normaliser; that is, N_G(P)
is a semidirect product P H. For brevity we refer to H as a 
parabolic complement.

If G is imprimitive, a parabolic complement is constructed using the methods
of <cit.>. If G is primitive, there is almost always a set of
roots of reflections which generate P and whose stabiliser is a complement;
the exceptions are certain parabolic subgroups of the Shephard and Todd
groups G_29 and G_31 and the parabolic subgroups of rank 1 excluded
by Lemma <ref>.

In most cases the existence of a parabolic complement is obtained without resort
to computer-assisted calculations.  However, for 17 of the 212 conjugacy classes
of parabolic subgroups of the primitive reflection groups a specific but
somewhat ad hoc choice of roots is given in Table <ref>.  The
choice of roots and the verification that their setwise stabiliser is a
parabolic complement was carried out using  <cit.> code.  

The  code is part of a general
package[<http://www.maths.usyd.edu.au/u/don/details.html#programs>]
which was written to facilitate further exploration and provide some of the
information in the tables of <ref>.  The root systems and 
generating reflections used by  satisfy the defining relations
corresponding to the diagrams in 
<cit.>.  For unitary reflection groups other 
choices of generators have been proposed 
<cit.> but 
except for the parabolic complements described in Table <ref>
our results do not depend on the particular choice of generators.

Throughout the paper H^∘ denotes the subgroup of H generated by the
elements that act as reflections on _V(P) and Q is the parabolic
subgroup G(U^⊥).

The complexification of a real reflection group is a Coxeter group and in
this case the structure of the normaliser of a parabolic subgroup is well
known <cit.>. Standard facts about unitary
reflection groups can be found in <cit.> and
<cit.>.



§ REDUCTION TO THE IRREDUCIBLE CASE

Suppose that G is a finite unitary reflection group on the vector space V
and that G preserves the positive definite hermitian form (-,-). From
<cit.>V is the direct sum of pairwise
orthogonal G-invariant subspaces V_1, V_2, …, V_k such that the
restriction of G to V_i is an irreducible reflection group  G_i on
V_i and G≃ G_1× G_2×…× G_k.


Suppose that P is a parabolic subgroup of G,
where G = G_1× G_2×…× G_k, as above. For 1≤ i≤ k,
let U_i = _V_i(P) and P_i = G_i(U_i). Then

  * _V(P) = U_1⊕ U_2⊕…⊕ U_k;

  * P = P_1× P_2×…× P_k;

  * P_i is the parabolic subgroup of G_i generated by the reflections
    in P whose roots belong to V_i;

  * N_G(P) = N_G_1(P_1)× N_G_2(P_2)×…×
    N_G_k(P_k);

  *  if H_i is a complement to P_i in N_G(P_i) for 1≤ i≤ k,
then H_1× H_2×…× H_k is a complement to P in N_G(P).

(i) Suppose that g∈ G and v = v_1 +… + v_k, where v_i∈ V_i.
Since V_i is G-invariant we have g(v) = v if and only if g(v_i) = v_i
for all i. This proves (i) and since g = g_1g_2⋯ g_k with g_i∈
G_i, the same calculation shows that g∈ P if and only if g_i∈ P_i
for all i, thus proving (ii). Then (iii) follows from <cit.>.

Finally, (iv) follows from the fact that g(U) = U if and only if g_i(U_i)
= U_i for all i and then (v) follows from (iv).


As a consequence of this theorem we may restrict our attention to parabolic
subgroups of irreducible unitary reflection groups.


According to Shephard and Todd <cit.> (see
<cit.> for more recent proofs), the
finite irreducible unitary reflection groups are:

  *  the symmetric groups, (n),

  *  imprimitive groups G(m,p,n) for m,n ≥ 2, except G(2,2,2),

  *  cyclic groups,

  *  34 primitive groups, of rank at most 8, labelled G_4, G_5, …,
    G_37.



A cyclic reflection group has no proper parabolic subgroups and we treat the
remaining cases separately, beginning with the symmetric and imprimitive
groups, culminating in Theorem <ref>.  The main results on
parabolic subgroups of the primitive reflection groups can be found in
Theorems <ref> and <ref>. Further information about the
parabolic complements is summarised in the tables of <ref>.



§ THE IMPRIMITIVE UNITARY REFLECTION GROUPS G(M,P,N)


A group G acting on a vector space V is imprimitive if, for some
k > 1, V is a direct sum of non-zero subspaces V_i (1≤ i ≤ k)
such that the action of G on V permutes V_1, V_2, …, V_k among
themselves; otherwise G is primitive.  The set Ω =
{V_1,V_2,…,V_k} is a system of imprimitivity for G. If G
acts transitively on Ω, then Ω is a transitive system of
imprimitivity.


The imprimitive unitary reflection group G(m,p,n) may be defined as
follows. Let V be a complex vector space of dimension n with a positive
definite hermitian form (-,-).  Let I = {1,2,…,n}, let Λ =
{ e_i | i∈ I } be an orthonormal basis for V, and let _m be
the group of mth roots of unity.  Given a function θ : I →_m,
the linear transformation that maps e_i to θ(i) e_i (1≤ i≤ n)
will be denoted by θ̂.

If p divides m, let A(m,p,n) be the group of all linear transformations
θ̂ such that ∏_i=1^n θ(i)^m/p = 1. For σ∈(n), the action of σ on V is defined by σ(e_i) =
e_σ(i).

The group G(m,p,n) is the semidirect product of A(m,p,n) by the symmetric
group (n) with the action on V given above.  When the elements of
G(m,p,n) are written as pairs (σ;θ), the multiplication becomes

    (σ_1;θ_1)(σ_2;θ_2) = (σ_1σ_2;
        θ_1^σ_2θ_2),

where for σ∈(n) and θ : I →_m we define
θ^σ(i) = θ(σ(i)).

The group G(m,1,n) is the wreath product _m≀ (n) and
G(m,p,n) is a normal subgroup of index p in G(m,1,n). If n > 1,
G(m,p,n) is imprimitive and {V_1,V_2,…,V_n} is a transitive
system of imprimitivity where V_i =  e_i. The group G(m,p,1) =
G(m/p,1,1) is cyclic of order m/p. The centre of G(m,p,n) consists of
scalar matrices and its order is m(p,n)/p.

The group G(1,1,n) is isomorphic to (n) and its elements are the
pairs (σ;1), where σ∈(n) and 1(i) = 1
for all i. In particular, if σ_i is the transposition (i,i+1), the
element r_i = (σ_i;1) is a reflection of order 2, which
interchanges e_i and e_i+1 and fixes e_j for j i, i+1.

Fix a primitive mth root of unity ζ and let t = (1;θ_ζ),
where θ_ζ(1) = ζ and θ_ζ(i) = 1 if i 1. Then
t is a reflection of order m and s = t^-1r_1t is a reflection of
order 2, which interchanges e_1 and ζ e_2.  These reflections
generate the groups G(m,p,n) (see <cit.>):

    G(1,1,n)    = r_1, r_2, …, r_n-1,
    
     G(m,m,n)    = s, r_1, r_2, …, r_n-1,
    
     G(m,1,n)    = t, r_1, r_2, …, r_n-1,
    
     G(m,p,n)    = s, t^p, r_1, r_2, …, r_n-1  for p1,m.


