Stabilisers of eigenvectors in complex reflection groups

Sinead Wilson B.Sc. (Hons), M.Sc.

A thesis submitted for the degree of Master of Philosophy at The University of Queensland in 2018 School of Mathematics and Physics Abstract

Eigenvectors of elements of real and complex reflection groups have been studied by Coxeter, Kostant, Springer, Lehrer and Bessis amongst others, due to their rich geometry, and the extensive information that they yield about the structure of the reflection group and related objects in Lie theory and braid theory. Recently, Kamgarpour proved that if G is an irreducible finite real reflection group of rank n, x is an eigenvector of any element of G with eigenvalue a primitive dth root of unity, and Φ and Φx denote the root systems of G and StabG(x) respectively, then |Φ| − |Φx| ≥ dn, with equality if and only if d is the Coxeter number. In this thesis, we prove the following generalisation of Kamgarpour’s inequality. If G is an irreducible complex reflection group of rank n and x is any eigenvector of an element of G with eigenvalue a primitive dth root of unity, then we have

`(π) − `(π x) ≥ dn, with equality if and only if G is well-generated and d = h. Here, π is the generator of the centre of the pure braid group of G, π x is the generator of the centre of the pure braid group of the stabiliser of x, and ` denotes the length function on the braid group of G. Our proof is case-by-case using the Shephard–Todd classification of complex reflection groups. We also investigate the case where `(π) − `(π x) − dn is as small as possible while still being positive, as well as a possible braid-theoretic explanation of our result. Declaration by author

This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. Publications included in this thesis

No publications included.

Submitted manuscripts included in this thesis

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Other publications during candidature

No other publications.

Contributions by others to the thesis

The uniform proof of Kamgarpour’s inequality (1.1.1) at the end of Section 4.7 (pp. 42–43) and the formulation of Conjecture 1.2 are due to Jean Michel.

Statement of parts of the thesis submitted to qualify for the award of another degree

No works submitted towards another degree have been included in this thesis.

Research involving human or animal subjects

No animal or human subjects were involved in this research. Acknowledgments

I am deeply grateful to my advisors, Masoud Kamgarpour and Ole Warnaar, for their extensive feedback and patience, and to Kamgarpour for suggesting this fascinating topic. I would also like to thank Jean Michel for our meeting in Sydney, and in particular, for pointing out the problem with [9], for showing me—and giving me permission to reproduce—the proofs in Section 4.7, and for suggesting Conjecture 1.2. This work was funded by a Research Training Scheme scholarship from the Australian Research Council. Financial support

This research was supported by an Australian Government Research Training Program Scholarship

Keywords complex reflection groups, eigenvectors of reflection group elements, parabolic subgroups, braid groups

Australian and New Zealand Standard Research Classifications (ANZSRC)

ANZSRC code: 010101 Algebra and Number Theory, 50% ANZSRC code: 010105 Group Theory and Generalisations , 50%

Fields of Research (FoR) Classification

FoR code: 0101, Pure Mathematics, 100% Contents

Abstract ...... ii

Contents vii

List of figures x

List of tables xi

1 Introduction 1 1.1 Stabilisers of eigenvectors in complex reflection groups ...... 1

2 Real reflection groups 5 2.1 Definition and examples ...... 5 2.2 Root systems ...... 7 2.3 Length ...... 8 2.4 Coxeter groups ...... 9 2.5 Classification ...... 10 2.6 Invariant theory ...... 11 2.6.1 The fundamental degrees ...... 12 2.7 Coxeter elements and exponents ...... 13

3 Complex reflection groups 17 3.1 Introduction ...... 17 3.2 Definitions and notation ...... 18 3.3 Example: generalised permutation matrices ...... 18 3.4 Classification ...... 19 3.5 Invariant theory ...... 20 3.5.1 M-exponents ...... 21

vii viii CONTENTS

3.5.2 The discriminant ...... 22 3.6 Braid groups ...... 22 3.6.1 Braid reflections ...... 24 3.6.2 Artin groups ...... 25 3.6.3 Length in braid groups ...... 26 3.6.4 The centres of P and B ...... 26 3.7 Parabolic subgroups ...... 28 3.7.1 Classification of parabolics in real reflection groups ...... 29 3.7.2 Classification of parabolics in complex reflection groups ...... 30 3.7.3 Parabolic subgroups of braid groups ...... 32

4 Eigenvectors of reflection group elements 35 4.1 Eigenvalues and fundamental degrees ...... 35 4.2 Regular elements ...... 36 4.3 Well-generated reflection groups and Coxeter elements ...... 37 4.4 Statement of the main theorem ...... 38 4.5 Necessity of conditions ...... 39 4.6 Braid group formulation ...... 40 4.7 The case of real reflection groups ...... 40 4.8 Cases where equality almost holds in (4.6.1) ...... 43 4.9 Towards a uniform proof ...... 44

5 Proof of the main theorem 47 5.1 The case G = G(m,1,1) ...... 47

5.2 The case G = G(1,1,n + 1) = Sn ...... 47 5.3 The groups G(m, p,n): structure of the stabiliser ...... 48 5.4 The groups G(m, p,n): proof of the inequality ...... 52 5.5 The exceptional groups ...... 55 5.5.1 Exceptional groups of rank 2 ...... 55 5.5.2 Exceptional groups of higher rank ...... 56 5.6 The case d = h ...... 57

Bibliography 59

A Sage code 63 CONTENTS ix

B Tables 67 List of figures

2.1 The symmetric group S3 as a reflection group...... 6 2.2 The dihedral groups of orders 10 and 12 as reflection groups...... 7

3.1 Some elements of B3 ...... 24 reg 3.2 The (unique) distinguished braid reflection in B(µm), as a path in V (left), and as a loop in V reg/G (right)...... 25

3.3 The generator π = β of Z(P3) = Z(B3)...... 27

3.4 A subdiagram of type A3 × A1 in the Coxeter diagram of type E6...... 29

x List of tables

2.1 Coxeter graphs of irreducible real reflection groups ...... 11

B.1 Groups of rank 2 ...... 67 B.2 Groups of higher rank ...... 68

xi

Chapter 1

Introduction

1.1 Stabilisers of eigenvectors in complex reflection groups

Finite real reflection groups play an important role in many areas of mathematics, arising for example, as Weyl groups of semisimple Lie algebras. Complex reflection groups are a generalisation of the more familiar real reflection groups, arising naturally in the study of eigenspaces of real reflection group elements, and also as automorphism groups of certain polytopes. The eigenspaces of elements of complex reflection groups reveal a rich pattern, studied exten- sively by Springer [34]. In this thesis, we continue in that direction by looking at the stabiliser of an eigenvector (which, by a theorem of Steinberg [35], is also a complex reflection group). Specifically, we prove the following theorem.

Theorem 1.1. Let (G,V) be an irreducible complex reflection group with dimV = n, w ∈ G any element, and x ∈ V an eigenvector of w with eigenvalue a primitive dth root of unity. Write N ∗ ∗ (respectively Nx) for the number of reflections and N (respectively Nx ) for the number of reflecting hyperplanes of G (respectively, of the stabiliser of x). Then, we have,

∗ ∗ N + N − Nx − Nx ≥ dn with equality if and only if G can be generated by n reflections and d is the largest degree of G.

