Reflection groups and polytopes over finite fields, II

0.31 MB·English

by  Barry Monson

Reflection Groups and Polytopes over Finite Fields, II
B. Monson∗
6
University of New Brunswick
0
0
Fredericton, New Brunswick, Canada E3B 5A3
2

n
and
a
J
Egon Schulte†
0
2 Northeastern University

] Boston, Massachussetts, USA, 02115
G
M
February 2, 2008
.
h
t
a
m
[

1 Abstract
v
2
When the standard representation of a crystallographic Coxeter group Γ is reduced
0
5 modulo an odd prime p, a finite representation in some orthogonal space over Z is
p
1 obtained. If Γ has a string diagram, the latter group will often be the automorphism
0
groupofafiniteregularpolytope. InPartIwedescribedthebasics ofthisconstruction
6
0 and enumerated the polytopes associated with the groups of rank 3 and the groups of
/
h spherical or Euclidean type. In this paper, we investigate such families of polytopes
t for more general choices of Γ, including all groups of rank 4. In particular, we study
a
m in depth the interplay between their geometric properties and the algebraic structure
: of the corresponding finite orthogonal group.
v
i Key Words: reflection groups, abstract regular polytopes
X
r AMS Subject Classification (2000): Primary: 51M20. Secondary: 20F55.
a
1 Introduction
The regular polytopes are a rich and ongoing source of mathematical ideas. Their combi-
natorial features, for instance, have been beautifully generalized in the theory of abstract
regular polytopes.
In [17], the precursor to this paper, we surveyed some of the essential properties of an
abstract regular polytope P, referring to [13] for details. Then, reframing the key results
∗Supported by NSERC of Canada Grant # 4818
†Supported by NSA-grants H98230-04-1-0116and H98230-05-1-0027
1
in [25], we outlined an abbreviated classification of finite, irreducible groups generated by
reflections in n-space V, over a field of odd characteristic p (see [17, Thm. 3.1]).
When G is a (possibly infinite) crystallographic Coxeter group with string diagram, re-
duction modulo an odd prime p of the standard real representation yields a finite reflection
group Gp, which we could then classify and which is often the automorphism group of a
finite, abstract regular n-polytope P. (If this is so, we say that Gp is a string C-group.)
Next we established two useful criteria for Gp to be a string C-group: Theorems 4.1 and
4.2 of [17] concern the features of V as an orthogonal space, as well as the action of standard
subgroups of Gp on V. With this, we were able to classify all groups Gp, and their polytopes,
whenever n ≤ 3, as well as when G is of spherical or Euclidean type, for all ranks n.
Here, we begin by summarizing in Section 2 some key notation. Next, in Section 3, we
extend our criteria for Gp to be a string C-group. Finally, in Sections 4 and 5, we discuss
and completely classify all 4-polytopes which arise from our construction.
2 Notation
We refer the reader to the notation and basic set up in [17]. Throughout, G = hr ,...,r i
0 n−1
will be a crystallographic Coxeter group [p ,p ,...,p ] with a string Coxeter diagram
1 2 n−1
∆ (G)(withbranches labeledp ,p ,...,p , respectively), obtainedfromthecorresponding
c 1 2 n−1
abstract Coxeter group Γ = hρ ,...,ρ i via the standard representation on real n-space V.
0 n−1
For any odd prime p, we may reduce G modulo p to obtain a subgroup Gp of GL (Z )
n p
generated by the modular images of the r ’s. We shall abuse notation by referring to the
i
modular images of objects by the same name (such as r , b , B = [b ], V, etc.). In particular,
i i ij
{b } will denote the standard basis for V = Zn. In any event, Gp is a subgroup of the
i p
orthogonal group O(Zn) of isometries for the (possibly singular) symmetric bilinear form
p
x·y, the latter being defined on Zn by means of the Gram matrix B; in particular, r is the
p i
orthogonal reflection with root b if b2 6= 0.
i i
Next we make a convenient definition: if p ≥ 5, or p = 3 but no branch of ∆ (G) is
c
marked 6, then we say that p is generic for G. In such cases, no node label of the diagram
∆(G) (for a basic system) is zero modp and a change in the underlying basic system for G
has the effect of merely conjugating Gp in GL (Z ). On the other hand, in the non-generic
n p
case, in which p = 3 and ∆ (G) has branches marked 6, the group Gp may depend essentially
c
on the actual diagram ∆(G) taken for the reduction mod p. (Note that p generic does not
necessarily mean that p ∤ |G|, or that certain subspaces of V are non-singular, etc.)