The support of a reflection subgroup is the subspace spanned by the
roots of its reflections. The group G(1,1,n) is imprimitive in its action
on V but for n ≥ 5 the action on its support is primitive.


We write λ⊢ n to mean that the sequence λ =
(n_1,n_2,…,n_d) is a partition of n; that is, n_1, n_2,
…, n_d are positive integers such that n_1≥ n_2≥⋯≥ n_d
and n = n_1+n_2 +⋯ + n_d. We also use the notation 1^b_1 2^b_2⋯ n^b_n⊢ n to denote a partition of n with b_i parts of
size i for 1≤ i≤ n.

Suppose that Ξ = (Λ_0,Π,ξ), where Λ_0 is a subset of
Λ, Π = (Λ_1, Λ_2, …,Λ_d) is a partition
of Λ∖Λ_0 and ξ : Λ∖Λ_0 →_m. Let n_i = |Λ_i| and define the subgroup P_Ξ of
G(m,p,n) by

    P_Ξ = P_0× P_1×⋯× P_d,

where P_0 is the reflection group G(m,p,n_0) acting on the subspace
spanned by Λ_0 and for 1≤ i≤ d, P_i is the reflection group
G(1,1,n_i)≃(n_i) permuting the vectors { ξ(e)e | e ∈Λ_i }. The factor P_0 is omitted if Λ_0 = ∅. If m
= 1, we require Λ_0 = ∅.

By construction P_Ξ is the pointwise stabiliser in G(m,p,n) of the
subspace spanned by the vectors ∑_e∈Λ_iξ(e) e for 1≤
i≤ d and therefore P_Ξ is a parabolic subgroup.

If n_0 = |Λ_0| and if λ is the partition of n - n_0
obtained by writing the quantities |Λ_i| (for 1≤ i≤ d) in
descending order, the type of P_Ξ is (n_0,λ).

Given n_0 ≤ n and λ = 1^b_1 2^b_2⋯ (n-n_0)^b_n-n_0⊢
n - n_0, the number of parabolic subgroups of G(m,p,n) of type
(n_0,λ) is

    n! m^n-n_0-d/n_0! b_λ

where d = b_1 + b_2 + … + b_n-n_0 and where

    b_λ = b_1! b_2! ⋯ b_n-n_0! (1!)^b_1(2!)^b_2⋯
      ((n-n_0)!)^b_n-n_0.
(cf. <cit.>)
We count the triples (Λ_0,Π,ξ) of type (n_0,λ). Triples
(Λ_0,Π,ξ) and (Λ_0,Π,ξ') define the same parabolic
subgroup if and only if, for 1 ≤ i≤ d, the vectors ∑_e∈Λ_iξ(e) e and ∑_e∈Λ_iξ'(e) e are equal up to a
non-zero scalar multiple.  Therefore, the number of parabolic subgroups is

    nn_0(n-n_0)!/b_λ m^n-n_0-d
      = n! m^n-n_0-d/n_0! b_λ.

A subgroup P of G(m,p,n) is parabolic if and only if P = P_Ξ for some
Ξ = (Λ_0,Π,ξ).

This follows from <cit.>.

Given n_0 such that 0≤ n_0 < n and λ =
(n_1,n_2,…,n_d)⊢ n - n_0 let k_-1 = 0 and for 0≤ i ≤ d
let k_i = n_0 + n_1 +⋯ +n_i. For 1≤ i≤ d define Λ_i = {
e_j | k_i-1 < j ≤ k_i} and let Π = (Λ_1,Λ_2,
…,Λ_d).

The standard parabolic subgroup of type (n_0,λ) is
P_(n_0,λ) = P_(Λ_0,Π,1), where 1(e) = 1
for all e∈Λ∖Λ_0. For consistency with Definition
<ref> we require n_0 = 0 if m = 1.


As a reflection group

    P_(n_0,λ) =  G(m,p,n_0) ×∏_i=1^d G(1,1,n_i)

where G(m,p,n_0) acts on the subspace spanned by Λ_0 and where
G(1,1,n_i) acts on the subspace of V spanned by Λ_i. If n_i =
1, the factor G(1,1,n_i) is trivial and if n_0 = 0 there is no factor
G(m,p,n_0).


Given α∈_m, define θ_α : I →_m by
θ_α(1) = α and θ_α(i) = 1 for i > 1. Then
θ̂_α∈ G(m,1,n) and we define

    P_(n_0,λ)^α =
      θ̂_α P_(n_0,λ)θ̂_α^-1.

A subgroup of G(m,p,n) is parabolic if and only if it is conjugate to
P_(n_0,λ)^α for some n_0, λ = (n_1,n_2,…,n_d)
and α. Furthermore, if n_0 0, then P_(n_0,λ)^α is
conjugate in G(m,p,n) to P_(n_0,λ) for all α, whereas the
groups P_(0,λ)^α and P_(0,λ)^β are conjugate in
G(m,p,n) if and only if αβ^-1∈_m/e where e =
(p,n_1,…,n_d).

If G = G(m,p,n) and P = P_(n_0,λ), where λ =
(n_1,n_2,…,n_d)⊢ n - n_0 has b_i parts of size i for 1≤ i≤
n-n_0, then

    |N_G(P)/P | = ∏_i=1^n-n_0 m^b_i b_i!   n_0  0
    ep∏_i=1^n m^b_i b_i!   n_0 = 0,

where e = (p,n_1,…,n_d).

We have |G(m,p,n)| = m^n n!/p and from (<ref>) we have

    |P | =  1p m^n_0 n_0!∏_i=1^d n_i!   n_0  0
    ∏_i=1^d n_i!   n_0 = 0.

Thus the corollary follows from equation (<ref>) combined with the
previous theorem.


Observe that if n_0  0 or p divides n_i for all i≥ 0,
|N_G(P)/P | = ∏_i,b_i 0|G(m,1,b_i)|. In this case we shall
see that the order coincidence can be lifted to an isomorphism of groups.


Suppose that λ = (n_1,n_2,…,n_d) = 1^b_1 2^b_2⋯
n^b_n⊢ n-n_0. The group G(1,1,n_i) in the decomposition
(<ref>) of the standard parabolic subgroup P =
P_(n_0,λ) of G(m,p,n) acts on the subspace spanned by Λ_i
and fixes the sum ΣΛ_i of the vectors in Λ_i.

Given k such that b_k 0, let M_k be the subspace spanned by the
vectors in the b_k blocks of size k of the partition Π defining P.
Then P acts on M_k as the reflection group P_k = G(1,1,k)^b_k and P
is the direct product of the P_k. We call the groups P_k the
isotypic components of P.

The vectors ΣΛ_i, where Λ_i is a block of size k, form
a basis for X_k = _M_k(P). The subspaces M_k and X_k are
invariant under the action of the normaliser of P in G(m,p,n).


