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IMPRIMITIVE COMPLEX
REFLECTION GROUPS G(m, p, n)
Jian-yi Shi
East China NormalUniversity, Shanghai
andTechnische Universitat
Kaiserslautern
Typeset by AMS-TEX
1
2 Jian-yi Shi
§1. Preliminaries.
1.1. V , n-dim space/C.
A reflection s on V : s ∈ GL(V ), o(s) < ∞,
codimV s = 1.
A reflection group G on V is a finite group
generated by reflections on V .
1.2. A reflection group G in V is imprimitive,
if
(i) G acts on V irreducibly;
(ii) V = V1 ⊕ ...... ⊕ Vr with 0 6= Vi ( V such
that G permutes {Vi | 1 6 i 6 r}.
For σ ∈ Sn, denote by [(a1, ..., an)|σ] the n × n
monomial matrix with non-zero entries ai in the
(i, (i)σ)-positions.
Complex Reflection Groups 3
For p|m (read “ p divides m ”) in N, set
G(m,p,n)=
{[(a1,...,an)|σ]
∣∣∣∣ai∈C, a
mi =1, σ∈Sn;
(∏
jaj)
m/p = 1
}
G(m, p, n) is the matrix form of an imprimitive
reflection group acting on V w.r.t. an orthonor-
mal basis e1, e2, ..., en.
4 Jian-yi Shi
1.3. Three tasks:
(1) A reflection group G can be presented by
generators and relations (not unique in general).
Classify all the presentations for G (J. Shi).
(2) Length function is an important tool in
the study of reflection groups.
Find explicit formulae for the reflection length
of elements of G (J. Shi).
(3) Automorphisms of G is one of the main
aspects in the theory of reflection groups.
Describe the automorphism groups of G (J.
Shi and L. Wang).
In this talk, we only consider G = G(m, p, n).
Complex Reflection Groups 5
1.4. For a reflection group G, a presentation of
G by generators & relations (or a presentation
in short) is by definition a pair (S, P ), where
(1) S is a finite set of reflection generators for
G with minimal possible cardinality.
(2) P is a finite relation set on S, and any
other relation on S is a consequence of the rela-
tions in P .
1.5. Two presentations (S, P ) and (S′, P ′) for
G are congruent, if ∃ a bijection η : S −→ S′
such that for any s, t ∈ S,
(∗) 〈s, t〉 ∼= 〈η(s), η(t)〉
“ Congruence ” to complex reflection groups
is an analogue of “ isomorphism ” to Coxeter
systems (only concerning relations involving at
most two generators).
6 Jian-yi Shi
§2. Graphs associated to reflection sets.
2.1. ∃ two kinds of reflections in G(m, 1, n):
∀ i < j in [n],
(i) s(i,j; k)=[(1,...,1,
ith︷︸︸︷ζ−km ,1, ...,1,
jth︷︸︸︷ζkm ,1,...,1)|(i,j)],
where (i, j) is the transposition of i and j, and
ζm := exp(
2πim
).
Call s(i, j; k) a reflection of type I. set s(j, i; k) =
s(i, j;−k).
(ii) s(i; k) = [(1, ..., 1,
jth︷︸︸︷ζkm , 1, ..., 1)|1] for some
k ∈ Z, m ∤ k.
Call s(i; k) a reflection of type II, having order
m/gcd(m, k).
All the reflections of type I lie in the subgroup
G(m,m, n).
Complex Reflection Groups 7
2.2. To any set X = {s(ih, jh; kh) | h ∈ J} of
reflections of G(m, 1, n) of type I, we associate
a digraph ΓX = (V,E) as follows. Its node set
is V = [n], and its arrow set E consists of all
the pairs {i, j} with labels k for any s(i, j; k) ∈
X. Denote by ΓX the underlying graph of ΓX
(replacing labelled arrows by unlabelled edges).
ΓX has no loop but may have multi-edges.