Recall from [17, Thm. 3.1] that an irreducible group Gp of the above sort, generated by
n ≥ 3 reflections, must necessarily be one of the following:
• an orthogonal group O(n,p,ǫ) = O(V) or O (n,p,ǫ) = O (V), excluding the cases
j j
O (3,3,0), O (3,5,0), O (5,3,0) (supposing for these three that disc(V) ∼ 1), and
1 2 2
also excluding the case O (4,3,−1); or
j
• the reduction mod p of one of the finite linear Coxeter groups of type A (p ∤ n+1),
n
B , D , E (p 6= 3), E , E , F , H or H .
n n 6 7 8 4 3 4
2
We shall say in these two cases that Gp is of orthogonal or spherical type, respectively,
although there is some overlap for small primes. Our description rests on the classification
of the finite irreducible reflection groups over any field, obtained in Zalesski˘ı & Sereˇzkin [25]
(see also [9, 20, 21, 24]). It is only a slight abuse of notation to let [p ,...,p ]p denote the
1 n−1
modular representation of a group [p ,...,p ], so long as p is generic for the group.
1 n−1
The generators r of Gp satisfy the Coxeter-type relations inherited from G. Our main
i
problemistodeterminewhenGp hastheintersection property(1)foritsstandardsubgroups.
For any J ⊆ {0,...,n−1}, we let Gp := hr |j 6∈ Ji; in particular, for k,l ∈ {0,...n−1}
J j
we let Gp := hr |j 6= ki and Gp := hr |j 6= k,li. Then Gp is a string C-group if and only
k j k,l j
if Gp satisfies the intersection property
hρ |i ∈ Ii∩hρ |i ∈ Ji = hρ |i ∈ I ∩Ji; (1)
i i i
and in this case Gp is the automorphism group of a finite regular polytope denoted by P(Gp)
(see [13, §2E]). Note as well that Gp is a string C-group if and only if Gp and Gp are string
0 n−1
C-groups and Gp ∩Gp = Gp . We also let V be the subspace of V = Zn spanned by
0 n−1 0,n−1 J p
{b |j 6∈ J}, and similarly for V ,V . Note that V is Gp-invariant.
j k k,l J J
3 The Intersection Property
The goal of this section is to assess when the modular reduction Gp of a crystallographic
string Coxeter group G satisfies the intersection property (1). In [17] we established a
number ofsufficient conditionsandverified thatGp, withp ≥ 3, hastheintersection property
whenever G has rank at most 3, or whenever G is of spherical or Euclidean type. However,
the situation changes drastically for more general groups of higher ranks, with obstructions
already occurring for rank 4. We already know that Gp is a string C-group if one of the
subgroups Gp or Gp is spherical and the other is a string C-group (see [17, Thm. 4.2]).
0 n−1
Moreover, Gp also is a string C-group if both Gp and Gp are C-groups, V is a non-
0 n−1 0,n−1
singular subspace of V, and Gp is the full orthogonal group O(n−2,p,ǫ) on V (see
0,n−1 0,n−1
[17, Thm. 4.1]).
The criteria established here will settle the Coxeter groups [k,l,m] of rank 4 completely.
Before we move on, note two simple cases. If l = 2 and k,m < ∞, then
G ∼= [k]×[m] ∼= Gp,
so Gp certainly is a C-group. Similarly, if say m = 2 (but k or l = ∞ is allowed), then
G ≃ [k,l]×C , and
2
Gp ≃ [k,l]p ×C .