Let Γ_1^k, Γ_2^k, …, Γ_b_k^k be the blocks of size
k in the partition of I corresponding to Π, labelled so that the
jth block is

    Γ_j^k = {i_1,j^k, i_2,j^k, …, i_k,j^k}.


The group (b_k)≃ G(1,1,b_k) is generated by transpositions and the
reflection σ∈ G(1,1,b_k) corresponding to the transposition (r,s)
induces

    v^(k)(r,s) = (τ;1) in G(1,1,n),

where τ = (i_1,r^k,i_1,s^k)(i_2,r^k,i_2,s^k) ⋯
(i_k,r^k,i_k,s^k). Then v^(k)(r,s) belongs to the normaliser of P
in G(m,p,n) and acts on X_k as the reflection σ.

Recall that in the paragraph preceding (<ref>) we chose a fixed
primitive mth root of unity ζ∈_m. Define θ_j^(k) :
Γ_j^k →_m by θ_j^(k)(g) = ζ if g∈Γ_j^k
and θ_j^(k)(g) = 1 otherwise.  Then

    h^(k)(j) = (1;θ_j^(k)) ∈ A(m,1,n)

normalises P and acts on X_k as (1;θ_j), where θ_j :
{1,2,…,b_k}→_m sends j to ζ and all other elements
to 1.


In effect we have M_k = U⊗ W, where U is a vector space of
dimension k and W is a vector space of dimension b_k. The action of
G(1,1,k) on U extends naturally to an action of G(1,1,k)^b_k on
U⊗ W, which is normalised by the action of G(m,1,b_k) on W.

Recall that when H is a complement to P in N_G(P), H^∘ denotes
the subgroup generated by the elements of H that act as reflections on
_V(P).  The identification of H^∘ as a reflection group in the
following theorem was previously obtained by Amend et 
al. <cit.>.


Suppose that G = G(m,p,n) and let P = P_(n_0,λ).  Let V be the
vector space on which G acts and let X_k be the fixed point space of the
kth isotypic component P_k of P acting on the space M_k spanned by
Λ_k as in Definition <ref>. If b_k 0, let φ_k
be the restriction homomorphism N_G(P) →(X_k). Then

  * N_G(P) has a complement H to P.

  *  If n_0 0, then φ_k(H) = G(m,1,b_k) and H = H^∘
    acts on _V(P) = ⊕_k X_k as the reflection group
    ∏_k,b_k 0 G(m,1,b_k).

  *  If n_0 = 0, then φ_k(H^∘) = G(m,p_k,b_k), where p_k =
    p/(p,k) and H^∘ acts faithfully on _V(P) as ∏_k,
    b_k 0G(m,p_k,b_k). Furthermore, for all k such that b_k 0,
    φ_k(H) = G(m,q_k,b_k) for some divisor q_k of p_k.

We begin by considering the case p = 1.

The transformations v^(k)(r,s) and h^(k)(j) defined in
(<ref>) and (<ref>) are image of the generators of
G(m,1,b_k) and thus for all g∈ G(m,1,b_k) there is an element ĝ∈ G(m,1,n) that acts on X_k as g and fixes M_ℓ pointwise for all
ℓ k.  Thus g↦ĝ is a monomorphism from G(m,1,b_k) into
the normaliser N of P = P_(n_0,λ) in G(m,1,n) and
φ_k(ĝ) = g.

On comparing orders, using Corollary <ref>, it follows that the
subgroup

    H = { ĝ| g ∈∏_k,b_k 0 G(m,1,b_k) }

is a complement to P in N. Furthermore, H acts on _V(P) as the
reflection group ∏_k,b_k  0 G(m,1,b_k).  This proves (i) and (ii)
for the case p = 1 and so from now on we may suppose that p 1.

Suppose that n_0 0. The reflections v^(k)(r,s) of
(<ref>) belong to G(1,1,n) ⊂ G(m,p,n).  Since n_0  0
we have e_1∈Λ_0 and therefore t^-kh^(k)(j)∈
G(m,m,n)⊆ G(m,p,n), where t is the generator of G(m,1,n) which
is used in (<ref>). The elements v^(k)(r,s) and
t^-kh^(k)(j) belong to the normaliser of P in G(m,p,n) and we
define H to be the subgroup generated by v^(k)(r,s) and
t^-kh^(k)(j) for all k such that b_k 0. Then H is isomorphic
to ∏_k,b_k  0 G(m,1,b_k) and φ_k(H) = G(m,1,b_k). A
comparison of orders shows that H is a parabolic complement.

For the remainder of the proof we may suppose that p 1 and n_0 = 0.
Let G̃ = G(m,1,n) and let X = _V(P). Because n_0 = 0, G̃(X) is the direct product of the same groups G(1,1,k) defining P and
therefore G̃(X) = P.  As shown above, the normaliser of P in
G̃ has a complement H̃. Then H = G∩H̃ is a
parabolic complement, completing the proof of (i).

We shall use the notation established in the first part of this proof. If g
= (1;θ)∈ A(m,1,b_k) where θ : {1,2,…,b_k}→_m,
then ĝ∈ A(m,1,n) belongs to A(m,p,n) if and only if

    ∏_j=1^b_kθ(j)^km/p = 1.

In this case ϕ_k(ĝ)∈ A(m,p_k,b_k) where p_k = m/(m,km/p) =
p/(p,k).

We have φ_k(H^∘) = G(m,p_k,b_k) and it follows that H^∘
acts faithfully on _V(P) as ∏_k=1^n G(m,p_k,b_k). In addition,
H̃ = ∏_k,b_k 0 G(m,1,b_k) and Ã = ∏_k,b_k 0
A(m,1,b_k) is a normal subgroup of H̃. Thus A = G∩Ã is
a normal subgroup of H and φ_k(A) contains A(m,m,b_k). It follows
that φ_k(H) = G(m,q_k,b_k) for some divisor q_k of p_k.

Suppose that G = G(2,2,n); that is, G is a Coxeter group of type D_n.
Let λ = 1^b_1 2^b_2⋯ n^b_n⊢ n. The complement H
to P = P_(0,λ) in its normaliser is generated by reflections in its
action on _V(P) if and only if there is at most one value of k such
that k is odd and b_k  0.

Suppose that p  1, n_0 = 0 and that P has a single isotypic
component G(1,1,k)^b_k. Then λ = k^b_k⊢ kb_k = n and the
subgroup H^∘ of the parabolic complement is
isomorphic to G(m,p/s,b), where s = (p,k). It follows from Corollary
<ref> that H^∘ is a parabolic complement for P.

Next suppose that p  1, n_0 = 0 and λ = 1^b_1 k^b_k⊢
b_1 + kb_k. Once again P is the direct product of b_k copies of
G(1,1,k). But now the subgroup H^∘ of the parabolic complement H
is H^∘ = H_1^∘× H_2^∘, where
H_1^∘≃ G(m,p,b_1), H_2^∘≃ G(m,p/s,b_k), where s =
(p,k) and the index of H^∘ in H is p/s.