Formally, a complex reflection group is a finite group G which acts on a finite-dimensional hermitian vector space V by pseudo-reflections—that is, finite-order unitary transformations that stabilise a complex hyperplane. Complex reflection groups arise, for example, as quotients of normalisers of maximal eigenspaces of elements of real reflection groups by the centralisers of the same eigenspaces. Thus, they can be considered a natural generalisation of real reflection groups

1 2 CHAPTER 1. INTRODUCTION

(which correspond to the case of an eigenvalue of 1). For example, if G = Smn, w is a product of n disjoint cycles of length m, and E is the eigenspace of w with eigenvalue exp(2πi/m), then the normaliser NG(E) acts on E as the group G(m,1,n) of generalised permutation matrices; see the beginning of Chapter 3 for a more detailed discussion. As with finite real reflection groups, there is a classification of complex reflection groups— namely, if G is irreducible, then it either belongs to the infinite series G(m, p,n), which consist of generalised permutation matrices, or it is one of 34 exceptions, G4 through G37. An important characterisation of complex reflection groups is the Chevalley–Shephard–Todd Theorem, which states that for G a finite subgroup of GL(V), G is a complex reflection group if G and only if the invariant algebra C[V] is generated by algebraically independent elements, whose degrees d1,...,dn are independent of the choice of generators. These numbers are called the degrees, and they contain a considerable amount of information about G. The study of eigenspaces of reflection group elements has a long history, going back at least as far as the discovery by Coxeter [15] and Coleman [14] of a relationship between the degrees of a real reflection group and the eigenvalues of what are now known as Coxeter elements. Later, a systematic study of eigenspaces of arbitrary elements was undertaken by Springer [34], who showed that there is a close relationship between the degrees and the orders of eigenvalues of elements. Springer also explored the case where an eigenvector is regular—that is, when it does not lie on any reflecting hyperplane—showing that certain properties of Coxeter elements are special cases of a more general pattern. In particular, we note that Springer’s work recovers the earlier result of Kostant [26], that if G is an irreducible real reflection group, then in the above setting the number of roots |Φ| is at least dn, with equality if and only if w is a Coxeter element. An interesting strengthening of this inequality was recently discovered by Kamgarpour [24], who proved that

|Φ| − |Φx| ≥ dn, (1.1.1) with equality if and only if w is a Coxeter element. Here, Φx denotes the set of roots of the stabiliser ∗ Gx of x in G. Notice that for G a real reflection group, we have |Φ| = N + N ; hence, (1.1.1) is a special case of Theorem 1.1. We can restate (1.1.1) more conceptually as follows. For any complex reflection group G, let V reg denote the complement of the hyperplane arrangement of G in V. Since by definition, G acts freely on V reg, the quotient space V reg/G is a manifold, and there is an exact sequence,

1 → P → B → G → 1, 1.1. STABILISERS OF EIGENVECTORS IN COMPLEX REFLECTION GROUPS 3

where the pure braid group P is defined to be the fundamental group of V reg and the braid group B is defined to be the fundamental group of V reg/G. Braid groups were studied extensively by Broue,´ Malle and Rouquier [9], who introduced a length function ` : B → Z generalising the length function on a real reflection group. They also show the pure braid group Px and the braid group Bx of Gx inject into P and B respectively, compatibly with the length functions. It was conjectured by [9], and proven by Digne, Marin and Michel [18], that the centre of P is a cyclic group generated by an element π of length N + N∗. Combining these results, it follows that Kamgarpour’s inequality can be restated as,

`(π) − `(π x) ≥ dn, where π x denotes the generator of Z(Px) of positive length. As this inequality makes sense for any irreducible complex reflection group, it is a natural question to ask whether it holds in this greater generality—a question that Theorem 1.1 answers in the affirmative. The braid-theoretic nature of the above inequality also raises the more difficult question (which we have been unable to answer) of whether there is a uniform proof of Theorem 1.1, for example using the topology of the hyperplane arrangement instead of the Shephard–Todd classification. However, as we show in Section 4.9, Theorem 1.1 is closely related to the following conjecture, which was suggested to us by Jean Michel.

Conjecture 1.2 (Michel, private communication [29]). In the setting of Theorem 1.1, there exists a d −e lift w of some element of Gxw to B such that w = ππππ x for some e ≥ 0.

We prove Theorem 1.1 via the classification of irreducible complex reflection groups. For the infinite series G(m, p,n), the key idea is to associate to x a G-invariant “characteristic polynomial”. ∗ We are then able to bound Nx +Nx by studying the action of Gx on the roots of this polynomial. For all but five of the 34 exceptional cases, Theorem 1.1 can be proven by comparing the largest value ∗ that Nx + Nx can take (which is easily deduced from the classification of parabolics by Taylor [37]), with the largest degree (which bounds d, by the work of Springer). However, the three cases G25,

G31 and G32 required an exhaustive verification using the computer algebra system SageMath [38]. This thesis is structured as follows. Background on real reflection groups is introduced in Chapter 2, while complex reflection groups and their braid groups are discussed in Chapter 3. In Chapter 4, we review previous work on eigenvalues and eigenvectors of reflection group elements, state the main theorem and its braid-theoretic interpretation, and discuss the prospects for a uniform proof using braid theory. Finally, in Chapter 5, we present our proof of Theorem 1.1 using the case-by-case approach sketched above.

Chapter 2

Real reflection groups

Finite real reflection groups have been studied extensively due to the pivotal role that they play in Lie theory. Here we give an overview of important facts about these groups. For a more detailed exposition, see [22] or [5].

2.1 Definition and examples

Definition 2.1. (i). Let V be a real vector space equipped with an inner product, (·,·).A reflection

in V is a transformation of the form rα : V → V defined by

(v,α) r (v) = v − 2 α, α (α,α)

def ⊥ for some α ∈ V. Notice that rα fixes the hyperplane Hα = α , which is called the reflecting

hyperplane, and that rα sends α to −α.

(ii). A (finite) real reflection group is a pair (G,V) consisting of a Euclidean space V and a finite subgroup G of the orthogonal group O(V) generated by reflections.

(iii). G is said to be irreducible if we cannot write V = V1 ⊥ V2 and G = G1 × G2 with Gi < O(Vi).

(iv). The rank of G is the codimension of V G in V.

Example 2.2. The symmetric group Sn can be realised as an irreducible real reflection group as n follows. Let Sn act on R by permuting the coordinates. The elements of Sn are then given by permutation matrices, that is, matrices with entries in {0,1} with a unique nonzero entry in each

5 6 CHAPTER 2. REAL REFLECTION GROUPS

row and column. For example,   0 1 0 0   0 0 0 1   1 0 0 0   0 0 1 0 is the permutation matrix corresponding to the permutation (1342). This action restricts to an action on the subspace,  n V = (x1,...,xn) ∈ R | x1 + ··· + xn = 0 , so the rank of Sn is n − 1. For example, the S3 case is depicted in Figure 2.1. To see that this is a reflection group, recall that Sn is generated by the transpositions (i,i + 1) for 1 ≤ i < j ≤ n.

Geometrically, any transposition (i, j) acts via the reflection rei−e j , where ei is the ith standard unity n vector in R .

s1

α2 α1 + α2

s1s2s1 = s2s1s2 s2

−α1 α1

−α1 − α2 −α2

Figure 2.1: The symmetric group S3 as a reflection group.

Example 2.3. For any integer m ≥ 3 let D2m (or I2(m)) denote the dihedral group of order 2m, i.e. the group of symmetries of a regular m-gon. This can be realised as an irreducible real reflection group of rank 2 as follows. Suppose that the vertices of the m-gon are (cos(2kπ/m),sin(2kπ/m)), and let s1 be the reflection about the x-axis, and s2 be the reflection through the line generated be the vector (cos(π/m),sin(π/m)), as depicted in Figure 2.2 (notice the different behaviours for even and odd m). Then, one can check that s2 ◦ s1 is a rotation by 2π/m, so that s1 and s2 generate D2m. 2.2. ROOT SYSTEMS 7

s2 s2 s 1 s1

Figure 2.2: The dihedral groups of orders 10 and 12 as reflection groups.

2.2 Root systems

Following [22], we define a root system as follows. Note that this is not the only definition in the literature: see Remark 2.5.

Definition 2.4. A (reduced) root system in a Euclidean vector space V is a finite subset Φ ⊂ V \{0} such that:

(i). The rα for α ∈ Φ permute Φ,

(ii). For all α ∈ Φ, we have Rα ∩ Φ = {±α}.

To any root system Φ we can associate a reflection group hrα | α ∈ Φi. On the other hand, we can associate to each reflection group G a root system Φ as the set of unit normal vectors to the reflecting hyperplanes of V. For example, a root system for S3 is given in Figure 2.1. Similarly, a root system for the dihedral group of order 2m is given by

 kπ  kπ   −sin ,cos 0 ≤ k ≤ 2m − 1 . m m

Remark 2.5. Some sources (such as [5]) use a more restrictive definition, which amounts to requiring that for each α ∈ Φ, the linear form1 2(·,α)/(α,α) takes integer values on Φ. Such root systems are called crystallographic. Assuming a crystallographic root system Φ for the group G def exists, it immediately follows that G acts on the lattice Q = hΦiZ.

