A linear transformation g of a vector space V over a field F of characteristic 0 is a
reflection if [V, g] := Im(Id−g) has dimension 1. Intuitively, these are transformations that fix some hyperplane in V, and groups generated by reflections are consequently
called reflection groups. Theroot of a reflection g is any non-zero vector in [V, g]. Let V be a complex vector space of dimension n, equipped with a positive definite hermitian form (−,−) : V × V → C. An isometry g is a transformation that preserves this form (that is, (u, v) = (g ·u, g ·v), and the unitary group, U(V) is the group of isometries of V. Any finite reflection group G of V preserves some hermitian form (take (u, v) :=P
g∈G(gu, gw)), so is a subgroup of U(V) after picking
an appropriate hermitian form. As any two forms are equivalent over C, U(V) is unique up to conjugacy in GL(V) so G appears as a subgroup of U(V) under the regular hermitian formh,i, given byh(u1, . . . , un),(v1, . . . , vn)i=u1v1+. . . unvn, where
vi denotes complex conjugation, and (u1, . . . , un), (v1, . . . , vn)∈Cn.
Groups G that are generated by complex reflections of some complex vector space V are called complex reflection groups, (sometimes unitary reflection groups in the literature) and if G is a finite subgroup of U(V) with V irreducible, then G is an irreducible complex reflection group. These are the building blocks of all complex reflection groups, and were classified by Shephard and Todd in 1954. It was shown that the irreducible complex reflection groups belong either to a family G(m, n, p) or are one of 34 exceptional groups. The description of the complex reflection groups can be found in [25], the reader is referred there and [4] for an introduction to the subject.
If N is the normalizer of a split torus T over a field k, char(k) 6= p in a semisimple algebraic group G, then define Tb as the integral representation of the Weyl group W ' N/T acting on the character lattice T∗. Theorem 1.10 in [30] asserts the following
and
ed(N;p)≤SymRank(Tb;p)−dim(T) + ed(F;p). (4.6)
Also, if the condition KF was satisfied, then the lower bound was an equality. Due
to the surjection of Galois cohomologyH1(k, N)H1(k, G), see [43, III.4.3 Lemma 6] together with [1, Lemma 1.3] the symmetric p-rank was calculated for each case, and this gave bounds on the essentialp-dimension of G.
The aim of the following is to attempt something similar for some complex reflection groups, though there does exist some issues upon stepping outside the realm of Weyl groups.
Firstly, there is not necessarily a canonical choice for a root lattice; complex root systems formcomplex lattices, which are freeJ-modules for some suitable ringJ ⊂
C. However, often complex reflection groups appear as low index subgroups inside an irreducible maximal finite subgroup of GLn(Z); this phenomenon is explored in Section 4.2.2.
Secondly, (4.5–4.6) relies on the construction of a representation of dimension equal to the symmetric p-rank; this was realised due to the representation of highest weight, which can only be applied in the case of algebraic groups. However, there does always exist a representation of this dimension for the split extensionT oW for aW action on T∗, and the arguments used in (4.5–4.6) can be applied to this group. For completeness, a proof is provided in Theorem 4.2.1, but note that the arguments are not the author’s own.
Moreover, there doesn’t exist an analogue of a semisimple algebraic group that has a complex reflection group as its Weyl group. Although there certainly does exist the groupToW of which one can attain bounds on the essentialp-dimension, this group
isn’t a normalizer of a split torus inside some larger algebraic group.
The story doesn’t quite end there however; the search for such analogues has lead to much fruitful mathematics, including the study of Spetses, where certain unipotent characters seem to evidence a mysterious algebraic structure that behaves like an analogue of semisimple algebraic groups for complex reflection groups, see [3] for more details.
Theorem 4.2.1. Let L be a W-lattice for some finite group W, with φW : W →
GL(L) the integral representation, and define the split torus T := Diag(L) over k,
char(k)6=p. There exists the following bounds on edk(T oW;p),
max{SymRank(φW;p)} −dim(T),edk(W;p)} ≤edk(T oW;p)
and
edk(T oW;p)≤SymRank(φW;p)−dim(T) + edk(W;p).
If the p-generating subset of L satisfiesKF (see Lemma 3.5.3), then the lower bound
is an equality.
Proof. Let Γp be a Sylow p-subgroup of W. Using Lemma 3.5.6 and replacing k by
k(p), means the value of the essential p-dimension of the groups remains unchanged, and the groups are smooth. Then by Lemma 3.3.2, edk(T oW;p) = edk(T oΓp;p),
and edk(W;p) = edk(Γp;p) so replaceW by Γp. For the lower bound, as the extension
is split, we can apply Proposition 3.5.1, so edk(Γp;p)≤edk(T oΓp;p). Also, suppose
V is a p-faithful representation of T o Γp, which decomposes into weight spaces {Vλ | λ ∈ ∆} for a subset ∆ ⊂ L = T∗. As V is p-faithful, the elements in ∆
generate a sublattice of L = T∗ of index that is finite and prime to p (as shown in the proof of Corollary 3.4.3). As Γp permutes the weight spaces, ∆ would also have
to be invariant under Γp, hence dim(V) = |∆| ≥ SymRank(φW;p). Applying the
lower bound of Theorem 3.5.7 gives SymRank(φW;p)−dim(T) ≤edk(T oW;p). If the subset ∆ satisfies the condition KF, then this is bound is an equality, by Lemma
3.5.3.