Put x = v^1(1)^s h^(k)(1)^ℓ where s = pr - kℓ for some r
and ℓ. The product of the diagonal entries of x is ζ^s+kℓ
and therefore x∈ G(m,p,n). By construction x normalises P,
H_1^∘ and H_2^∘ and on comparing orders we see that H =
H^∘,x.

The space of fixed points of P has a decomposition _V(P) = X_1⊕
X_2 corresponding to the isotypic components of P. The image of the
restriction of H_1 = H_1^∘,x to X_1 is the reflection group
G(m,s,b_1).  Recall that t = (1;θ) is the reflection defined prior
to (<ref>) and that its determinant is ζ.  We have t^p/s∈
H_2^∘ and therefore the restriction of H_2 = H_2^∘,x to X_2
is the reflection group G(m,1,b_k) because (ζ^ℓ)^k/s =
ζ^(p/s)rζ.

Suppose that p  1, n_0 = 0 and λ = k_1^b_1 k_2^b_2⊢
n, where n = k_1b_1 + k_2b_k.  Then P = P_1× P_2, where P_i is
the direct product of b_i copies of G(1,1,k_i). The subgroup H^∘ of
the parabolic complement H is H^∘ = H_1^∘×
H_2^∘, where H_i^∘≃ G(m,p_i,b_i) and where p_i =
p/(p,k_i). The index of H^∘ in H is
p(p,k_1,k_2)/(p,k_1)(p,k_2).


§ ROOT SYSTEMS AND PARABOLIC SUBGROUPS OF COXETER TYPE

Let P be a proper parabolic subgroup of the reflection group G and let X
= _V(P). Then N_G(P) = G_X = G_Y, where Y = X^⊥ is the
orthogonal complement of X in V.  If r∈ N_G(P) is a reflection and
a is a root of r, then a∈ X∪ Y and therefore the reflection
subgroup of N_G(P) is P× Q, where Q is the parabolic subgroup
G(Y). This symmetry between P and Q is echoed in the tables in
<ref>.

When G is a real reflection group (namely a finite Coxeter group) a
parabolic complement to P can be obtained as the setwise stabiliser of a
set of (simple) roots corresponding to reflections that generate P (see
<cit.>). In this section we show that in many cases
there is a generalisation of this result to parabolic subgroups of unitary
reflection groups. Consequently, the complement contains the subgroup Q.

However, there are unitary reflection groups which contain parabolic
subgroups P such that Q cannot be extended to a parabolic complement. 
Therefore, in these cases, even though it turns out that the
normaliser of P is a semidirect product, a complement cannot be realised as
the stabiliser of a set of roots of P.



Let F be a finite abelian extension of , let A be the ring of integers
of F, let  be the group of roots of unity in A and let U be a vector
space over F with an hermitian product (-,-) such that V = ⊗_F U.
An A-root system for the reflection group G, as defined in
<cit.>, is a pair (Σ,f) such that
Σ spans U, 0∉Σ, and f : Σ→. Furthermore,
G is generated by the reflections { r_a,f(a)| a ∈Σ}, and

  *  for all a∈Σ and all λ∈ F we have λ a∈Σ
    if and only if λ∈;

  *  for all a∈Σ and all λ∈ we have f(λ a) = 
    f(a)  1;

  *  for all a,b∈Σ we have (1-f(b))(a,b)/(b,b)∈ A;

  *  for all a,b∈Σ we have r_a,f(a)(b)∈Σ and 
    f(r_a,f(a)(b)) = f(b).


Let ζ_m denote a primitive mth root of unity and put _m = ζ_m.
A [ζ_m]-root system for G(m,m,n) is Σ(m,m,n) = { ξ e_i - η e_j
|ξ,η∈_m and i j }, where f(a) = -1 for all roots.
To obtain a root system Σ(m,p,n) for G(m,p,n) where p  m include
the roots ±ξ e_i for 1≤ i≤ n and ξ∈_m and define f(±ξ
e_i) = ζ_m^p. If f(a) = -1, let r_a denote the reflection with root a.
See <cit.> for descriptions of root systems for the primitive
reflection groups.


A parabolic subgroup P of G is of Coxeter typeX if it is
generated by a set S of reflections such that (P,S) is a Coxeter system
and X is the standard name of (P,S). The standard names are described
in <ref>.


Suppose that P is a parabolic subgroup of Coxeter type in the reflection
group G and that Σ is an A-root system for G.  Let Σ_P be
the set of roots in Σ corresponding to the reflections in P.   Let
Φ denote the root system of P as in <cit.>.  At this
point we work with the following hypotheses.


The root system Φ is a subset of Σ_P.


It is clear that it is possible to choose a root system Φ for the
standard parabolic subgroup P_(0,λ) of G(m,p,n) so that Hypothesis 1
is satisfied.  Later we shall show that it holds for many other groups.

 The group of roots of unity of A is = -1×γ where the order h of γ is odd.



Let Δ be a set of simple roots in Φ and let Φ^+ be the
corresponding set of positive roots. Define

    Δ= ⋃_i=0^h-1γ^iΔ,
        Φ^+ = ⋃_i=0^h-1γ^iΦ^+
        and  Φ^- = -Φ^+.

For g∈ N_G(P), define N(g) = { a∈Φ^+| g(a)∈Φ^- }. For a∈Δ, if g(a)∈Φ^-, then
r_a N(gr_a) = N(g)∖{a}, whereas if g(a)∈Φ^+,
then N(gr_a) = r_aN(g)∪̇{a}.

This is a direct consequence of the well-known fact <cit.> that for Coxeter groups
r_a(Φ^+∖{a}) = Φ^+∖{a}.


The proof of the following theorem is based on <cit.>.


Suppose that P is a parabolic subgroup of Coxeter type in the reflection
group G and that G has an A-root system such that Hypotheses 1 and 2
hold. Then G_Δ̃ = { g∈ G| g(Δ) =
Δ } is a complement to P in its normaliser in G.

If g∈ N_G(P) and a∈Σ_P, then gr_ag^-1 = r_g(a) and
g(a)∈Σ_P = Φ^+∪Φ^-. Define ℓ(g) =
|N(g)| and set H_0 = { g ∈ N_G(P) |ℓ(g) = 0 }. For g∈ P,
ℓ(g) coincides with the usual length function for Coxeter groups and
therefore P∩ H_0 = 1.

For each coset of P in N_G(P) choose an element g in the coset such
that ℓ(g) is minimal.  If g(a)∈Φ^- for some
a∈Δ, then from Lemma <ref>ℓ(gr_a) = ℓ(g) - 1,
contradicting the choice of g.  Thus g(a)∈Φ^+ for all
a∈Δ.