2
Thus the intersection property of Gp follows directly from that of its subgroup [k,l]p (see
[17, Thm. 5.1]). More generally, if [p ,...,p ]p is a string C-group, then so is
1 n−1
[p ,...,p ,2]p ≃ [p ,...,p ]p ×C .
1 n−1 1 n−1 2
After a preliminary lemma, we shall continue to build upon the known results concerning
C-groups mentioned above.
3
Lemma 3.1 Suppose that G has rank n and that (rad V)∩V = {o}. Then the subgroup Gp
j j
is, by restriction to the invariant subspace V , isomorphic to Hp, the reduction modulo p of
j
the group of rank n−1 defined from the subdiagram of ∆(G) which results from the deletion
of node j. In particular, when j = 0 or n−1 this holds if p is generic for G.
Proof. This result is well known in characteristic 0 [8, §5.5]. Here we restrict g ∈ Gp to the
j
invariant subspace V , and so obtain a homomorphism
j
ϕ : Gp −→ O(V )
j j
g 7−→ g .
|Vj
Of course, as a subspace of V, V is isometric to Zn−1, with the metric structure obtained
j p
from the subdiagram of ∆(G) obtained by deleting node j. Clearly the image group ϕ(Gp)
j
is isomorphic to the reflection group Hp of rank n−1 defined directly from the subdiagram.
Suppose g ∈ kerϕ. Then g(b ) = b for all k 6= j, whereas g(b ) = b +x for some x ∈ V .
k k j j j
Thus for any k 6= j
b ·b = g(b )·g(b ) = (b +x)·b = b ·b +x·b ,
j k j k j k j k k
so that x·b = 0, and x ∈ rad V . But then x·x = 0 and so
k j
b ·b = g(b )·g(b ) = b ·b +2x·b +x·x ,
j j j j j j j
whence x·b = 0. Thus x ∈ (rad V)∩V , so that x = o when this subspace is trivial. Hence
j j
ϕ is injective. When p is generic for G, a direct calculation in coordinates along the string
diagram shows that (rad V)∩V = {o} for j = 0,n−1. (cid:3)
j
Remark. Informally, the Lemma asserts that reduction by a generic prime commutes with
the deletion of a node from ∆(G). Note that
Gp ≃ [p ,...,p ]p ×[p ,...,p ]p ≃ [p ,...,p ,2,p ,...,p ]p .
j 1 j−1 j+2 n−1 1 j−1 j+2 n−1
Concerning the non-generic cases, there are examples showing the necessity of the hy-
potheses. For example, the group G ≃ [4,3,6] with diagram
2 1 1 3
• • • •
yields, as we observe below, a C-groupG3. Here the subgroup G3 is the automorphism group
0
of order 108 for the toroidal polyhedron {3,6} . However, the subdiagram
(3,0)
1 1 3
• • •
yields the smaller group of order 36 for {3,6} . Thus the map ϕ of the Lemma is here not
(1,1)
injective.
Theorem 3.1 Suppose that G ≃ [k,l,m] is crystallographic and that the subgroup [k,l] or
[l,m] is spherical. Then Gp is a C-group for any prime p ≥ 3.
4
Proof. Let [k,l] (say) be spherical, so that Gp ≃ G = [k,l]. First suppose that p is generic
3 3
for G. Since Gp is a C-group by Lemma 3.1, the proof follows directly from [17, Thm. 4.2].
0
Moreover, even in non-generic cases of rank 4 (so that p = 3), G3 turns out to be a C-group
when [k,l] is spherical. This is routinely verified using the computer algebra system GAP
[7]. The pertinent examples are G ≃ [3,3,6],[3,4,6] or [4,3,6], each with two essentially
distinct diagrams ∆(G) (for the basic systems). (cid:3)
We now establish two general results, of which the first allows us to reject large classes
of groups Gp as C-groups because of the size of their subgroups Gp ∩ Gp . First we deal
0 n−1
with the fully non-singular case.