For the upper bound, let ∆ be a minimal p-generating, Γp-invariant subset of L.
Then there exists ap-faithful representation V∆ ofT oΓp of dimension |∆|, given by
V∆ := Spank{(vλ) | λ ∈ ∆}. The finite group group Γp acts by permuting the basis
elements, g :vλ 7→ vg·λ, for all g ∈Γp, λ ∈∆, and T acts by t : vλ 7→λ(t)vλ for any
t∈T and vλ ∈Vλ (see [35, pp. 473]). This action on the basis {vλ | λ ∈∆}, is then
extend linearly to the whole of V∆. If KF is satisfied, then by Lemma 3.5.3 it is p-
generically free, and if not, as Γp is ap-group, there exists a faithful representationVΓ of Γp of dimension edk(Γp;p) (from Theorem 3.3.1) and by Proposition 3.5.2,V∆×VΓ is p-generically free of dimension |∆|+ edk(W;p).
Example 4.2.2 shows how to use the tables for calculating these bounds.
The symmetric p-ranks of lattices associated to complex reflection groups of rank 3 or greater (those which can be easily attributed to complex root systems) can be found in Table 4.2. and the essential p-dimension of the complex reflection groups themselves was studied in [15]. Remarkably it turns out to be equal to the number of fundamental invariants ofW that divide p. The list for each of the complex reflection groups considered here is given in Table A.1 in Appendix A.
gives the lattice that contains that complex reflection group via the construction outlined in 4.2.2, along with the index of the complex reflection group in the full automorphism group of that lattice. The column labelledKF denotes at whichp the
condition KF is satisfied (see Lemma 3.5.3). The condition is not satisfied mostly in
the cases where SymRank(φL;p) = rank(L) so 0 ≤ edk(G;p) ≤ 1, for G satisfying
(2.31). As the only algebraic groups with essential dimension 0 are connected [40, Theorem 5.4], in this case edk(G;p) = 1. To elucidate how to use Tables 4.2 and A.1
in conjunction with Theorem 4.2.1, consider the following example.
Example 4.2.2. Take the complex reflection group J3(4). For each prime considered, assume char(k) 6=p. The corresponding lattice Q6(1) contains W(J
(4)
3 ) as an index 2 subgroup (more information on this group is given in 4.3.2). The symmetric 2- rank of both the Weyl group and the larger full automorphism group is 16, and as KF is satisfied for p = 2, ed(T oW(J3(4)); 2) = 10. The symmetric 3-rank is equal to the rank, and as mentioned previously T oW(J3(4)) is not connected, its essential 3-dimension is therefore 1. Finally, the symmetric 7-rank is 7, so 1 ≤
ed(T oW(J3(4)); 7) ≤ 2. In fact, as a C7-lattice, Q6(1) ' A6 (the A6 root lattice) so edk(T oW(J
(4)
3 ); 7) = edk(Diag(A6)oC7; 7) = 2 (see the beginning of 5.3 for an explanation of this figure).
Remark 4.2.3. Comparing Table 4.2 of symmetric p-ranks with Table A.1, some relationships appear. Firstly, edk(W;p) = 0 precisely when the group W has no
p-symmetry and if edk(W;p) = 1, then the symmetric p-rank is either equal to the
rank, or in the case of p = 7, the nearest multiple of p greater than the rank. For edk(W;p) =n ≥2, the value of the symmetricp-rank is always greater than pn.
Σ Λ(Σ)real [Aut(Λ) : W(Σ)] SymRank(φL;p) KF p= 2 p= 3 p= 5 p= 7 H3 Q6(4) 2 16 o o - 2 Q6(4)+2 12 o o - - Q6(4)+4 16 o o - 2 J3(4) Q6(1) 2 16 o - 7 2 L3 E6 25·5 16 27 - - 2, 3 M3 24·5 16 27 - - 2, 3 J3(5) Q12 22 32 54 o - 2, 3 F4 D4 1 16 9 - - 2, 3 H4 Q8(1) 2 64 18 25 - 2, 3, 5 N4 E8 25·34·5·7 64 o 10 - 2 O4 24·33·5·7 128 18 10 - 2 L4 27·5·7 16 81 o - 2, 3 K5 Q10 2 128 81 o - 2, 3 K6 K12 2 128 243 o 14 2, 3 E6 E6 2 32 27 o - 2, 3 E6+3 32 27 o - 2, 3 E7 E7 1 64 27 o o 2, 3 E7+2 40 27 o o 2, 3 E8 E8 1 128 81 25 o 2, 3, 5
4.2.1
Complex lattices and root systems
LetF be a finite abelian extension ofQ, and J the intersection of F and the set of all algebraic integers, called thering of integers of F. A J-lattice is a collection ofn linearly independent vectors inFn, along with all theirJ-linear combinations. Any
submodule of a free J-module is itself free if J is a principal ideal domain, and although this isn’t necessarily the casea priori in practice, the specificJ considered are indeed principal ideal domains. This leads to a natural generalisation of root systems for complex reflection groups, see [25, Def. 1.43].