The set Φ^+ is the disjoint union of pairwise orthogonal irreducible
components Φ_1^+, Φ_2^+, …, Φ_k^+ and Φ^+
is the disjoint union of the sets γ^jΦ_i^+ for 1≤ j≤ h and
1≤ i≤ k. If a,b∈Δ and (a,b) 0, then
g(Δ)⊆Φ^+ implies g(a), g(b)∈γ^jΦ_i^+
for some i and j. A connectedness argument now shows that
g(Δ∩Φ_i^+)⊆γ^jΦ_i^+. Every element of
a∈Φ_i^+ is a positive linear combination of the roots in
Δ∩Φ_i^+ and so g(Δ)⊆Φ^+ implies
g(Φ^+) = Φ^+. Therefore g∈ H_0.

To prove that H_0 is a subgroup we show that h_1^-1h_2∈ H_0 for all
h_1,h_2∈ H_0.  By way of contradiction suppose that
h_1^-1h_2(a)∈Φ^- for some a∈Φ^+. Then
h_1^-1h_2(a) = λ b for some b∈Φ^+, where
-λ∈γ. But then h_2(a) = λ h_1(b) and h_1(b),
h_2(a) ∈Φ^+. This contradiction completes the proof that
H_0 is a subgroup.

If g∈ G_Δ̃, then g∈ N_G(P) and g(Δ) =
Δ whence ℓ(g) = 0 and so g∈ H_0.  Conversely, by
definition, if g∈ H_0 and a∈Φ^+, then g(a)∈Φ^+
and it follows that g(Φ^+) = Φ^+. Then
Δ∩γ^jΦ_i^+ is the set roots in γ^jΦ_i^+
that cannot be written as a nontrivial positive combination of other roots in
γ^jΦ_i^+.  Therefore g(Δ) = Δ and
hence H_0 =  G_Δ̃.

Suppose that P is the standard parabolic subgroup P_(0,λ) in G =
G(m,p,n) and that m is odd or twice an odd integer. Then N_G(P) is the
semidirect product of P by the stabiliser in G of a set of roots of P.

The condition on m ensures that Hypothesis 2 is satisfied. The parabolic
subgroup P is of Coxeter type and for the simple roots we may take vectors
e_i-1-e_i for suitable values of i.  Thus Hypothesis 1 is satisfied and
therefore the stabiliser of Δ is a parabolic complement.



Next consider the standard parabolic subgroup P = P_(n_0,λ), where
n_0  0. We may write P = P_0× P_1, where P_0 = G(m,p,n_0) and
where P_1 is the parabolic subgroup P_(0,λ) in G_1 =
G(m,p,n-n_0). Assume Hypothesis 2 and let Δ be the roots of
P_1 defined in (<ref>). If λ = (n_1,…,n_d) and e =
(p,n_1,…,n_d), let θ = ζ_m^me/p and set

    Δ_0 = { θ^j e_1 - e_2,θ^j e_1 - ζ_m e_2
       | 0≤ j < p/e  }∪{  e_i-1-e_i | 3≤ i ≤ n_0  }.

If p m, adjoin the roots { θ^j e_1 |  0≤ j < p/e  }∪{ e_i | 2≤ i≤ n_0 } to Δ_0. The reflections whose roots belong
to Δ_0∪Δ generate P.


Suppose that Hypothesis 2 holds and that P is the standard
parabolic subgroup P_(n_0,λ) in G = G(m,p,n) where n_0  0,
λ = (n_1,…,n_d) and e = (p,n_1,…,n_d). If m/p and
p/e are coprime, N_G(P) is the semidirect product of P by the setwise
stabiliser of Δ_0∪Δ in G.

In the notation established above we have N_G_1(P_1) = P_1H_1, where
H_1 stabilises Δ and fixes every root in Δ_0.
It follows from Corollary <ref> that the index of PH_1 in
N_G(P) is p/e. If p = e, then H_1 = G_Δ_0∪Δ
is a complement to P in N_G(P).

If p e, let H = G_Δ_0 ∪Δ̃ and suppose that m/p
and p/e are coprime. Then H_1⊆ H⊆ N_G(P) and H∩ P = 1.
It follows from the proof of Theorem <ref> that N_G(P)
contains an element g such that e_1 is an eigenvector of g with
eigenvalue θ = ζ_m^me/p and g(e_i) = e_i for 2≤ i≤ n_0. The
construction of g in the proof of Theorem <ref> utilised
the elements h^(k)(j), which act on the blocks Γ_j^k of
size k as scalar matrices ζ I_k.  However, if m is even, the
construction needs to be modified so that g fixes Δ.
This is achieved by replacing the scalar matrix ζ I_k by the matrix with
ζ along the anti-diagonal.  Then g fixes Δ_0 ∪Δ̃
and since m/p and p/e are coprime, H = H_1,g and N_G(P) = PH.


§ THE PRIMITIVE UNITARY REFLECTION GROUPS

From now on suppose G is a finite primitive unitary reflection group of
rank at least 2 acting on V.



 §.§ Parabolic subgroups of rank 1

Throughout this section P denotes a rank 1 parabolic subgroup of G. Thus
P = r, where r is a reflection of order k, where k is 2, 3, 4 or
5 and r is not the square of an element of order 4. Reflections of orders 4
or 5 occur only when the rank of G is 2.


If P is a parabolic subgroup of G of rank 1, then N_G(P) = C_G(P).

If g∈ N_G(P) and P = r, then g^-1rg = r^ℓ for some ℓ.
If a is a root of r, then ra = λ a for some root of unity
λ 1. Then rg a = λ^ℓ ga and it follows that ℓ = 1.
Thus g∈ C_G(r) = C_G(P).  The converse is obvious.



Our first approach to finding a complement to P = r in its normaliser
is to characterise those rank 1 parabolic subgroups whose normaliser is the
semidirect product of the parabolic by the stabiliser of a set of roots.

To this end, let a be a root of the reflection r, let k be the order of
r and set

    = { ξ∈|ga = ξ a for some g∈ N_G(P) }  and  
      B = { ξ^k a|ξ∈ }.

Then  consists of roots of unity and it is a cyclic group whose order
m is a multiple of k.


If m/k and k are coprime, the stabiliser in G of the set B
is a complement to P in N_G(P). Conversely, if the stabiliser of a set of
scalar multiples of a is a complement to P in N_G(P), then m/k and
k are coprime.

We have |B| = m/k and the set { ξ a|ξ∈ } is the
disjoint union of the images of B under the action of N = N_G(P). Thus
H = G_B is contained in N and its index in N is k. The assumption
that m/k and k are coprime implies H ∩ P = 1 and since k = |P |
it follows that N = P H.

Conversely, suppose that H = G_D is a complement to P in N, where D
is a set of scalar multiples of a. Without loss of generality we may
suppose that a∈ D and that D is an orbit of H.  Then D = _0 a,
where

    _0 = { λ∈| ha = λ a for some h∈ H }

is a subgroup of , hence cyclic. Let θ be a generator of
_0 and suppose that ra = ζ a.  The elements of  are the
products ζ^iθ^j. If ζ^iθ^j = 1, then r^i preserves
D and hence r^i = 1 since P∩ H = 1.  Thus m = k|D|.  If k and
m/k are not coprime, ζ and θ have non-trivial
intersection, contrary to what we have just proved.  This completes the
proof.