Let φ : V → R be any linear form such that φ(α) 6= 0 for all α ∈ Φ. Then, we can write

Φ = Φ+ t Φ−, where Φ+ = {α ∈ Φ | φ(α) > 0} and Φ− = −Φ+ = {α ∈ Φ | φ(α) < 0}. Elements of Φ+ are called positive roots.

1In fact, Bourbaki does not fix a Euclidean form, and instead talks about coroots in V ∗. 8 CHAPTER 2. REAL REFLECTION GROUPS

If G is crystallographic, and Φ is a crystallographic root system, the lattice Q has a unique basis + of roots ∆ = {α1,...,αn} ⊂ Φ , called the simple roots such that all positive roots are nonnegative def linear combinations of simple roots; furthermore, the corresponding simple reflections si = rαi generate G [5, Ch. V §3.2]. In the non-crystallographic case, we cannot expect something this strong. Nevertheless, there is a weaker notion of simple roots in this case, which can be found in [22].

Theorem 2.6 ( [22, §1.3, 1.5]). Let Φ be any root system, and Φ+ as above. Then, there exists a unique subset + ∆ = {α1,...,αn} ⊆ Φ , called the set of simple roots, such that for any β ∈ Φ we have n β = ∑ aiαi, i=1

+ def and the ai are nonnegative for β ∈ Φ . Furthermore, the simple reflections si = rαi are a minimal set of generators for G.

Remark 2.7. Notice that (αi,α j) < 0, since otherwise at least one out of si(α j) and s j(αi) will be positive, but not a nonnegative linear combination of simple roots.

2.3 Length

Definition 2.8. For any element w ∈ G, denote by `(w) the shortest length of a word for w in the simple reflections. Then, `(w) is called the length of w, and any word for w of length `(w) is called a reduced word.

We have the following standard facts about the length function ` : G → Z≥0.

Proposition 2.9 (Exchange relation, [5, Ch. 4, §1, Prop. 4]). Let w ∈ G and α ∈ ∆, and let w = si1 ···sik be a reduced word. Then,

(i). `(rα w) = k ± 1,

(ii). If `(rα w) = k + 1, then rα si1 ···sik is a reduced word.

(iii). If `(rα w) = k − 1, then there exists 1 ≤ j ≤ k such that

rα w = si1 ···si j−1 si j+1 ···sik

is a reduced word. 2.4. COXETER GROUPS 9

Proposition 2.10 ( [22, §1.7]). For any w ∈ G, `(w) is equal to the number of positive roots α such that w(α) is negative.

+ Proposition 2.11 ( [22, §1.8]). There is a unique element w0 ∈ G of length |Φ |, called the longest element. The order of w0 is 2.

The terminology longest element is justified by Proposition 2.10, in that the length of an element cannot exceed |Φ+|. Furthermore, we can define a partial order ≤ on G by declaring2 that for w1,w2 ∈ G, we have w1 ≤ w2 if and only if there exists g ∈ G such that gw1 = w2 and

`(g) + `(w2) = `(w2). The element w0 is then the unique maximal element with respect to ≤. There is a second length function on G: the reflection length. We define T to be the collection of all reflections. Since T is stable under G-conjugation, we see that G has a presentation given by the generators T modulo all relations of the form st = tu; this is called the Steinberg presentation.

Definition 2.12. The reflection length `T : G → Z≥0 is defined by taking w ∈ G to the minimal length of any word for w in T.

Proposition 2.13 was proven by Carter in [10, §2 Lemma 3] in the context of Weyl groups, but as observed by Bessis in [2], Carter’s proof works more generally.

Proposition 2.13 (Carter). For any w ∈ G, we have,

`T (w) = codimV ker(w − Id) .

As with the classical length function, define a partial order on G by declaring that for w1,w2 ∈ G, we have w1 ≤T w2 if and only if there exists g ∈ G such that gw1 = w2 and `T (g)+`T (w2) = `T (w2). Since the Steinberg relations are homogeneous, this is equivalent to the existence of g0 ∈ G such 0 0 that w1g = w2 and `T (g ) + `T (w2) = `T (w2).

2.4 Coxeter groups

We now turn to the more general concept of a Coxeter group.

Definition 2.14. (i). A Coxeter matrix M = (mi j) is a symmetric n × n matrix with entries from N ∪ {∞} whose diagonal entries are all equal to 1, and whose non-diagonal entries are all at least 2.

2This is slightly different from the description given by Humphreys, but one can easily show that the two definitions are equivalent by the strong exchange relation given in loc. cit. 10 CHAPTER 2. REAL REFLECTION GROUPS

(ii).A Coxeter group is a group of the form

mi j hs1,...,sn | (sis j) = 1,for all mi j ∈ Mi.

If mi j = ∞, then no relation is imposed between si and s j. Letting S = {s1,...,sn}, we say that the pair (G,S) is a Coxeter system.

2 Note that in particular that, since mii = 1 for all i, we have si = 1 for all i. To any finite real reflection group we can associate a Coxeter matrix as follows. Let α1,...,αn be the simple roots; recall that we normalised these to have norm 1. For 1 ≤ i < j ≤ n, we set π mi j = −1 . In other words, mi j is the order of rαi rα j . cos (αi,α j) Proposition 2.15 ( [22, §6.4]). With the above notations,

(i). The map rαi 7→ si is an isomorphism from G to the Coxeter group of M.

(ii). In this way, the finite real reflection groups coincide with the finite Coxeter groups.

2.5 Classification

A convenient way of depicting a Coxeter matrix is via its Coxeter graph. To each simple reflection, we associate a vertex, and to each pair of vertices we associate an edge if and only if the corre- sponding entry mi j in the Coxeter matrix is at least 3, in which case we label the edge with the number mi j. In analogy with the Dynkin diagrams of semisimple Lie algebras, one usually omits the labels on edges labelled 3, while edges labelled 4 and 6 are denoted by double and triple edges, respectively. For example, the diagram for the dihedral group Dm is,

m

while the diagram for Sn+1 is,

where the total number of vertices is n. Coxeter graphs classify finite Coxeter groups. For example, the irreducible finite Coxeter groups can be classified into four infinite families and six exceptional groups, whose Coxeter diagrams are ∼ ∼ ∼ given in Table 2.1. Notice that I2(3) = A2, I2(4) = B2, and D3 = A3; these are the only cases in which the families overlap. 2.6. INVARIANT THEORY 11

Table 2.1: Coxeter graphs of irreducible real reflection groups

Name Diagram Degrees

An (n ≥ 1) 2,3,4,...,n + 1 Bn (n ≥ 2) 2,4,6,...,2n

Dn (n ≥ 3) 2,4,...,2(n − 1),n m I2(m) (m ≥ 3) 2,m 5 H3 2,6,10 5 H4 2,12,20,30 F4 2,6,8,12

E6 2,5,6,8,9,12

E7 2,6,8,10,12,14,18

E8 2,8,12,14,18,20,24,30

2.6 Invariant theory

Definition 2.16. Let k be a field. A (finitely-generated) graded k-algebra A is said to be a polynomial algebra if it is generated by algebraically independent generators, or equivalently, if it is isomorphic to a graded k-algebra of the form k[x1,...,xn].

Proposition 2.17 ( [27, Prop. 3.25]). If A is a polynomial algebra over a field of characteristic 0, with two sets of algebraically independent generators f1,..., fn and g1 ...,gm, then n = m and the n m multisets {deg( fi)}i=1 and {deg(gi)}i=1 coincide.