Let Y = X∪{s(i; k)}. we define another kind
of graph ΓrY , which is obtained from ΓX by root-
ing the node i, i.e., ΓrY is a rooted graph with
the rooted node i. Sometimes we denote ΓrY by
([n], E, i).
Use notation ΓY for ΓX .
8 Jian-yi Shi
Examples 2.3. Let n = 6.
(1)X = {s(1, 2; 4), s(3, 4; 2), s(4, 6; 0), s(3, 4; 3)}.
Then ΓX is
1 2 3 4 564
2
3 0
ΓX is
1 2 3 4 56
(2) Let Y = X ∪ {s(6; 3)}. Then ΓrY is
1 2 3 4 56
Note: reflections of type I are represented by
edges, rather than nodes.
Complex Reflection Groups 9
2.4. We described the congruence classes
of presentations (c.c.p. in short) for two spe-
cial families of imprimitive complex reflection
groups G(m, 1, n) and G(m,m, n) in terms of
graphs.
Theorem 2.5. The map (S, P ) → ΓrS induces
a bijection from the set of c.c.p.’s of G(m, 1, n)
to the set of isom. classes of rooted trees with
n nodes.
Theorem 2.6. The map (S, P ) → ΓS induces
a bijection from the set of c.c.p.’s of G(m,m, n)
to the set of isom. classes of connected graphs
with n nodes and n edges (or equivalently with
n nodes and exactly one circle).
10 Jian-yi Shi
Examples 2.7. Let n = 4.
(1) There are 4 isomorphic classes of rooted
trees of 4 nodes:
•——◦——◦—— ◦ ◦——•——◦——◦
•|
◦——◦——◦
◦|
◦——•——◦
HenceG(m, 1, 4) has 4 congruence classes of pre-
sentations.
(2) There are 5 isomorphic classes of con-
nected graphs with 4 nodes and exactly one cir-
cle:
◦===◦——◦—— ◦ ◦——◦===◦——◦
◦‖
◦——◦——◦
◦� �◦——◦——◦
◦——◦| |◦——◦
Hence G(m,m, 4) has 5 congruence classes of
presentations.
Complex Reflection Groups 11
Now we consider the imprimitive complex re-
flection group G(m, p, n) for any m, p, n ∈ N
with p|m (read “ p divides m ”) and 1 < p < m.
Lemma 2.8. The generator set S in a presen-
tation (S,P) of the group G(m,p,n) consists of n
reflections of type I and one reflection of order
m/p and type II. Moreover, the graph ΓS is
connected with exactly one circle.
2.9. Assume thatX is a reflection set ofG(m, p, n)
with ΓX connected and containing exactly one
circle, say the edges of the circle are {ah, ah+1},
1 6 h 6 r (the subscripts are modulo r) for
some integer 2 6 r 6 n. Then X contains the
reflections s(ah, ah+1; kh) with some integers kh
for any 1 6 h 6 r (the subscripts are modulo
r). Denote by δ(X) := |∑r
h=1 kh|.
12 Jian-yi Shi
Now we can characterize a reflection set of
G(m, p, n) to be the generator set of a presenta-
tion as follows.
Theorem 2.10. Let X be a subset of G(m, p, n)
consisting of n reflections of type I and one re-
flection of order m/p and type II such that the
graph ΓX is connected. Then X is the generator
set in a presentation of G(m, p, n) if and only if
gcd{p, δ(X)} = 1.
Complex Reflection Groups 13
2.11. Define the following sets:
Σ(m, p, n): the set of all S which form the gen-
erator set in some presentation of G(m, p, n).
Λ(m, p): the set of all d ∈ N such that d|m and
gcd{d, p} = 1.
Γ(m, p, n): the set of all the connected rooted
graphs with n nodes and n edges.
Γ1(m, p, n): the set of all the rooted graphs in
Γ(m, p, n) each contains a two-nodes circle.