Definition 4.2.4. [25] AJ-root system in a vector spaceV overF, with hermitian inner product (−,−) is a pair (Σ, f), where Σ is a finite subset of V, f : Σ →J×
a function such that
1. Σ spans V and 0∈/ Σ,
2. for all α∈Σ and λ∈F, λα∈Σ if and only ifλ ∈J× ,
3. for all α∈Σ and λ∈J×, f(λα) =f(α)6= 1,
4. for all α, β ∈Σ, the Cartan coefficient
ha | bi= (1−f(β))(α, β) (β, β) belongs toJ, 5. for all α, β ∈Σ,rα,f(α)(β) :=β−(1−f(α)) (v, f(α)) (α, α) a∈Σ and f(rα,f(α)(β)) =f(β).
The group generated by the reflections {rα | α ∈ (Σ, f)}, is denoted W(Σ, f),
and is called (in slightly confusing nomenclature) the Weyl group of the system. It will be shown later that many of the irreducible complex reflection groups are indeed Weyl groups of some complex root system. A notable difference between this and the “real” case is the fact that these reflections can have order greater than 2.
The complex lattice Λ(Σ) is formed of theJ-linear combinations of the root system Σ; this gives a suitable generalisation of the root lattice of a real root system for complex reflection groups. From these “complex lattices” it is often possible to define a real counterpart; a free Z-module, exhibiting the same symmetry as its complex counterpart. Throughout this chapter, defineλ= 12(−1 +√−7) andσ = 12(−1 +√5), and the instances of J used in this chapter will be Z[α] ∈ C, where α is one of
{λ, σ, ζp}, where ζp is a primitivepth root of unity. To each of theseα is assigned an
involution; for a non real α, it is complex conjugation x 7→ x, otherwise x ∈ Z[σ], and the involution isx7→x0, where (a+b√5)0 =a−b√5.
J ζ3 =ω=e2πi3 λ= 1 2(−1 +
√
−7) σ= 12(−1 +√5) Involution ω=ω2 =−1−ω λ=−1−λ σ0 =−σ−1
Table 4.3: Reference table for lattices associated to complex reflection groups.
4.2.2
Λ(Σ)
realThere are different ways to obtain a real lattice L'Zn from a complex root system,
or complex lattice Λ. Ideally, there would be a canonical choice, such that the auto- morphisms GL(L) are the same as GL(Λ). The following construction from [8, §2.6]
defines a real lattice Λreal, from a complex lattice Λ. Firstly, for ω or λ, a complex
J-lattice vector (x1, . . . , xn)∈Λ, defines a lattice vector
(Re(x1),Im(x1), . . . ,Re(xn),Im(xn))∈Z2n. (4.7)
Forα =σ ∈R, this lattice vector is instead
(x1, . . . , xn, x01, . . . x
0
n). (4.8)
The integer combinations of these vectors give the elements of the rank 2n Z-lattice, Λreal, with inner product given by the Hermitian form
(x1, . . . , xn)·(y1, . . . , yn) = Re(x1y1+. . . xnyn) = 1 2(x1y1+x1y1+. . .+xnyn+xnyn) (4.9) for α=ζp or λ, and (x1, . . . , xn)·(y1, . . . , yn) = 1 2(x1y1+. . .+xnyn+x 0 1y 0 1+. . . x 0 ny 0 n) (4.10) for α=σ.
The lattice Λreal is very closely related to its complex counterpart, and notably if G is a group of automorphisms of a complex J-lattice Λ, (so comprised of elements of GLn(J)), then G is a group of automorphisms of Λreal as a subgroup of GL2n(Z). However these groups are often not isomorphic, as extra automorphisms of Λreal creep in, for instance complex conjugation.
order 3 automorphism of Λ that leaves no non-zero fixed points of Λ, which induces an order 3 automorphism of Λreal. In fact ifLis a real lattice, with a fixed-point-free automorphism of order 3, thenL= Λreal for a Z[ω]-lattice Λ.
Now define θ :=ω−ω =√−3. The element θ generates a prime ideal, indeed there exists an isomorphism of rings
Z[ω]/θZ[ω]'Z/3Z. (4.11)
The modulo 3 structure of Λreal is the modulo θ-structure of Λ.
As a concrete example, the root latticeA2 is Λreal for Λ :=Z[ω]. The automorphisms of Z[ω] are multiplication by −1 and ω, which generate the cyclic group of order 6. This group sits as an index 2 subgroup of Aut(A2), with the extra symmetry coming from complex conjugation.