For any finite-dimensional vector space V over a field k, we define the algebra of polynomial functions on V to be the k-algebra k[V] def= Sym(V ∗) of symmetric tensors of the dual space. More concretely, if we chose a basis {x1,...,xn} of V, with ∗ ∗ dual basis {x1,...,xn}, then k[V] consists of all finite sums of the form

n ∗ ri f (v) = ∑ ar1,...,rn ∏xi (v) . r1,...,rn≥0 i=1 12 CHAPTER 2. REAL REFLECTION GROUPS

∗ ∗ def ∗ ∗ By evaluating such expressions via xi x j (v) = xi (v)x j (v), we see that they are indeed polynomial functions of the variables xi. If V carries an action of a group G, then V ∗ carries the contragredient G-action3,

(g,φ) 7→ g(φ) : v 7→ φ(g−1(v))
.

By the functoriality of Sym, this induces a G-action on k[V], via (g,φ1 ···φn) 7→ g(φ1)···g(φn).

n Example 2.18. Let G = Sn acting on V = R as in Example 2.2. Then, we can identify R[V] ∗ with R[x1,...,xn] via ei 7→ xi. The G-action is given as follows: A permutation σ ∈ Sn takes a G polynomial function f (x1,...,xn) to f (xσ −1(1),...,xσ −1(1)) Thus, R[V] is simply the algebra of symmetric polynomials in n variables. By the Jacobian criterion, a set of algebraically independent generators for this algebra is given by the elementary symmetric polynomials, e1,...,en, where

def ei(x1,...,xn) = ∑ ∏x j. (2.6.1) |I|=i j∈I I⊆{1,...,n}

n i In other words, ei(x1,...,xn) is the coefficient of t in ∏(1 +tx j). These polynomials have degrees j=1 1,...,n; notice that we could alternatively have taken the Newton power-sums pi(x1,...,xn) = n i ∑ j=1 x j, which have the same degrees, as generators. n ∼ n−1 Since the irreducible reflection module of Sn is R /(x1 + ··· + xn) = R , we see that the invariant polynomials on this space are generated by e2,...,en, which have degrees 2,...,n.

Theorem 2.19 (Chevalley). Let G be a real reflection group on the vector space V. Then, the G algebra of G-invariant polynomial functions R[V] is a polynomial algebra of dimension dim(V).

2.6.1 The fundamental degrees

Definition 2.20. If G is a real reflection group of rank n, then the degrees of a set of algebraically G independent generators of R[V] are called the fundamental degrees of G. They are denoted d1 ...,dn, in ascending order.

The degrees of the irreducible real reflection groups are given in Table 2.1. The multiset of degrees of a product G1 × G2 of two reflection groups is the union of the multisets of the degrees of

G1 and G2, since if V1 and V2 are the corresponding reflection modules, then

G1×G2 ∼ G1 G2 k[V1 ⊕V2] = k[V1] × k[V2] . 3This action is chosen in order to ensure compatibility with the evaluation pairing, V ∗ ×V → k. 2.7. COXETER ELEMENTS AND EXPONENTS 13

The degrees of a reflection group G encode a significant amount of information about the group. For example, a generating series argument (see e.g. [22]) shows the following.

Proposition 2.21. Let G be a real reflection group with degrees d1,...,dn and root system Φ. Then,

(i). |G| = d1 ···dn.

n (ii). |Φ| = ∑i=1(di − 1).

Remark 2.22. If G is a split, simply connected, semisimple algebraic group over a finite field Fq, whose Weyl group has root system Φ and degrees d1,...,dn, then we have the following q-analogue of Proposition 2.21 (i), which can be found in [36, Thm. 25 (a)]: n |G| = q|Φ| ∏(qdi − 1). i=1 The Betti numbers of a compact semisimple Lie group G are determined by the degrees d1,...,dn of the corresponding Weyl group. More precisely, Chevalley [12] showed that the Poincare´ polynomial of G is given by n ∏(1 +t2di−1). i=1

2.7 Coxeter elements and exponents

It is not immediately obvious how to calculate the degrees of an irreducible real reflection group. One important insight, which was discovered empirically by Coxeter [15] and later proven by Coleman [14], is that the degrees can be read off from the eigenvalues of certain elements called Coxeter elements, which are easy to calculate. This was the historical impetus for later studies of the eigenvalues of reflection group elements, discussed in Chapter 4.

Definition 2.23. A Coxeter element in a real reflection group is the product of all of the simple reflections corresponding to any choice of Φ+, in any order.

It is shown in [22, §3.16] that any two Coxeter elements are conjugate.

n−1 Example 2.24. Let G = Sn acting on V = R . Then, the Coxeter elements in G are the cycles of order n. That they form a single conjugacy class is evident from the fact that the cycle type of a permutation is invariant under conjugation. From now on, we will fix ! 0 1 c = s1 ···sn−1 = . I n−1 0 V 14 CHAPTER 2. REAL REFLECTION GROUPS

The eigenvalues of this matrix are the nth roots of unity, and those of the restriction to V are the nontrivial nth roots of unity, since V is orthogonal to the 1-eigenspace. n−1 n−2 T The eigenspace for ω ∈ µn \{1} spanned by the vector (ω ,ω ,...,1) . Notice that this vector is regular—that is, it does not lie in any reflecting hyperplane—as its coordinated are distinct.

The following theorem was discovered empirically by Coxeter [15], and later proven by Cole- man [14]. Amongst other things, it provides an easy method for finding the degrees of an irreducible real reflection group.

Theorem 2.25 (Coxeter, Coleman). Let G be any irrreducible real reflection group with highest degree h, and c a Coxeter element in G. Let ζ be a primitive hth root of unity. Then, the order of c is h, and its eigenvalues are ζ di−1, 1 ≤ i ≤ n.

Definition 2.26. With notation as in Theorem 2.25, h is called the Coxeter number of G, and the def mi = di − 1 are called the exponents of G.

One interesting property of the exponents is the following duality. Later, we shall see that a generalisation of this duality characterises those complex reflection groups for which there is an analogue of Coxeter elements.

Proposition 2.27 ( [22, §3.17]). Let G be an irreducible real reflection group with exponents m1 ...,mn. Then, for all 1 ≤ i ≤ n, we have

mi + mn+1−i = h.

Proof. Because c and c−1 = cT commute, they are simultaneously diagonalisable. On the other hand, they have the same multisets of eigenvalues, and their product is the identity, so we see that for each i there must be some j for which mi +m j ∈ hZ. But this can only happen if mi +mn+1−i = h.

Combining Proposition 2.27 and Proposition 2.21 (ii), one obtains the following fact, which we will use extensively later.

Corollary 2.28 ( [22, §3.18]). Let G be any irreducible real reflection group with root system Φ. Then, |Φ| = nh. (2.7.1)

Theorem 2.25 has following converse due to Kostant. As we shall see in Section 4.1, the assumption that G is crystallographic is unnecessary. 2.7. COXETER ELEMENTS AND EXPONENTS 15

Theorem 2.29 (Kostant, [26, Corollary 9.2]). Let G be a crystallographic real reflection group with Coxeter number h, w ∈ G any element, and d the order of any eigenvalue of G. Then, d ≤ h, with equality if and only if w is a Coxeter element. Furthermore, in the latter case, the4 corresponding eigenvector does not lie on any reflecting hyperplane.

4By Theorem 2.25, Coxeter elements have distinct eigenvalues.

Chapter 3

Complex reflection groups

3.1 Introduction

We now turn to the more general notion of complex reflection groups. To motivate what follows, we consider the following example. In what follows, we will denote by µm the group of mth roots of unity in C.

Example 3.1. Let m,n ≥ 2, and denote by W the symmetric group Smn, which acts on V = mn mn C /Span(1,1,...,1) by permuting the standard basis vectors in C . Consider the mth power of a Coxeter element, say,

cm = (1,2,...,m)(m + 1,m + 2,...,2m)···((n − 1)m + 1,(n − 1)m + 2,...,nm).