Γ2(m, p, n): the complement of Γ1(m, p, n) in
Γ(m, p, n).
Γ(m, p, n), resp., Γi(m, p, n):
the set of the isomorphism classes in the
set Γ(m, p, n), resp., Γi(m, p, n) for i = 1, 2.
Σ(m, p, n): set of congruence classes in Σ(m, p, n).
14 Jian-yi Shi
Now we describe all the congruence classes of
presentations for G(m, p, n) in terms of rooted
graphs.
Theorem 2.12. (1) The map ψ : S 7→ ΓrS from
Σ(m, p, n) to Γ(m, p, n) induces a surjection
ψ: Σ(m, p, n) ։ Γ(m, p, n).
(2) Let Σi(m, p, n) = ψ−1(Γi(m, p, n)) for i =
1, 2.
Then the map ψ gives rise to a bijection:
Σ2(m, p, n)←→ Γ2(m, p, n);
also, S 7→ (ΓrS , gcd{m, δ(S)})
induces a bijection:
Σ1(m, p, n)←→ Γ1(m, p, n)× Λ(m, p).
Complex Reflection Groups 15
Example 2.13. Let n = 4, m = 6 and p =
2. Then Λ(6, 2) = {1, 3}. ∃ 13 isomorphic
classes of rooted connected graphs with 4 nodes
and exactly one circle, 9 of them contain a two-
nodes circle. So G(6, 2, 4) has 22 = 9× 2 + 4
congruence classes of presentations.
•===◦——◦—— ◦ ◦===•——◦——◦
◦===◦——•—— ◦ ◦===◦——◦——•
•——◦===◦—— ◦ ◦——•===◦——◦
•‖
◦——◦——◦
◦‖
•——◦——◦
◦‖
◦——•——◦
•� �◦——◦——◦
◦� �◦——•——◦
◦� �◦——◦——•
•——◦| |◦——◦
16 Jian-yi Shi
§3. The relation set of a presentation for
G(m, p, n).
3.1. Let S = {s, th | 1 6 h 6 n} be in Σ(m, p, n),
where
s = s(a; k);
all the th’s are of type I;
a is the rooted node of ΓrS .
Complex Reflection Groups 17
3.2. The following relations hold:
(A) sm/p = 1;
(B) t2i = 1 for 1 6 i 6 n;
(C) titj = tjti if the edges e(ti) and e(tj) have
no common end node;
(D) titjti = tjtitj if the edges e(ti) and e(tj)
have exactly one common end node;
(E) stisti = tistis if a is an end node of e(ti);
(F) sti = tis if a is not an end node of e(ti);
(G) (titj)m/d = 1 if ti 6= tj with e(ti) and
e(tj) having two common end nodes, where d =
gcd{m, δ(S)};
(H) ti · tjtltj = tjtltj · ti for any triple X =
{ti, tj , tl} ⊆ S with ΓX having a branching node
(I) s · titjti = titjti · s, if e(ti) and e(tj) have
exactly one common end node a;
18 Jian-yi Shi
Call shj := thth+1...tj−1tjtj−1...th
a path reflection in ΓrS :
◦th
——◦th+1——◦—– · · · · · · –—◦
tj
——◦
(J) (s1jsj+1,r)m
gcd{m,δ(S)} = 1 for p < j < q.
a
a
a
a
0 1x= tt1
r r−1r
a
a
a
a
q
p+1
t
p+1 t
q
ta
q−1
t j+1
j
j−1
j+1
ajap
(K) ss1jsj+1,r = s1jsj+1,rs for p < j < q
(L) (sj+1,rs1j)p−1 = s−δ(S)s1js
δ(S)sj+1,r for
p < j < q.