The values of k and m for all primitive unitary reflection groups can be
obtained from Tables D.1 and D.2  of <cit.> and the
descriptions of the rank 2 reflection groups in
<cit.>.

The lemma just proved provides a parabolic complement for 22
of the 36 conjugacy classes of parabolic subgroups in the primitive
reflection groups of rank 2 and for the parabolic subgroups of rank 1 in all
but two of the primitive groups of rank at least 3. The exceptions (for rank
at least 3) are the Shephard and Todd groups G_29 and G_31, each of
which has a single class of reflections. In both cases the reflections are
involutions and the order of  is 4.


The Shephard and Todd group G = G_25 is the semidirect product of an
extraspecial group of order 27 and exponent 3 by the group (2,3) (see
<cit.>). A rank 1 parabolic subgroup P is
generated by a reflection r of order 3 and P has 12 conjugates, whence
|N_G(P)| = 54. Let a denote a root of r. The order of the group 
of (<ref>) is 6 and therefore, by Lemma <ref>, the
stabiliser of {a,-a} is a parabolic complement. Since
N_G(P) = C_G(P) it can be seen that N_G(P) has an elementary abelian
group of order 27 and three complements to P in N_G(P).  The complements
not equal to P act transitively on the 6 roots (-ω)^j a (0≤ j<
6).  Thus H is the unique complement which is the stabiliser of a set of
roots.  The elements of order 6 in H fix no non-zero vectors and therefore
they are not 1-regular (in the sense of Springer <cit.>).  In
particular, the fact that the elements of a parabolic complement in a
Coxeter group are 1-regular does not generalise to the unitary case.

In order to deal with the groups not covered by Lemma <ref> we
pursue other approaches to finding a complement.


Suppose that P = r, where r is a reflection of order k and let d
be the order of the image of : N_G(P)→^×. Then N_G(P) =
P× H where H = { g∈ N_G(P)|(g)^d/k = 1 } except when
P is a parabolic subgroup of order 2 in G_9 or G_11.

The image of : N_G(P) → is cyclic and its order is divisible by
k. It follows immediately that the index of H = { g∈ N_G(P)|(g)^d/k = 1 } in N_G(P) is k and P∩ H = 1 if and only if
k and d/k are coprime. To complete the proof observe that d divides
the least common multiple of the orders of the reflections in G and
therefore, from the tables in <cit.>, d/k and k are
coprime except when k = 2 and 4 divides d.  Among the primitive
reflection groups only G_9 and G_11 contain parabolic subgroups of
orders 2 and 4.


The exceptions in G_9 and G_11 to Lemma <ref> are also
exceptions to Lemma <ref>.  We deal with these exceptions by
showing that in general, for almost all maximal parabolic subgroups P in a
primitive reflection group G, the centre of G is a parabolic complement
for P.



If P is a maximal parabolic subgroup of the primitive reflection group G,
then N_G(P) = P× Z(G) except in the following cases. (See
<ref> for the definition of the type of a parabolic
subgroup.)

  * G is G_13 or G_15 and P is a parabolic subgroup
of order 2 with 6 conjugates in G.

  * G = G_25 and the type of P is 2L_1.

  * G = G_33 and the type of P is D_4.

  * G = G_35 and the type of P is A_5 or A_1+2A_2.

The index of a parabolic subgroup in its normaliser is tabulated in
<cit.> and the order of Z(G) can be
obtained from <cit.> using <cit.>.  Since P∩ Z(G) = 1, it can be seen that the
only exceptions to N_G(P) = P× Z(G) are those listed.

  *  Parabolic complements for the the parabolic subgroups of types D_4 in 
    G_33 and A_5 and A_1+2A_2 in G_35 follow from Theorem 
    <ref> below.

  *  Suppose that P is a parabolic subgroup of type 2L_1 in G = G_25
    and let Q be the pointwise stabiliser of _V(P)^⊥. From
    Table <ref> of <ref> the type of Q is L_1,
    |N| = 54 and there is an involution t∈ N, which commutes with Q
    and interchanges the root lines of P. Then H = Qt is a
    complement to P in N.








Suppose that P is a parabolic subgroup of rank 1 in a primitive unitary
reflection group G of rank at least 2 and let Q be the pointwise
stabiliser of _V(P)^⊥.

  *  There exists a complement H to P in its normaliser.

  *  Let k = |P | and let m be the order of the group  defined
    in (<ref>).  If k is coprime to m/k, H can be chosen to
    be the stabiliser of a set of roots and consequently Q⊆ H.
    In particular this holds if the rank of G is at least 3 and G is
    neither G_29 nor G_31.

  *  Except when G is G_27, G_29, G_33 or G_34, the
    group H acts on _V(P) as a unitary reflection group.  If the
    rank of G is 2, then H is cyclic. If the rank is at least 3, the
    isomorphism types of Q, H^∘ and H are given in the tables of
    <ref>.

(i) The existence of H is a consequence of Lemma <ref>
and Theorem <ref>.

(ii) Suppose that the rank of G is at least 3. Then the
order of a reflection in G is either 2 or 3. From
<cit.> the order of the group of roots of unity
of the ring of definition of G is:

  *  2 for G_23, G_24, G_28, G_30, G_35, G_36 and
    G_37;

  *  6 for G_25, G_26, G_27, G_32, G_33 and G_34;

  *  4 for G_29 and G_31.


Thus the order of the group  defined in (<ref>) is 2 or 6,
except for reflections in G_29 and G_31 where it is 4. Therefore,
from Lemma <ref>, except for G_29 and G_31, a complement
to P may be chosen to contain Q.

(iii) If the rank of G is 2, then H is cyclic and
thus it acts as a reflection group on _V(P). Furthermore, in all
cases except when the order of P is 3 and G is G_4 or G_7, the
groups P and Q have the same order (see <cit.>).
In the two exceptional cases Q = 1.

If the rank of G is at least 3, full details of the structure of H can
be found in <cit.> and verified using the  <cit.>
code located at <http://www.maths.usyd.edu.au/u/don/details.html#programs>.


From now on we suppose that the rank of G is at least 3 and the rank of
P is at least 2.



 §.§ Parabolic subgroups of Coxeter type

From the tables in <cit.> or
<cit.> there are 159 conjugacy classes
of parabolic subgroups of rank at least 2 in the 15 primitive reflection
groups of rank at least 3. All except 19 of these are of Coxeter type. We
shall see that, except for some instances when the ring of definition is the
Gaussian integers [i], the existence of a parabolic complement for a
parabolic subgroup of Coxeter type is a consequence of Theorem <ref>.

The primitive unitary reflection groups which contain reflections of order 3
do not have parabolic subgroups of Coxeter type and rank at least 2.
Therefore, throughout this section we may suppose that G is generated by
reflections of order 2.

From <cit.>, except for G_28—the
Coxeter group of type F_4—there is an A-root system Σ for G
such that (a,a) = 2 for all a∈Σ, where A is the ring of definition
of G.  For the groups G under consideration  is the group of mth
roots of unity, where m is 2, 4 or 6.