This element has m − 1 distinct eigenspaces, Vζ ,...,Vζ m−1 of dimension n, with eigenvalues the m−1 nonzero mth roots of unity ζ,...,ζ , as well as a (n−1)-dimensional 1-eigenspace V1. Explicitly, n−1 m − j Vζ = Span{vi}i=0 , where vi = ∑ j=1 ζ eim+ j. The normaliser NW (Vζ ) is therefore of the form n ∼ n µm o Sn, where the µm act on Vζ = C by multiplication on the respective basis vector vi, and Sn acts by permuting the vi. Because not every element of this group has real determinant, it is not generated by reflections on any real form of Vζ . However, as we shall see, it is generated by a more general type of trans- formation called pseudo-reflections. In fact, this is a special case of a more general phenomenon, whereby a complex reflection group can be constructed from a maximal eigenspace arising from another complex reflection group—see [27, Ch. 11] or [8, Ch. 5].

17 18 CHAPTER 3. COMPLEX REFLECTION GROUPS

3.2 Definitions and notation

Definition 3.2. Let V be a complex hermitian vector space. A pseudo-reflection in V is a unitary transformation r of finite order which fixes a complex hyperplane H pointwise. In this case, H is called the reflecting hyperplane of r. Equivalently, r is a transformation of the form,

(v,x) r : v 7→ v − (1 − ω) x, x,ω (x,x) where x ∈ V and ω is a root of unity.

Notice that if we put ω = −1, we recover the usual notion of reflection.

Definition 3.3. A complex reflection group is a pair (G,V) consisting of a complex hermitian vector space V and a finite subgroup G of the unitary group U(V) which is generated by pseudo-reflections.

Where there is no risk of confusion, we shall simply refer to the group G as a complex reflection group. We say that G is reducible if we can write V = V1 ⊥ V2 ⊥ ··· ⊥ Vs in such a way that

G = G1 × G2 × ··· × Gs, where Gi = G ∩ U(Vi); otherwise, we say that G is irreducible.

As a first example, notice that if (G,V) is a real reflection group, then (G,V ⊗R C) is a complex reflection group. However, these are not the only complex reflection groups: for example, for any integer m ≥ 2, the group µm of mth roots of unity is a complex reflection group of rank 1, acting on C by multiplication. For m > 2, this is not a real reflection group. For any complex reflection group (G,V) we denote by R the set of reflections in G, and A the set of reflecting hyperplanes in V. We write N = |R| and N∗ = |A|; note that in the case of real reflection groups, these numbers are equal, since every reflection has degree 2. However, this is not true in general, for example if G is the reflection group µm of rank 1, then N = m − 1, whereas ∗ N = 1. Finally, for any H ∈ A, we write eH for the order of the cyclic group StabG(H).

3.3 Example: generalised permutation matrices

An important class of examples comes from the following generalisation of the symmetric group.

Definition 3.4. A generalised permutation matrix is a matrix of the form

diag(ω1,...,ωn) · σ, where ω1,...,ωn are roots of unity, and σ is a permutation matrix. We denote the group of all generalised permutation matrices with ωi ∈ µm by G(m,1,n). 3.4. CLASSIFICATION 19

For example,      i 0 0 i 0 0 1 0 0      0 0 −1 = 0 −1 00 0 1      0 i 0 0 0 i 0 1 0 is an element of G(4,1,3).

Remark 3.5. Abstractly, G(m,1,n) is the wreath product

def n µm o Sn = µm o Sn, where Sn acts by permuting the copies of µm. In other words, it is the group from Example 3.1.

Definition 3.6. Let m, p and n be natural numbers such that p|m. We define G(m, p,n) to be the subgroup of G(m, p,n) given by

 m/p G(m, p,n) = diag(ω1,...,ωn)σ ∈ G(m,1,n) | (ω1 ···ωn) = 1 .

n Let V = C with the natural action of G. Then, (G,V) is a complex reflection group, as one can easily check that it is generated by the reflections,   Ii−1 p !  − j  ω 0  0 ω  p re1,ω = and re −ω je ,−1 =  , 0 I i i+1  ω j 0  n−1   In−i−1 where ω is a generator of µm, 1 ≤ i ≤ n−1, and 0 ≤ j ≤ m−1 [27]. However, this is not a minimal generating set.

Proposition 3.7 ( [27, §2.3]). Let G = G(m, p,n). Then,

 m  n
(i).N = n p − 1 + m 2 ,  n + mn
if p 6= m (ii).N ∗ = 2 . n
m 2 if p = m

3.4 Classification

Definition 3.8. Let V be a group and V a representation of G. We say that G acts imprimitively on V if V has a vector space decomposition

V = V1 ⊕ ··· ⊕Vn, such that n ≥ 2, dimVi = 1 for all i, and G permutes the Vi. 20 CHAPTER 3. COMPLEX REFLECTION GROUPS

Theorem 3.9 (Shephard–Todd, [33]). Every irreducible complex reflection group belongs to exactly one of the following families:

(i). The cyclic groups Cn = G(m,1,1), m ≥ 2;

(ii). The symmetric groups, Sn = G(1,1,n), n ≥ 2, with (n − 1)-dimensional reflection module as per Example 2.2;

(iii). The imprimitive groups G(m, p,n), m,n ≥ 2, (m, p,n) 6= (2,2,2);

(iv). 34 exceptional groups, labeled G4 through G37.

Furthermore, G is imprimitive if and only if it is of type (iii).

Remark 3.10. The irreducible real reflection groups fit into this picture as follows [27].

∼ (i).W (An) = G(1,1,n + 1). ∼ (ii).W (Bn) = G(2,1,n). ∼ (iii).W (Dn) = G(2,2,n).

1 ∼ (iv). After a change of basis ,W(I2(m)) = G(m,m,2). ∼ ∼ ∼ ∼ ∼ ∼ (v).W (H3) = G23, W(H4) = G30, W(F4) = G28, W(E6) = G35, W(E7) = G36, andW(E8) = G37.

3.5 Invariant theory

In the 1950s, Shephard and Todd [33] discovered the following generalisation of Theorem 2.19, which they used in their proof of Theorem 3.9

Theorem 3.11 (Shephard–Todd). Let V be a finite-dimensional complex vector space, and let G be G a subgroup of GL(V). Then, C[V] is a polynomial algebra if and only if G is a complex reflection group on V.

Remark 3.12. In fact, the same statement is true over any field whose characteristic does not divide |G|. On the other hand, if the characteristic divides |G|, then the “only if” part still holds, but there are reflection groups whose rings of invariants are not polynomial algebras. See [23] for more details.

1 If the dihedral group I2(m) is realised as the symmetries of the regular m-gon with vertices {cos(2πk/m)e1 + sin(2πk/m)e2 | 1 ≤ k ≤ m − 1}, then the necessary change of basis is given by e1 7→ e1 + e2 and e2 7→ e2. 3.5. INVARIANT THEORY 21

Definition 3.13. As with real reflection groups, we define the degrees of a complex reflection group G G to be the degrees of a set of algebraically independent generators of C[V] , in ascending order.

Example 3.14 ( [27, §2.8]). Let G = G(m, p,n). Then, a set of algebraically independent generators G m m m/p m/p for C[V] is given by the polynomials ei(x1 ,...,xn ) for 1 ≤ i ≤ n − 1, and en(x1 ,...,xn ). Hence, the degrees are mn {m,2m,...,(n − 1)m} ∪ . p

3.5.1 M-exponents

In general, complex reflection groups do not admit an analogue of Coxeter elements (but see Theorem 4.8), so the definition of the exponents of a real reflection group does not carry over. Nevertheless, we have the related concept of M-exponents introduced by Springer in [34, §2.5]. G Let S = C[V], and let m denote the ideal of elements of positive degree in S . We call the def algebra SG = S/mS the coinvariant algebra of G; this is a finite-dimensional graded G-module.

Chevalley showed in [13] that SG is isomorphic as a G-module to the regular representation, so by [32, §2.4], we have [SG : M] = dimM for any irreducible representation M of G.

Definition 3.15. Let M be any irreducible G-module. We define the fake degree of M to be the polynomial i qM(t) = ∑[(SG)i : M]t . i We define the M-exponents of G to be the integers q1 ≤ q2 ≤ ··· ≤ qdimM such that

q q qM(t) = t 1 + ··· +t dimM .

For example, if M = V, then the M-exponents are the di − 1, 1 ≤ i ≤ n (see [27, Cor 10.23]). In particular, if G is a real reflection group, then they coincide with the exponents defined in Section 2.7.