Theorem 3.2 Let G = hr ,...,r i be a crystallographic linear Coxeter group with string
0 n−1
diagram. Suppose that n ≥ 3 and that the prime p is generic for G. Let the subspaces V,
V , V and V be non-singular, and let Gp, Gp be of orthogonal type. Suppose as well
0 n−1 0,n−1 0 n−1
that there is a square among the labels of the nodes 1,...,n−2 of the diagram ∆(G) (this
can be achieved by readjusting the node labels).
(a) Then Gp ∩Gp acts trivially on V⊥ , and
0 n−1 0,n−1
O (V ) ≤ Gp ∩Gp ≤ O(V ),
1 0,n−1 0 n−1 0,n−1
where we have identified the (n−2)-dimensional groups O(V ) and O (V ) with the
0,n−1 1 0,n−1
pointwise stabilizers of V⊥ in the n-dimensional groups O(V) and O (V), respectively.
0,n−1 1
(b) If Gp = O(V ) and Gp = O(V ), with a similar interpretation as stabilizers, then
0 0 n−1 n−1
Gp ∩Gp = O(V ).
0 n−1 0,n−1
Proof. Since all four subspaces of V = hb ,...,b i are non-singular, we have the orthog-
0 n−1
onal sums
V = V ⊕hvi = V ⊕hv′i, V = V ⊕hwi, V = V ⊕hw′i,
n−1 0 n−1 0,n−1 0 0,n−1
for non-isotropic vectors v,v′,w,w′. Then,
hv,v′i = V⊥ = hw,w′i.
0,n−1
Since p is generic for G, each reflection r actually has b as a root, and v ⊥ b for j ≤ n−2,
j j j
while v′ ⊥ b for j ≥ 1. Hence the subgroups Gp , Gp and Gp ∩Gp stabilize the vectors
j n−1 0 0 n−1
v, v′ or v,v′, respectively. In particular,
Gp ∩Gp ≤ O(V ),
0 n−1 0,n−1
with O(V ) identified with the pointwise stabilizer of V⊥ in O(V). Note that the
0,n−1 0,n−1
restrictions of Gp , Gp and Gp∩Gp to the subspaces V , V or V , respectively, are
n−1 0 0 n−1 n−1 0 0,n−1
faithful, by Lemma 3.1.
Since Gp, Gp are of orthogonal type and there is a square among the labels of the
0 n−1
nodes 1,...,n−2, we must have O (V ) ≤ Gp and O (V ) ≤ Gp (that is, a group merely
1 n−1 n−1 1 0 0
of type O cannot occur). Now, if g ∈ O (V ), then g(v) = v, so that g ∈ O(V );
2 1 0,n−1 n−1
but the spinor norm is invariant under orthogonal embedding ([1, Thm. 5.13]), so actually
5
g ∈ O (V ). Similary, g ∈ O (V ), and hence g ∈ Gp ∩Gp . This completes the proof of
1 n−1 1 0 0 n−1
part (a).
Now let Gp = O(V ) and Gp = O(V ). Once again, if g ∈ O(V ), then g(v) = v,
0 0 n−1 n−1 0,n−1
so now g ∈ O(V ) = Gp . Similarly, g ∈ O(V ) = Gp, and hence g ∈ Gp ∩ Gp , as
n−1 n−1 0 0 0 n−1
(cid:3)
required.
We note an immediate corollary to Theorem 3.2. It shows that many groups Gp of rank
4 fail to satisfy the intersection property for large primes p. However, those primes for which
Gp actually is a C-group lead to interesting polytopes, which we investigate in later sections.
Corollary 3.1 Suppose the prime p is generic for the crystallographic group G = [k,l,m].
Let V, V , V , V be non-singular, and let Gp, Gp be of orthogonal type.
0 3 0,3 0 3
(a) Then Gp is not a C-group if p > 2l + ǫ(V ), where ǫ(V ) = ±1 is the parameter
0,3 0,3
associated with the plane V .