The set ℒ of 1-dimensional subspaces spanned by the vectors in
Σ is the line system of G. Let P be a parabolic subgroup of
G of Coxeter type, let ℳ be the subset of ℒ spanned
by the roots of reflections in P and let Σ_P be the roots in
Σ which belong to the lines in ℳ. Let Φ be the
standard root system of P as defined, for example, in <cit.>.
The roots of P are eigenvectors of its reflections and so we may assume
that the lines in ℳ are spanned by roots in Φ. Our first task
is to determine when Hypothesis 1 is satisfied.


If P does not contain reflections r and s such that the
order of rs is 4, the root system Φ may be realised as a subset of
Σ_P.

It is a consequence of our assumptions that for a,b∈Σ_P the order
of r_ar_b is 1, 2, 3 or 5.  From <cit.> if
|r_ar_b| = 2, then (a,b) = 0;  if |r_ar_b| = 3, then (a,b)∈
whereas if |r_ar_b| = 5, then (a,b) = ατ or (a,b) =
ατ^-1 for some α∈ and where τ^2 = τ+1.

Let ℳ = {ℓ_1,ℓ_2,…,ℓ_t} and for 1≤ j≤ t
choose a_j∈ℓ_j∩Σ_P. Taking each root a_i in turn, if
(a_i,a_j)  (a_j,a_i) replace a_j by α a_j, where α =
(a_i,a_j)/|(a_i,a_j)|.  It follows from <cit.> that this
process results in roots a_i∈ℓ_j∩Σ_P such that all inner
products (a_i,a_j) are real. These roots with their negatives form the
standard root system Φ of P.

If P contains reflections r and s such that the order of rs is 4, the
standard root system Φ of P is the disjoint union of its long roots
Φ_ℓ and its short roots Φ_s.  If G = G_28, then Φ =
Σ_P. In all other cases Φ_ℓ may be realised as a subset of
Σ_P and there exists γ in A, the ring of definition, such
that γγ̅= 2 and Φ_s may be realised as a  subset
of (γ/2)Σ_P.

We may suppose that G  G_28. It follows from <cit.> that A is either [λ], [ω,τ]
or [i], where λ^2 + λ + 2 = 0, ω^2+ω+1 = 0,
τ^2 - τ - 1 = 0 and i^2 + 1 = 0. We may take γ to be
λ, ω+τ or i+1, respectively.

For all a∈Σ_P such that a spans a short root line in ℳ,
replace a by (γ/2)a.  We complete the proof by applying the
construction of the previous lemma.

Of the 15 primitive reflection groups of rank at least 3 only G_24,
G_27, G_28 and G_29 contain parabolic subgroups of Coxeter type
that satisfy the hypotheses of this lemma and only types B_2 and B_3
occur.


From now on suppose that Φ is realised as in the previous two lemmas and
let Δ, Δ and Φ^+ be defined as in
(<ref>).


Suppose that G is a primitive reflection group of rank at least 3 and that
the group of units of its ring of definition has order 2 or 6. If P is a
parabolic subgroup of Coxeter type in G, then G_Δ̃ is a
complement to P in N_G(P).

It is clear that Hypothesis 2 holds. If there are no reflections r,s in P
such that the order of rs is 4, Hypothesis 1 is a consequence of
Lemma <ref> and the result follows from Theorem <ref>.
This is also the case for G_28, the Coxeter group of type F_4.

Thus we are reduced to showing that the result holds when P has type B_2
and G is either G_24 or G_27.  In this case Δ = {a,b}
where a is a long root and b is a short root.  As in the proof of Theorem
<ref>, { g∈ G| g(Δ) =
Δ } is a complement to P in N_G(P).

If = _4, then G is G_29 or G_31 and if P is a
parabolic subgroup of Coxeter type in G, we have N_G(P) = P H for
some subgroup H. The complement H can be chosen to be the setwise
stabiliser of a set of roots of reflections which generate P if and only if
the type of P is A_2, B_2 or A_3 (two classes) in G_29 or A_2
or A_3 in G_31.

Let N = N_G(P) and suppose at first that P = r× P_1 where r is
a reflection and P_1 is generated by reflections. We shall show that the
existence of a set J of roots of P whose setwise stabiliser in N is a
complement to P in N leads to a contradiction. If J does not contain a
root of r, then r fixes J and hence N_J∩ P  1. If J contains
a root a of r, then N contains an element g such that g(a) = ia and
if N = N_JP we may write g = hg_1g_2, where h∈ N_J, g_1∈ P_1 and
g_2∈r. It follows that h(a) = ± ia and hence -a∈ J. But then
r(J) = J and again we reach the contradiction N_J∩ P 1.  This
proves that if P is a parabolic subgroup of type 2A_1 or A_1+A_2, a
complement to P in its normaliser cannot be the stabiliser of a set of
roots of P.

If G is G_29 and the type of P is B_3, the pointwise stabiliser Q
of _V(P) has order 2 and there are no elements g∈ N of order 4 such
that g^2∈ Q. It follows that there is no set of roots of P whose
setwise stabiliser is a complement to P in N.

For all other parabolic subgroups of Coxeter type we shall show that a
complement can be obtained as the stabiliser of a set of roots.

If the type of P is A_2 and if a_1 and a_2 are simple roots we set J
= {± a_1,± ia_2}. If the type of P is A_3 and if a_1, a_2 and
a_3 are simple roots with Cartan matrix
([  2 -1  0; -1  2 -1;  0 -1  2 ]), we set J = {± a_1,±
a_2,± ia_2, ± ia_3}.

Finally, if G is G_29 and the type of P is B_2, we may choose
simple roots a_1 and a_2 for P with Cartan matrix
([  2 -2; -1  2 ])
such that a_1, (-i+1)a_2 belong to the root system Σ_P and then
set

    J = {a_1,(-i+1)a_2,i(a_1+2a_2),-(i+1)(a_1+a_2)}.

In all cases a direct calculation shows that N = P N_J.




 §.§ Parabolic subgroups of non-Coxeter type

There are 20 conjugacy classes of non-Coxeter parabolic subgroups P of rank
at least 2 in the primitive reflection groups of rank at least 3.  Thirteen
of these are maximal and we have shown in Theorem <ref> that
except for the parabolic subgroups of type 2L_1 in G_25, the centre of
G is a parabolic complement for P. However, it turns out that in
almost all cases there is a complement to P which is the stabiliser of a
set of roots and which therefore contains the pointwise stabiliser Q of the
orthogonal complement of the space of fixed points of P.


In preparation for the proofs that follow we represent the roots of
reflections as row vectors and the elements of G as matrices acting on the
right.  This is consistent with the conventions of the <cit.> code for unitary reflection groups.  If G is generated by
reflections r_1, r_2, …, r_ℓ with roots a_1, a_2,
…, a_ℓ, there are corootsb_1, b_2, …, b_ℓ such
that for all row vectors v we have

    v r_j = v - vb_j^⊤ a_j.