∗ ∗ Definition 3.16. We define the coexponents mi of G to be the V -exponents, and the codegrees to ∗ def ∗ be di = mi − 1.

Example 3.17. Let G be a real reflection group. Then, the pairing (·,·) induces an isomorphism of G-modules V =∼ V ∗, so for all i we have,

∗ mi = mi.

(By contrast, if G is a complex reflection group, then V ∗ is only isomorphic to V with the complex conjugate of the action that we started with.) 22 CHAPTER 3. COMPLEX REFLECTION GROUPS

3.5.2 The discriminant

For each orbit O in A, choose a linear form αH vanishing on some H ∈ O. For every other 0 0 hyperplane H ∈ O, choose some g ∈ G such that H = g(H), and let αH0 = g(αH).

Definition 3.18. The discriminant is the polynomial,

eH ∆ = ∏ αH ∈ C[V]. H∈A

Remark 3.19. Equivalently, we can write

∆ = ∏ ∏ g(αH), O g∈G where H ∈ O is the distinguished hyperplane chosen above. This shows that ∆ is uniquely determined up to scalar multiplication. Notice also that ∆ is G-invariant, and that it is nonvanishing on the ∗ complement V reg of the hyperplane arrangement A. Hence we obtain a morphism, ∆ : V reg/G → C . ∗ Finally, notice that ∆ is homogeneous of degree N + N (as an element of C[V]).

Example 3.20. The discriminant of Sn is the square of the Vandermonde determinant:

2 ∆(x1,...,xn) = ∏ (x j − xi) . 1≤i< j≤n

G ∼ n Proposition 3.21 ( [31, Prop. 6.106]). Identifying V/G with Spec(C[V] )(C) = A (C), we have,

reg ∼ n V /G = A \ Z(∆).

3.6 Braid groups

An important class of groups related to complex reflection groups consists of the generalised braid groups. We discuss these in this section, following [9]; the relationship with classical braid groups is discussed in Example 3.24. Throughout this section we take the convention that if X is a space and f ,g : [0,1] → X are paths with g(1) = f (0), then their composition is given by  g(2t) if 0 ≤ t ≤ 1 f g(t) = 2 1  f (2t − 1) if 2 ≤ t ≤ 1; note that this is the opposite convention to that used in [21]. Consider the hyperplane arrangement A consisting of all of the reflecting hyperplanes in G. 3.6. BRAID GROUPS 23

Definition 3.22. (i). We define the hyperplane complement to be

V reg = V \ [ H. H∈A Vectors in V reg are said to be regular.

reg (ii). Fix a base point x0 ∈ V . We define the pure braid group of G as the fundamental group of the hyperplane complement, def reg P = π1(V ,x0).

(iii). Similarly, we define the braid group of G to be the fundamental group of the quotient of the hyperplane complement by G, def reg B = π1(V /G,x0).

reg Remark 3.23. If v ∈ V , then StabG(v) is trivial, since by Steinberg’s theorem (see Section 3.7), it is generated by the reflections which it contains. Hence, G acts freely on V reg, so by [21, Prop. 1.40], we have an exact sequence,

p 0 → P → B → G → 0.

∗ ∗ n Example 3.24. In the case where G = Sn and V = ker(x1 +···+xn) ⊆ C , there is a more concrete description, as discussed for example in [4, Chapter 1]. In this case, the configuration space,

reg n V = {(x1,...,xi) ∈ C | xi 6= x j∀i 6= j}, parametrises n-tuples of distinct points in C, while V reg/G parametrises n-tuples of distinct points up to permutation. A loop in V reg/G is thus given by a braid, that is, an n-tuple of disjoint curves in C × [0,1], each of which maps bijectively onto [0,1] under the obvious bijection, such that the end points are given by {(i, j) | i ∈ {1,...,n}, j ∈ {0,1}} in some order (see Figure 3.1a). Similarly, Pn is the subgroup of pure braids, i.e. those which preserve the order of the strands (Fig. 3.1b). reg Two braids correspond to the same element of π1(V /G) if and only if they are isotopic—that is, if there is a homotopy between them which preserves the second coordinate (this is not the same as simply being homotopic, as shown by Goldsmith [19]). The group structure on Bn and Pn is 1 given by glueing C × {1} in the first braid to C × {0} in the second, and scaling by 2 . By projecting braids onto R × [0,1] we see that Bn is generated by the crossings σ i of the ith strand over the (i+1)st (and their inverses, the crossings of the (i+1)st strand over the ith), subject to the relations σ iσ i+1σ i = σ i+1σ iσ i+1 and σ iσ j = σ jσ i if |i − j| > 1. We shall see later that, in fact, the braid group of every real reflection group has a similar presentation. 24 CHAPTER 3. COMPLEX REFLECTION GROUPS

Figure 3.1: Some elements of B3

(a) A braid (b) A pure braid (c) A positive braid

3.6.1 Braid reflections

If U is a linear subspace of V, then we denote by prU : V → V the projection onto U with kernel U⊥.

reg Definition 3.25. (i). A braid reflection is an element in π1(V /G,x0) given by a path of the form s(γ)−1σσγγ, where:

(a) s is a reflection with reflecting hyperplane H;

reg reg (b) γ is a path in V from x0 to some xH ∈ V close enough to H that a ball centred at

prH(xH) containing H does not intersect any other reflecting hyperplane;

(c) and σ is the arc from xH to s(xH) of the circle t 7→ prH⊥ (xH) + exp(2πit)prH(xH).

(ii). A reflection s about a hyperplane H is said to be distinguished if its nontrivial eigenvalue is

exp(2πi/eH). A braid reflection is said to be distinguished if its image in G is distinguished.

We denote the distinguished reflections sH, and the distinguished braid reflections σ H,γ .

Proposition 3.26 (Broue-Malle-Rouquier)´ . The braid group is generated by the distinguished braid eH reflections σ H,γ (for all H and γ), and the pure braid group is generated by the σ H,γ .

The following is a generalisation of the definition given by Bessis for real reflection groups.

Definition 3.27 (Bessis, [2]). Let G be a complex reflection group. A reflecting hyperplane H is 0 said to be visible from x0 if the real line segment γ from x0 to H orthogonal to H with respect 3.6. BRAID GROUPS 25

2πi
exp m x0

V(∆) x0 x0 H = (0,0)

reg Figure 3.2: The (unique) distinguished braid reflection in B(µm), as a path in V (left), and as a loop in V reg/G (right). to the Euclidean inner product Re(·,·) does not intersect any reflecting hyperplanes other than H. + The local monoid B (x0) is defined to be the submonoid of B generated by all σ H,γ where H is 0 + visible from x0, and γ is the segment from x0 to H in γ as above. Note that B (x0) (and even its isomorphism class) depends on the choice of x0.

3.6.2 Artin groups

If G is a real reflection group, then we have the following generalisation of Example 3.24.

n Definition 3.28. Let M = (mi, j)i, j=1 be any Coxeter matrix. The corresponding Artin group is defined as * +

A = σ 1,...,σ n σ iσ jσ i ··· = σ jσ iσ j ···, ∀i, j . | {z } | {z } mi j times mi j times

+ The Artin monoid A is the submonoid of A generated by the σ i.

There is an obvious surjection A → G to the corresponding Coxeter group, given by σ i 7→ si. Part (i) of the following proposition was proven by Brieskorn in [6], while part (ii) was proven by Bessis in [2, Prop. 3.4.3].

Proposition 3.29 (Brieskorn, Bessis). Let G be a real reflection group, and take x0 to be a point in the Weyl chamber C ⊂ V reg. R

(i). There is an isomorphism A → B, compatible with the corresponding surjections to G, given

by taking σ i to σ Hi,γ , where γ is a path in C, as per Definition 3.25.

+ + (ii). The above isomorphism identifies A with B (x0). 26 CHAPTER 3. COMPLEX REFLECTION GROUPS

From now on, we will talk about the Artin group and the braid group of a finite Coxeter group interchangeably. We can lift any element w ∈ G to B by considering a reduced word w = si1 ···sik and taking the element w = σ i1 ···σ ik ∈ B. A priori, this may depend on the choice of reduced word, but the following Lemma shows that it does not.