0,3
(b) If Gp = O(V ) and Gp = O(V ), then Gp is a C-group if and only if p = l+ǫ(V ).
0 0 3 3 0,3
Proof. We apply Theorem 3.2 with n = 4. The subgroups Gp and Gp are known to be
0 3
C-groups ([17, Thm. 5.1]), so it suffices to determine when
Gp ∩Gp = Gp .
0 3 0,3
Now Gp = hr ,r i is a dihedral group of order 2l = 6, 8 or 12; the case l = ∞ is excluded,
0,3 1 2
as V is then a non-singular plane. Note that we may assume that there is a square (in fact,
0,3
a 1) among the labels of the nodes 1 or 2 of the diagram; this can be achieved by readjusting
the node labels as described earlier. Then, by Theorem 3.2 we have
O (V ) ≤ Gp ∩Gp,
1 0,3 0 3
so Gp∩Gp is larger than Gp if the order of O (V ), which is p−ǫ(V ), exceeds 2l. Hence
0 3 0,3 1 0,3 03
the intersection property certainly fails if p > 2l + ǫ(V ). Moreover, by Theorem 3.2, if
0,3
Gp = O(V ) and Gp = O(V ), then
0 0 3 3
Gp ∩Gp = O(V ),
0 3 0,3
so Gp ∩Gp = Gp if and only if 2(p−ǫ(V )) = 2l, or equivalently, p = l+ǫ(V ). (cid:3)
0 3 0,3 03 0,3
Corollary 3.1 immediately implies (in fully non-singular cases) that Gp is not a C-group
if p > 13. However, for the primes p = 5,7,11,13 (and 3), the outcome is less predictable
and actually depends on the group G = [k,l,m] as well as the diagram ∆(G) chosen for the
reduction modulo p. For example, G13 can only be a C-group if l = 6 and G13 = O (V ) or
0 1 0
G13 = O (V ). Similarly, if Gp = O(V ) and Gp = O(V ), then Gp is not a C-group if p > 7;
3 1 3 0 0 3 3
moreover, G7 can then be a C-group only if l = 6.
NextwestudythecasewhenthemiddlesectionofthediagramforGdeterminesasingular
space V , while V, V and V still are non-singular, again with Gp,Gp of orthogonal
0,n−1 0 n−1 0 n−1
type. In a singular space W over a field K, the isometry group O(W) leaves invariant the
radicalsubspaceradW,therebyprovidinganaturalepimorphismη : O(W) → O(W/radW).
Since W/radW is non-singular, we may define a ‘spinor norm’ θ on W, sufficient for our
needs, by
θ(g) := θ (η(g)), g ∈ O(W) .
W/radW
6
Now let O(W) denote the subgroup of O(W) consisting of those isometries g which act
trivially on radW. It is not hard to show that O(W) contains all reflections (with non-
b
isotropic roots in W) and is even generated by them. The key observation here is that any
b
transvection in the kernel of the action of O(W) on radW can be factored as a product of
reflections (cf. [1, Thm. 3.20 and p. 133]).
It is easy to see that if g is the product of reflections with non-isotropic roots a ,...,a ,
1 k
then θ(g) = a2···a2K˙2. Naturally, by O (W) (or O (W)) we mean the subgroup of O(W)
1 k 1 2
generated by the reflections in O(W) whose spinor norm is a square (or non-square, respec-
b b b
tively).
Theorem 3.3 Let G = hr ,...,r i be a crystallographic linear Coxeter group with string
0 n−1
diagram, and suppose the prime p is generic for G. Let V, V , V be non-singular, let
0 n−1
V be singular, and let Gp,Gp be of orthogonal type. Suppose there is a square among
0,n−1 0 n−1
the labels of the nodes 1,...,n−2 of the diagram ∆(G) (this can be achieved by readjusting
the node labels).