a
a
a
a
0 1x= tt1
r r−1r
a
a
a
a
q
p+1
t
p+1 t
q
ta
q−1
t j+1
j
j−1
j+1
ajap
Complex Reflection Groups 19
(M) For p < j < q
(a) us1ju · vsj+1,rv = vsj+1,rv · us1ju,
(b) us1jsj+1,rus1jsj+1,r =s1jsj+1,rus1jsj+1,ru,
(c) vs1jsj+1,rvs1jsj+1,r =s1jsj+1,rvs1jsj+1,rv,
a
a
a
a
0 1x= tt1
r r−1r
a
a
a
a
q
p+1
t
p+1 t
q
ta
q−1
t j+1
j
j−1
j+1
aj
u v
ap
Call all the relations (A)-(M) above
the basic relations on S.
20 Jian-yi Shi
Then we have.
Theorem 3.3. Let S ∈ Σ(m, p, n) and let PS
be the set of all the basic relations on S. Then
(S, PS) forms a presentation of G(m, p, n).
Remark 3.4. There are too much basic rela-
tions on S in general. We can get a much smaller
subset P ′
S from PS such that (S, P ′
S) still forms
a presentation of G(m, p, n). Under the assump-
tion of relations (A)–(F), we can reduce the size
of relation set (J) by replacing it by (J′), the lat-
ter consists of any single relation in (J). Similar
for (K), (L) and (M). The size of the relation
sets (I) and (J) can also be reduced.
Complex Reflection Groups 21
3.4. Two kinds of presentations have simpler
relation sets:
(i) ΓrS is a string:
•===◦——◦——◦— · · · · · ·—◦——◦
(ii) ΓS is a circle:
22 Jian-yi Shi
§4. Reflection length.
4.1. T , the set of all the reflections inG(m, p, n).
Any w ∈ G(m, p, n) has an expression w =
s1s2 · · · sr with si ∈ T . Denote by lT (w) the
smallest possible r among all such expressions.
Call lT (w) the reflection length of w.
lT (w) on G(m, p, n) is presentation-free. We
have lT (w) 6 lS(w) for any presentation (S, P )
of G(m, p, n).
Except for the case of G(m, 1, n) with one
special presentation (see 5.1), so far we have no
close formula of the length function lS(w) on
G(m, p, n) with 1 < p 6 m, where (S, P ) is any
presentation of G(m, p, n).
Complex Reflection Groups 23
4.2. Given m, p, r ∈ P with p|m. Let C =
[[c1, c2, ..., cr]] be a multi-set of r integers. P =
{P1, ..., Pl} a partition of [r].
Call E ⊆ [r]
(C,m)-perfect if∑
h∈E ch ≡ 0 (mod m); and
(C,m, p)-semi-perfect, if∑
h∈E ch ≡ 0 (mod p)
and∑
h∈E ch 6≡ 0 (mod m).
Call P
(C,m)-admissible if Pj is (C,m)-perfect for any
j ∈ [l]; and
(C,m, p)-semi-admissible if Pj is either (C,m)-
perfect or (C,m, p)-semi-perfect for any j ∈ [l].
24 Jian-yi Shi
Let Λ(C;m) (resp., Λ(C;m, p)) be the set of
all the (C,m)-admissible (resp., (C,m, p)-semi-
admissible) partitions of [r].
When Λ(C;m) 6= ∅ (resp., Λ(C;m, p) 6= ∅),
denote by
t(P ) (resp., u(P )) the number of (C,m)-perfect
(resp., (C,m, p)-semi-perfect) blocks of P for
any P ∈ Λ(C;m) (resp., P ∈ Λ(C;m, p)), and
define
t(C,m) = max{t(P ) | P ∈ Λ(C;m)}.
Define
v(P ) = 2t(P ) + u(P )
for any P ∈ Λ(C;m, p). Define
v(C,m, p) = max{v(P ) | P ∈ Λ(C;m, p)}
if Λ(C;m, p) 6= ∅.