Proposition 3.30 (Matsumoto’s Lemma, [28]). Let

si1 ...sik = s j1 ...s jk be two reduced words for the same element of G. Then, the following equality holds in B:

σ i1 ...σ ik = σ j1 ...σ jk .

3.6.3 Length in braid groups

In general, there is no length function on a complex reflection group. However, there is a length function on the braid group, which we discuss in this section.

Definition 3.31. The length function on B is defined to be the homomorphism π1(∆) : B → × ∼ reg × π1(C ) = Z induced by ∆ : V /G → C . Concretely, the length of the homotopy class of × b :S1 → V reg/G is the homotopy class of ∆ ◦ b : S1 → C .

−1 Example 3.32. Let σ H,γ = sH(γ) σσγγ be a distinguished braid reflection. Then, the corresponding reg −1 path in V /G is of the form p(γ) p(σ )p(γ), so `(σ H,γ ) = `(σ ). But by definition, ∆(σ ) has winding number 1, so `(σ H,γ ) = 1.

Remark 3.33. The length function on the braid group relates to the length function defined in Chapter 2 as follows. Suppose G is a real reflection group. Then, for any element w, consider the lift w ∈ B of w defined by Matsumoto’s Lemma. Since distinguished braid reflections have length 1, the length of w is then `(w).

3.6.4 The centres of P and B

Springer [34, Cor. 3.3] proved an easy description of the centre of a complex reflection group in terms of its degrees. By contrast, it is less obvious what the centres of P and B are. In the case of real reflection groups, these have been understood since Deligne [16] and Brieskorn-Saito [7], but in general they were not fully understood until this century.

Proposition 3.34 (Springer). Let G be a complex reflection group with degrees d1,...,dn. Then,

Z(G) is cyclic of order gcd(d1,...,dn). 3.6. BRAID GROUPS 27

Figure 3.3: The generator π = β of Z(P3) = Z(B3).

Proof. By Schur’s Lemma, Z(G) acts on V by scalars, so by finiteness it must be cyclic. If we denote by w a generator of Z(G) acting by ζ, and by d its order, then Theorem 4.3 implies that di|d for each 1 ≤ i ≤ n, since the ζ-eigenspace of w is V. On the other hand, if d > gcd(d1,...,dn) then there is some 1 ≤ i ≤ n with d - di, so applying Theorem 4.3 again, we obtain that V ⊆ Z( fi), so fi = 0, a contradiction.

Define the loop π : [0,1] → V reg and the path β : [0,1] → V reg by

π(t) = exp(2πit)x0, β (t) = exp(2πit/|Z(G)|)x0.

The following was conjectured by Broue,´ Malle and Rouquier [9, §2]; part (i) was proven be Digne, Marin and Michel [18, Thm 1.2], while part (ii) was proven by Bessis [3, Thm 12.8] (part (iii) is then immediate).

Proposition 3.35. (i). The group Z(P) is infinite cyclic and generated by π.

(ii). The group Z(B) is infinite cyclic and generated by β .

(iii). There is an exact sequence, 1 → Z(P) → Z(B) → Z(G) → 1.

Example 3.36. If G = Sn+1, then h = n + 1, a Coxeter element is given by s1s2 ···sn, and by n+1 Schur’s Lemma, we have |Z(G)| = 1. Therefore, π = β is the element (σ 1σ 2 ···σ n) , which braid-theoretically is given by rotating all of the strands by a full circle, as depicted in Figure 3.3.

Proposition 3.37 (Broue-Malle-Rouquier,´ [9, Cor. 2.21]). We have, `(π) = N + N∗. 28 CHAPTER 3. COMPLEX REFLECTION GROUPS

Proof. The loop π, and hence also ∆ ◦ π, is positively oriented. By the Cauchy residue theorem, the length of π is therefore given by:

1 Z dz `(π) = 2πi π1(∆)(π) z 1 Z d(∆) = 2πi π ∆ 1 Z dαeH
= H ∑ eH 2πi H∈A π αH 1 Z dαH = ∑ eH 2πi H∈A π αH = ∑ eH H∈A = N + N∗, where the αH are chosen as in Definition 3.18.

The following corollary was originally proven in 1972 by Brieskorn and Saito [7, Satz 7.1], and Deligne [16, Thm. 4.21]. However, Propositions 3.35 and 3.37 enable the following alternative proof.

Proposition 3.38. If G is a real reflection group, and x ∈ V reg, then π = w2, where w is the 0 R 0 0 reduced word lift of w0.

reg Proof. From the definition, we see that the loop π in V intersects VR at ±x0. But −x0 is in the + same chamber as w0(x0), so we see that π is homotopic to the square of some lift w ∈ B of w0. By Proposition 3.37, we have `(w) = |Φ+|, so by positivity we can write w as a product of |Φ+| braid + reflections which are lifts of simple reflections. But since |Φ | = `(w0), this descends to a reduced expression for w0, and therefore we have w = w0.

3.7 Parabolic subgroups

An important class of subgroups of reflection groups are the subspace stabilisers, or parabolic subgroups. The name parabolic subgroup comes from the fact that the Weyl group of a Levi factor is a parabolic in the Weyl group. However, in contrast to the case of reductive groups, parabolic subgroups of reflection groups are not self-normalising. For example, let G = S4 = hs1,s2,s3i, and

H = hs1i. Then, we will see later that H is a parabolic subgroup, but s3 ∈ NG(Gx). 3.7. PARABOLIC SUBGROUPS 29

Definition 3.39. Let (G,V) be a complex reflection group. A parabolic subgroup of G is the def stabiliser GU = StabG(U) of a C-linear subspace U ⊆ V (which we may without loss of generality assume to be an intersection of reflecting hyperplanes).

Equivalently, one could define a parabolic subgroup as the stabiliser of an intersection of reflecting hyperplanes, or as the stabiliser of a single vector. A foundational theorem about parabolic subgroups is the following.

Theorem 3.40 (Steinberg, [35]). Any parabolic subgroup is generated by the reflections which it contains. In particular, it is a reflection subgroup.

3.7.1 Classification of parabolics in real reflection groups

Let G be a real reflection group, and let r1,...,rn be a set of simple reflections. A standard parabolic subgroup is a subgroup generated by any subset of the ri; such a group is equal to the stabiliser of the intersection of the reflecting hyperplanes for the remaining simple reflections, and is therefore a parabolic subgroup. We have the following classification, which can be considered analogous to the classification of parabolic subalgebras of a reductive Lie algebra.

Theorem 3.41 ( [22, §1.15]). Every parabolic subgroup in a real reflection group G is conjugate in G to a standard parabolic subgroup.

In other words, parabolic subgroups are classified up to conjugacy by subdiagrams of the Coxeter diagram. Note however that two standard parabolics may still be conjugate (for example, the two simple reflections in S3 are conjugate by the longest element). Thus, a conjugacy class of parabolics need not correspond to a unique subdiagram. Furthermore, it follows from the classification of Coxeter groups that the Coxeter diagram of a parabolic is simply the corresponding subdiagram, for example the parabolic subgroup of E6 highlighted in black in Figure 3.4 is a copy of S4 × S2.

Figure 3.4: A subdiagram of type A3 × A1 in the Coxeter diagram of type E6.

Example 3.42. Let G = Sn. Let U be an intersection of reflecting hyperplanes. Then, U is given by system of equations of the form xi = x j; this partitions the set {1,...,n} into subsets S1,...,Sk 30 CHAPTER 3. COMPLEX REFLECTION GROUPS

such that whenever i, j ∈ Sl, xi = x j for all x ∈ V. We can then find some permutation σ such that the elements of σ (Sl) are adjacent for all l, and the σ (Sl) are arranged in descending order of cardinality; thus, we get a partition λ of n, and GU is conjugate by σ to the standard parabolic of type Aλ1−1 × ··· × Aλk−1. We shall see in the next subsection that there is a similar classification for parabolic subgroups of the groups G(m, p,n).