(a) Then Gp ∩Gp acts trivially on V⊥ , and
0 n−1 0,n−1
O (V ) ≤ Gp ∩Gp ≤ O(V ),
1 0,n−1 0 n−1 0,n−1
where O(V ) has been ibdentified with the pointwisbe stabilizer of V⊥ in O(V), and
0,n−1 0,n−1
O (V ) with the subgroup of O (V) generated by the reflections with roots in V and
1 0,n−b1 1 0,n−1
square spinor norm.
b
(b) If Gp = O(V ) and Gp = O(V ), then also
0 0 n−1 n−1
O(V ) = Gp ∩Gp .
0,n−1 0 n−1
b
Proof. As before we have
V = V ⊕hvi = V ⊕hv′i
n−1 0
with non-isotropic vectors v,v′. The subspace V⊥ still is 2-dimensional, so necessarily
0,n−2
V⊥ = hv,v′i. Moreover,
0,n−1
V ∩V⊥ = radV 6= {o},
0,n−1 0,n−1 0,n−1
so the vectors in V ∪V⊥ span a singular hyperplane U in V with 1-dimensional radical
0,n−1 0,n−1
radU = radV . For the same reason as before, Gp , Gp and Gp ∩ Gp stabilize the
0,n−1 n−1 0 0 n−1
vectors v, v′ or v,v′, respectively, and yield faithful restrictions to the subspaces V , V
n−1 0
and V . In particular,
0,n−1
Gp ∩Gp ≤ H := {g ∈ O(V)|g(x) = x, ∀x ∈ V⊥ }.
0 n−1 0,n−1
Note also that Gp ∩Gp leaves U invariant because it leaves V invariant.
0 n−1 0,n−1
We claim that we may identify H with O(V ); this would settle the inclusion on the
0,n−1
right in part (a) of the theorem. Now, since each element of H leaves V invariant, while
b 0,n−1
fixing radV , we can consider the restriction mapping to V ,
0,n−1 0,n−1
κ : H −→ O(V )
0,n−1
(2)
g −→ g .
b|V0,n−1
7
We prove that κ is an isomorphism. If g ∈ ker(κ), then g acts trivially on V , and hence
0,n−1
also on U because g ∈ H. It follows that g = e, the identity mapping on V. Here we have
used the fact that an isometry of a non-singular space is uniquely determined by its effect
on a hyperplane, H in this case, if this hyperplane is singular ([1, Thm. 3.17]). This shows
that κ is injective. Now let h ∈ O(V ). Then h acts trivially on radV , so we can
0,n−1 0,n−1
extend h to an isometry h′ (say) of U = V ⊕hvi by setting h′(v) := v. We now apply
b 0,n−1
Witt’s extension theorem for isometries between subspaces of a non-singular space ([1, Thm.
3.9]) and conclude that h′ extends further to an isometry g of the entire space V. Then g
must be in H and so κ is also surjective. By our earlier remarks, O(V ) is generated by
0,n−1
all reflections r , for non-isotropic roots a ∈ V ; pulling back, a similar claim is true for
a 0,n−1 b
H.
Continuing along these lines, we now prove the inclusion on the left in part (a) of the
theorem. For a non-isotropic vector a ∈ V , let r , r , r and r denote
0,n−1 a,V a,Vn−1 a,V0 a,V0,n−1
the reflections with root a in V, V , V or V , respectively. Then r ∈ H because
n−1 0 0,n−1 a,V
a ⊥ V⊥ , and
0,n−1
r = κ(r ).
a,V0,n−1 a,V
It follows that the subgroup H of H generated by the reflections r , with a ∈ V and
1 a,V 0,n−1
a2 a square, is isomorphic, under κ, to O (V ). We have to show that H ≤ Gp ∩Gp .