Complex Reflection Groups 25
4.3. For w = [ζa1m , ..., ζan
m |σ] ∈ G(m, p, n), write:
σ = (i11, i12, ..., i1m1)......(ir1, ir2, ..., irmr
)
with∑
j∈[r]mj = n. Denote
(i) r(w) = r.
Let Ij = {ij1, ij2, ..., ijmj} for j ∈ [r]. Then
I(w) = {I1, ..., Ir} is a partition of [n] deter-
mined by w. Let cj =∑
k∈Ijak and let C(w) =
[[c1, c2, ..., cr]]. Denote
Λ(w;m, p) := Λ(C(w);m, p).
For w ∈ G(m, p, n), we always have
Λ(w;m, p) 6= ∅.
Denote
(ii) t(w) := t(C(w),m) if p = m and
(iii) v(w) = v(C(w),m, p) if p|m.
(iv) t0(w) = #{j ∈ [r] | cj ≡ 0 (mod m)}.
26 Jian-yi Shi
Theorem 4.4. lT (w) = n− t0(w).
for any w ∈ G(m, 1, n).
Theorem 4.5. lT (w) = n+ r(w)− 2t(w)
for any w ∈ G(m,m, n).
Theorem 4.6. Let m, p, n ∈ P be with p|m.
Then
lT (w) = n+ r(w)− v(w)
for any w ∈ G(m, p, n).
When w ∈ G(m, 1, n), we have
t0(w) = v(w)− r(w);
when w ∈ G(m,m, n), we have
v(w) = 2t(w).
So Theorems 4.4–4.5 are special cases of
Theorem 4.6.
Complex Reflection Groups 27
4.7. For any y, w ∈ G(m, p, n), denote by y ⋖ w
and call w covers y (or y is covered by w), if
yw−1 is a reflection with lT (w) = lT (y) + 1.
The reflection order � on G(m, p, n) is the
transitive closure of the covering relations ⋖.
28 Jian-yi Shi
4.8. For any cyclic permutation σ ∈ Sn, the set
B(σ) = {τ ∈ Sn | τ � σ} can be described in
terms of circle non-intersecting partitions.
Put the nodes 1, 2, ..., k on a circle clockwise.
Partition these k nodes into h blocks X1, ...,Xh
with Xj 6= ∅, j ∈ [h], such that the convex hulls
Xj , j ∈ [h], of these blocks are pairwise disjoint.
The partition X = {X1, ...,Xh} is called a circle
non-intersecting partition of [k]. Reading the
nodes of each Xj clockwise along the boundary
of Xj , we get a cyclic permutation τj . Then set
τ(X) = τ1τ2 · · · τh.
Complex Reflection Groups 29
Example 4.9. Let σ = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) ∈
Sn. Take a partition X of [11] as in Figure 1.
Then τ(X) ∈ B(σ) is (1, 2, 9, 10)(4, 5, 8)(6, 7)(3)(11).
11
10
9
8
7
6
5
4
3
21
Figure 1.
τ � σ if and only if τ = τ(X) for some circle
non-intersecting partition X of [11].
30 Jian-yi Shi
The relation x � y can also be described com-
binatorially in the group G(m, 1, n).
Theorem 4.10.
Let w = [ζa1m , ..., ζan
m |σ] ∈ G(m, 1, n) be with σ =
(1, 2, ..., r) a cyclic permutation and aj = 0 for
j > r.
(1) If∑
j∈[r] aj ≡ 0 (mod m), then |B(w)| =
Cr.
(2) If∑
j∈[r] aj 6≡ 0 (mod m), then |B(w)| =
(r + 1) · Cr,
where Cr =1
r+1
(2r
r
), the rth Catalan number.
Complex Reflection Groups 31
§5. Auto. group Aut(m, p, n) of G(m, p, n).
Assume m > 2 and (p, n) 6= (m, 2)
(i.e. G(m, p, n) is not Coxeter).
