imprimitive complex reflection groups g m,p,n jian-yi shiag-algebra/sgw1/shi_slides.pdf · complex...

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).


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).


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.


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.


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.


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).


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 .


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


(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.


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).


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= ∅.


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.


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.


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.


§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.


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


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


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.

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