1 0,n−1 1 0 n−1
Now, by assumption, the subgroupsbGp,Gp of G are of orthogonal type, and there is
0 n−1
a square among the labels of the nodes 1,...,n−2 of the diagram. It follows that Gp and
0
Gp , when restricted to the subspaces V or V , respectively, must contain the groups
n−1 0 n−1
O (V ) or O (V ). Hence, if we identify the restricted groups with the stabilizers of v′ or
1 0 1 n−1
v, respectively, in O (V), then we see that O (V ) and O (V ) are actually subgroups of
1 1 0 1 n−1
Gp and Gp . In particular, if a ∈ V and a2 is a square, then r belongs to O (V ), for
0 n−1 0,n−1 a,V 1 j
j = 0,n−1, so r ∈ Gp ∩Gp . Now it follows that H is a subgroup of Gp ∩Gp . This
a,V 0 n−1 1 0 n−1
settles part (a).
Finally, suppose that Gp = O(V ) and Gp = O(V ). Much of the same analysis
0 0 n−1 n−1
carries over, but now it is applied to all reflections, including those whose spinor norm is a
non-square. In particular, if a is any non-isotropic vector in V , then r ∈ Gp ∩Gp .
0,n−1 a,V 0 n−1
Hence we also have
O(V ) ≤ Gp ∩Gp ,
0,n−1 0 n−1
sinceO(V )isgeneratedbyitsrbeflections. Nowpart(a)gives theequalityofthesegroups.
0,n−1
(cid:3)
b
Remark. It is not difficult to show that O (V ) can also be identified with the pointwise
1 0,n−1
stabilizer of V⊥ in O (V).
0,n−1 1 b
For a crystallographic Coxeter group [k,l,m], the middle section of the diagram deter-
mines a singular subspace if and only if l = ∞. In this case, the reduced group Gp is always
a C-group:
Corollary 3.2 Let G ≃ [k,∞,m] be crystallographic. Then Gp is a C-group for any prime
p ≥ 3.
Proof. We know that Gp and Gp are C-groups ([17, Thm. 5.1]), so again it suffices to check
0 3
8
that
Gp ∩Gp = Gp .
0 3 0,3
Suppose for the moment that p is generic for G, so that we may apply Theorem 3.3 with
n = 4. Then, with few exceptions, V is still non-singular. Moreover, V and V correspond
0 3
to the subgroups [∞,m] and [k,∞] of G, and hence are known to be non-singular as well
(except in the non-generic case with p = 3, k,m = 6; see [17, Sect. 5]). However, V is a
0,3
singular plane, so Theorem 3.3 implies that
O (V ) ≤ Gp ∩Gp ≤ O(V ),
1 0,3 0 3 0,3
provided V is non-singular. If bthe labels of the nodesb1 and 2 of the diagram are 1 and 4,
respectively, as we may assume, then V is a singular plane in which the squared norm of
0,3
each non-isotropic vector is a square. In particular,
O (V ) = O(V ) ∼= [p] ∼= Gp ,
1 0,3 0,3 0,3
and hence Gp ∩ Gp = Gp . Inbfact, sinceb2b + b spans radV , a matrix representing an
0 3 0,3 1 2 0,3
element of O(V ) in the basis b ,2b +b must necessarily have the form
0,3 1 1 2
b ±1 0
(µ ∈ Z ),
(cid:20) µ 1 (cid:21) p
so there are at most 2p of them. On the other hand, the restrictions of r and r to V
1 2 0,3
alreadygenerateadihedralgroup[p]containedinO (V ),sothethreegroupsmustcoincide.
1 0,3
The space V is singular for the following groubps G (up to duality) and primes p(> 3):
[4,∞,3] and [6,∞,6], with p = 5; [3,∞,3] and [4,∞,6], with p = 7; and [3,∞,6], with
p = 13. In each of these cases, as well as for p = 3 for any of the crystallographic groups
[k,∞,m], computations in GAP confirm that Gp also is a C-group. (cid:3)
We now concentrate entirely on the groups G = [k,l,m] which are not yet covered by
the previous results. These groups have a Euclidean subgroup [k,l] or [l,m]. The following
theorem settles the case when l = 4 or 6.
Theorem 3.4 Let G ≃ [k,l,m] be crystallographic. Suppose the subgroup [k,l] or [l,m] is
Euclidean, and that l = 4 or 6. Then Gp is a C-group for any prime p ≥ 3.