5.1. G(m, p, n) has a generator set S0:
(i) {s0, s′
1, si | i ∈ [n− 1]} if 1 < p < m;
(ii) {s0, si | i ∈ [n− 1]} if p = 1;
(iii) {s′1, si | i ∈ [n− 1]} if p = m,
where s0 = s(1; p), s′1 = s(1, 2;−1) and
si = s(i, i+ 1; 0).
s’ s ss0
1 2 n−1
s s2 n−1
s s ss0
1 2 n−1
s’1
s1
s1
G(m,1,n)
G(m,m,n)
G(m,p,n)
32 Jian-yi Shi
5.2. By an automorphism φ of a reflection group
G, it means that φ is an automorphism of the
group G as an abstract group which sends re-
flections of G to reflections.
5.3. Two presentations (S, P ), (S′, P ′) of
G(m, p, n) are called strongly congruent,
if there exists a bijective map η : S → S′ such
that P ′ = η(P ), where η(P ) is obtained from P
by substituting any s ∈ S by η(s).
strongly congruent=⇒6⇐=
congruent.
A strongly congruent map η can be extended
uniquely to an automorphism of G.
Complex Reflection Groups 33
5.4. Let τg : h 7→ ghg−1 be the inner automor-
phism of G(m, p, n). Let
Int(m, p, n) = {τg | g ∈ G(m, p, n)}.
5.5. Set Φ(m) := {i ∈ [m− 1] | gcd(i,m) = 1}.
For any k ∈ Φ(m) and any matrix w = (aij),
define
ψk(w) = (akij).
If
w = [ζa1m , ..., ζan
m |σ] ∈ G(m, p, n),
then
ψk(w) = [ζka1m , ..., ζkan
m |σ] ∈ G(m, p, n).
We have ψk ∈ Aut(m, p, n).
Define
Ψ(m) := {ψk | k ∈ Φ(m)}.
34 Jian-yi Shi
5.6. Let λ ∈ Aut(m, p, n), 1 < p 6 m, be
determined by
λ(s0) = s−10 ,
λ(s′1) = s1,
λ(s1) = s′1 and
λ(si) = si for 1 < i < n
s
s
0
1
s’1
s0
s1
s’1
−1
λ
s2 sn−1
s2 sn−1
1 23 n−1 n
1 23 n−1 n
Complex Reflection Groups 35
Let λ′ ∈ Aut(3, 3, 3) be determined by
λ′(s′1) = s(2, 3;−1),
λ′(si) = si for i = 1, 2.
s1
s’1
s21
23
s2
s1
s(2,3;−1)
1 2 3
λ’
36 Jian-yi Shi
Let η ∈ Aut(4, 2, 2) be determined by
(η(s0), η(s1), η(s′
1)) = (s1, s0, s′
1).
s’1
1 2s0
s1
1s’
1 2s1
s0
η
Complex Reflection Groups 37
Theorem 5.7.
(1) If gcd(p, n) = 1, then
Aut(m, p, n) = Int(m, p, n) ⋊ Ψ(m);
(2) If gcd(p, n) > 1 and
(m, p, n) 6= (3, 3, 3), (4, 2, 2), then
Aut(m, p, n) = 〈Int(m, p, n),Ψ(m), λ〉;
(3) Aut(3, 3, 3) = 〈τs1, λ, λ′〉.
(4) Aut(4, 2, 2) = 〈ψ3, λ, η〉.
Theorem 5.8. The order of Aut(m, p, n) is
mn−1n!φ(m) if (m, p, n) 6= (3, 3, 3), (4, 2, 2),
432 if (m, p, n) = (3, 3, 3),
48 if (m, p, n) = (4, 2, 2).
38 Jian-yi Shi
Some structural properties of Aut(m, p, n) are
studied. For example, we show that
the center Z(Aut(m, p, n)) of Aut(m, p, n) is
trivial if n > 2;
while Z(Aut(m, p, 2)) contains 2 · gcd(m, 2)
elements.