The matrices A and B with rows a_1, a_2, …, a_ℓ and b_1,
b_2, …, b_ℓ are basic root and coroot matrices for G.  The
complex Cartan matrix of A and B is AB^⊤.  We may suppose that the
roots a_1, a_2, …, a_n, where n is the rank of G, are the
standard basis vectors. If ℓ = n, then C = B^⊤ and the construction
of <cit.> produces the reflections r_1, r_2,
…, r_n.  In this case it follows from <cit.> that every
subset of {r_1,r_2,…,r_n} generates a parabolic subgroup.


For G_31 we have n = 4 and ℓ = 5 and the root and coroot matrices
are given in Table <ref>. The root system may be obtained by
taking the images of the basic roots under the action of G.  The Cartan
matrices in Table <ref> and the roots and coroots in Table
<ref> have been chosen so that the reflections satisfy the
defining relations corresponding to the diagrams in
<cit.>.


If P is a maximal parabolic subgroup of a primitive reflection group G of
rank at least 3 and if P is not of type A_1+A_2 or B_3 in G_29
nor of type A_1+A_2 or G(4,2,3) in G_31, there is a set J of roots
of P such N_G(P) is the semidirect product of P by the setwise stabiliser
of J in G.

From Theorems <ref> and <ref> we may suppose that P is
not of Coxeter type.  If the ring of definition of G is the Eisenstein
integers [ω], a suitable set J  of roots is given in
Table <ref>.  Direct computation or the  code referred
to in <ref> can be used to check that the stabiliser of J is a
parabolic complement for P.

This leaves the groups G_29 and G_31 to be considered.  If P is a
parabolic subgroup of type G(4,4,3) in G_29 we may suppose that P is
generated by the reflections r_1, r_2 and r_3 obtained from the root
and coroot matrices of Table <ref>.  The stabiliser of J = {±(0,0,i,0), ±(i,0,1,0), ±(0,i,i-1,0), ±(i,1,i+1,0)} is a parabolic
complement.

Finally suppose that P is a parabolic subgroup of type G(4,2,3) in
G_31. Then N_G(P) = P× Z(G) and Z(G) is cyclic of order 4.
Furthermore, _V(P) is spanned by a root and the pointwise stabiliser
Q of _V(P)^⊥ has order 2.  It follows from Lemma <ref>
that C_G(Q) = Q × K for some K. If g is a complement to P in
N_G(P) and Q⊂g, then g∈ C_G(Q) and this contradiction
completes the proof.

Suppose that P is a parabolic subgroup of a primitive reflection group G
of rank at least 3. Then the normaliser N of P is the semidirect product of
P and a subgroup H. Furthermore, there is a set J of roots of P such
that H is the setwise stabiliser of J in G except when

  * G = G_29 and the type of P is A_1, 2A_1, A_1+A_2 or
    B_3;

  * G = G_31 and the type of P is A_1, 2A_1, A_1+A_2,
    G(4,2,2) or G(4,2,3).

From results proved so far we may assume that P is neither rank 1, maximal,
nor of Coxeter type.  This leaves seven types to be considered. Sets of roots
J for the parabolic subgroups of types 2L_1 and L_2 in G_32, type
G(3,3,3) in G_33 and types G(3,3,3), G(3,3,4) and A_1+G(3,3,3) in
G_34 are given in Table <ref>. It can be checked directly that
the stabiliser of J is a parabolic complement for P.

The remaining case is a parabolic subgroup P of type G(4,2,2) in
G_31. Then Q is also a parabolic subgroup of type G(4,2,2).
Reflections r_1, r_2, …, r_5 which generate G_31 can be
obtained from the root and coroot matrices in Table <ref> and
we may suppose that P = r_1,r_3,r_5. The subgroup H generated by

    [    -1     0     0     0;     0    -i     1     1;     i     0    -i     0;    -1     0 i + 1     1 ]  and  [     -i      0     -1      0;  i + 1      0     -i     -i;      0      0     -1      0;  i - 1     -1 -i + 2 -i + 1 ]

is a complement to both P and Q in N = N_G(P). Thus H acts faithfully
on _V(P) and _V(P)^⊥ and its restrictions to these subspaces
are isomorphic to the Shephard and Todd group G_8, of order 96.

Let S be a Sylow 3-subgroup of N.  Then S normalises Q and the index
of M = PQS in N is 2.  A direct calculation shows that the orders of the
elements in N∖ M are 4 and 8 and for x∈ N∖ M the order
of QSx is either 384 or 1 536.  Thus no complement to P in N
contains Q, hence there is no complement that stabilises a set of roots of
P.

Let P be a parabolic subgroup of type G(3,3,3) in G = G_33. Then
N = N_G(P) acts faithfully on the support of P as the reflection group
G_26 and it acts on the space of fixed points of P as G_4.  There
are three conjugacy classes of complements to P in N only one of which
is represented by the setwise stabiliser H of a set of roots (see 
Table <ref>) of generators of P and every element of H is 1-regular.

For a parabolic subgroup P of type G(3,3,4) in G_33 we have
N = N_G(P) = P× Z, where the centre Z of G_33 has order 2. 
In this case Z is the setwise stabiliser of the roots {± a_1,± a_2,
± a_3, ± a_4} but it is not 1-regular.  There are two other 
conjugacy classes of complements to P in N, one of size 18, the other
of size 27; all of these complements are 1-regular and stabilisers of
a set of roots of P.


§ THE TABLES

The labels for the conjugacy classes of parabolic
subgroups of rank at least 3 use the notation introduced by Cohen
<cit.>. The captions on the tables use both the Cohen and the
Shephard and Todd naming schemes.

A parabolic subgroup which is the direct product of irreducible reflection
groups of types T_1, T_2,…, T_k will be labelled
T_1+T_2+⋯+T_k and if T_i = T for all i we denote the type by kT.

For the imprimitive reflection groups which occur in the tables we use the
Coxeter notation B_n for the group G(2,1,n), D_n for G(2,2,n) (n ≥
4) and I_2(m) for G(m,m,2) but otherwise use G(m,p,n) as in
<ref>.  For the Coxeter group of type G_2 we use I_2(6)
to avoid confusion with the Shephard and Todd notation G_k. The cyclic
reflection group of order n is denoted by _n except that A_1 and
L_1 denote the groups of orders 2 and 3 respectively. If there are two
conjugacy classes of parabolic subgroups of type T the classes are labelled
T and T'.

If A and B are groups, A B denotes the semidirect product of A
by B, where A is normalised by B.  The symbol A∘ B denotes a
central product of A and B.

The notation H^∘ in the column headed H indicates that the parabolic
complement H coincides with the reflection group H^∘.

Tables of conjugacy classes of the parabolic subgroups of the Coxeter groups
of types E_6, E_7, E_8, F_4, H_3 and H_4 can also be found in
<cit.>.  The identification of the reflection subgroup
H^∘ of the complement H was verified using  code
independently of the tables in <cit.> and
<cit.>.



§ ACKNOWLEDGEMENTS


This work is based on a portion of the first author's Ph. D. thesis, which
was supervised by the second author at the University of Sydney. We thank 
an anonymous referee for several useful comments.

abbrv