Proof. Suppose G = [k,l] (say) is Euclidean. The case m = 2 was already settled, so let
3
m 6= 2. For the moment, let p be generic for G. Then V = hb ,...,b i is non-singular and
0 3
V is singular, so that every isometry of V is uniquely determined by its effect on V . Let
3 3
radV = hci (say).
3
Let g ∈ Gp ∩ Gp. Then g leaves V and V invariant, and fixes c. Since l = 4 or 6,
0 3 3 0,3
we necessarily have G = [3,6] or [4,4], so Gp is a semi-direct product of its “translation
3 3
subgroup” Tp (of order p2) by Gp . In particular, since we are allowed to multiply g by an
0,3
element in Gp , we may assume that g ∈ Tp. We prove that this forces g = e, hence the
0,3
desired conclusion.
When G = [4,4], we may assume that the labels of the nodes 0, 1, 2 of the diagram of
3
G are 1, 2, 1, respectively. Then c = b +b +b and
0 1 2
Tp = hr r r r ,r r r r i ∼= Z ×Z .
0 1 2 1 1 0 1 2 p p
9
Let Mp denote the group of p2 matrices of the form
1 0 0
M(λ,µ) :=  0 1 0  (λ,µ ∈ Z ). (3)
p
λ µ 1
 
In the basis b ,b ,c of V , each element of Tp, when restricted to V , is represented by a
0 1 3 3
matrix in Mp. (Tp acts faithfully on V by Lemma 3.1.) For example, we have
3
(r r r r )j(r r r r )k 7→ M(2j,2(k −j))
0 1 2 1 1 0 1 2
so Tp and Mp are clearly isomorphic. Now suppose that M(λ,µ) is the matrix for g. Then
g(b ) = b +µc = µb +(1+µ)b +µb ,
1 1 0 1 2
so we must have µ = 0 because V is invariant under g. Similarly,
0,3
g(b ) = g(c−b −b ) = g(c)−g(b )−g(b )
2 0 1 0 1
= c−(b +λc)−b = −λb −λb +(1−λ)b ,
0 1 0 1 2
so also λ = 0, for the same reason. It follows that g acts trivially on V and hence also on
3
V, that is, g = e.
When G = [3,6], we may take the labels of nodes 0, 1, 2 of the diagram to be 1, 1, 3,
3
respectively. Then c = b +2b +b and
0 1 2
Tp = hr r (r r )2,r r (r r )2i ∼= Z ×Z .
0 1 2 1 1 0 1 2 p p
The elements of Tp still are represented by the matrices M(λ,µ) in Mp (in the basis b ,b ,c
0 1
of V ), so we can proceed in a similar fashion as above. In particular, if M(λ,µ) is the matrix
3
for g, then we obtain µ = 0 from
g(b ) = b +µc = µb +(1+2µ)b +µb ,
1 1 0 1 2
and λ = 0 from
g(b ) = g(c−b −2b ) = g(c)−g(b )−2g(b )
2 0 1 0 1
= c−(b +λc)−2b = −λb −2λb +(1−λ)b ,
0 1 0 1 2
in each case using the invariance of V under g. Hence g = e, as required.
0,3
Only a few non-generic cases remain, which are not covered by previous theorems: p = 3
for [4,4,6], [3,6,3], [3,6,4], [3,6,6] or [3,6,∞]. But for each of the eleven possibly distinct
(cid:3)
basic systems here, we easily verify the intersection condition with the help of GAP.
This, then, leaves us with the groups G = [6,3,m]. If m = 3 or 4, then G is spherical,
0
so Theorem 3.1 applies and proves that Gp is a C-group. It remains to investigate the cases
m = 6 or ∞.
Theorem 3.5 Let G = [6,3,m] with m = 6 or ∞. Then for m = 6, Gp is a C-group only
for p = 3; and for m = ∞, Gp is a C-group if and only if p = 3 or p ≡ ±5 (mod 12).
10