Definition 3.43. We say that w ∈ G is a parabolic Coxeter element if it is a Coxeter element in a parabolic subgroup of G (where, by a Coxeter element in a reducible real reflection group, we mean a product of Coxeter elements in its irreducible components).

For example, if G = Sn+1, then by Example 3.42, every element is a Coxeter elements of the parabolic subgroup corresponding to its cycle decomposition. By contrast, the element s1s2s1s2 ∈

W(Bn), which acts by the scalar −1, has no nontrivial fixed points, and therefore, cannot be a parabolic Coxeter element. We will need the following proposition in Chapter 4.

Proposition 3.44 (Bessis, [2, Lemma 1.4.3]). Let G be a real reflection group, and w ∈ G. Then, w is a parabolic Coxeter element if and only if w ≤T c for some Coxeter element c.

3.7.2 Classification of parabolics in complex reflection groups

The situation with parabolics in complex reflection groups is more complicated. There are diagrams analogous to the Coxeter graphs of real reflection groups, albeit with some non-Coxeter like relations imposed; see [9] for one class of such diagrams based on the structure of the corresponding braid group. However, contrary to what is written in [9], not all conjugacy classes of parabolic subgroups are classified by subdiagrams of these diagrams, even for the imprimitive groups.2 A classification of all conjugacy classes of parabolic subgroups (and reflection subgroups more generally) is given in [37]; however, this classification is case-by-case, and does not use diagrams. We summarise this classification here, as we will need to rely heavily on it for the proof of our main theorem. For the groups G(m, p,n), Taylor introduces the concept of augmented partitions, and classifies reflection subgroups using these.

Definition 3.45 (Taylor). Let (m, p,n) be a triple of positive integers such that p|m.

(i).A feasible triple for (m, p,n) is a triple (m0, p0,n0) of positive integers such that

2Note however that for the groups G(m, p,n), the results of [37] imply that all parabolics are conjugate in the larger group G(m,1,n) to parabolics given by admissible subdiagrams of the diagrams in [9]; in particular, the latter do give a complete classification up to isomorphism. 3.7. PARABOLIC SUBGROUPS 31

(a) n0 ≤ n; (b) p0|m0|m; m0 m (c) p0 | p .

(ii). A feasible triple (m0, p0,n0) is called thin if m0 = 1; otherwise it is called thick.

(iii). The set of feasible triples (m0, p0,n0) for (m, p,n) is ordered lexicographically with the convention that n0 > m0 > p0.

(iv). An augmented partition of (m, p,n) is a (weakly) decreasing sequence

Λ = {(m1, p1,n1),...,(mr, pr,nr)}

of feasible triples, such that (n1,...,nr) is a partition of n.

(v). If Λ is an augmented partition for (m, p,n) and α ∈ µm, we define

GΛ = ∏G(mi, pi,ni), i and α −1 GΛ = θα GΛθα ,

where θα = diag(α,1,...,1) ∈ G(m,1,n).

Theorem 3.46 (Taylor). (i). Every reflection subgroup of G(m, p,n) is conjugate to a subgroup α of the form GΛ.

(ii). Every parabolic subgroup of G(m, p,n) is conjugate to either

(a) a subgroup of the form GΛ where Λ contains exactly one thick feasible triple, which is

of the form (m, p,n0), or α (b) a subgroup of the form GΛ where Λ only contains thin feasible triples.

T
Example 3.47. To see why case (b) is necessary, let G = G(4,2,2), and V1 = Span (i,1) . Then,

StabG(V1) is the group of order 2 generated by the matrix, ! ! ! !−1 0 i i 0 0 1 i 0 = . −i 0 0 1 1 0 0 1

i In other words, it is GΛ, where Λ consists of the single triple, (1,1,2). However, one can easily check that this matrix is not conjugate in G to a permutation matrix, so StabG(V1) is not of the form

GΛ. 32 CHAPTER 3. COMPLEX REFLECTION GROUPS

0 On the other hand, the parabolic Stab(V1 ⊕ C) in G(4,2,3) is isomorphic to GΛ0 , where Λ = (1,1,2),(4,1,1) , since we can now conjugate in G(4,2,3):      −1 1 0 0 i 0 0 1 0 0 i 0 0       0 0 i = 0 i 00 0 10 i 0 .       0 −i 0 0 0 1 0 1 0 0 0 1

For the exceptional groups, Taylor determines all of the reflection subgroups. They can be found in the tables in [37, §6].

3.7.3 Parabolic subgroups of braid groups

Let U be a subspace of V, and let AU ⊆ A denote the collection of reflecting hyperplanes containing 0 reg U. Then by Steinberg’s Theorem, A is the hyperplane arrangement for the stabiliser, GU . Let VU denote the complement of AU .

Definition 3.48. A parabolic subgroup of P (respectively B) is the fundamental group of the reg reg complement of some VU as above (respectively, of VU /GU ). We denote these groups by PU and BU , respectively.

reg reg reg reg Remark 3.49. The inclusions ι : V ,→ VU and κ : V ,→ VU induce surjections ι∗ : P  PU and κ∗ : B  BU . Broue,´ Malle and Rouquier show that ι∗ and κ∗ are in fact split, so it makes sense to refer to PU and BU as parabolic subgroups. Note however that the resulting inclusions of PU into

P and of BU into B are canonical only up to P-conjugation. Furthermore, while the image of PU is

P is independent of any choices, the image of BU in B is not. reg S We briefly recall the construction of these inclusions given in [9]. We let U =U \ U*H∈A H ∩ reg reg U, and choose a path γ in V from x0 = γ(0) to γ(1) ∈ U . We let Ω denote a ball with centre γ(1) with sufficiently small radius that Ω does not intersect any hyperplane not containing U, and 0 choose x1 to be any point on γ in Ω. Write γ for the rescaled path from x0 to x1 along γ . We define

reg reg λ : π1(V ∩ Ω,x1) → π1(V ,x0) w 7→ γ 0−1wγ 0.

reg reg Because V ∩ Ω = VU ∩ Ω, the composition,

reg reg reg ι∗ ◦ λ : π1(V ∩ Ω,x1) → π1(V ,x0) → π1(VU ,x0) is an isomorphism. Hence we obtain an injective group homomorphism,

−1 reg reg reg λ ◦ (ι∗ ◦ λ) : PU = π1(VU ,x0) → π1(V ∩ Ω,x1) → π1(V ,x0) = P. 3.7. PARABOLIC SUBGROUPS 33

Since Ω is G-stable, a similar argument gives a splitting

reg reg µ : π1((V ∩ Ω)/G,x0) → π1(V /G,x0)

−1 of κ∗, and hence an injection µ ◦ (κ∗ ◦ µ) : BU → B; but again, note that a different choice of the path γ will give a P-conjugate injection. We obtain the commutative diagram,

1 PU BU GU 1

1 P B G 1.

Remark 3.50. It is not true that if b ∈ B descends to an element of a parabolic of G, then b is an element of the corresponding parabolic of B (even allowing for conjugation by P). For example, let 2 G be a real reflection group. Then, π = w0 descends to the identity in G, but it is not contained in any proper parabolic, because a word for w0 contains all of the generators.

Chapter 4

Eigenvectors of reflection group elements

In Section 2.7, we saw that when w is a Coxeter element in a real reflection group, its eigenvalues are determined by the fundamental degrees, and that furthermore, the corresponding eigenvectors lie in V reg. The question of what happens for an arbitrary element w ∈ G was studied by Springer in [34]; we recall some of Springer’s results in the first two sections. The topic of this thesis, the stabilisers of eigenvectors, is discussed beginning in Section 4.4

4.1 Eigenvalues and fundamental degrees

For any root of unity ζ and any element w ∈ G we denote by V(w,ζ) the eigenspace of w with eigenvalue ζ.

n Theorem 4.1 (Springer, [34, §3]). Let (G,V) be a complex reflection group, and let { fi}i=1 be a n set of fundamental invariants with degrees {di}i=1. Let x ∈ V, d ∈ N, and let ζ be a primitive dth root of unity. Then, [ \ V(w,ζ) = Z( fi). w∈G d-di In particular, ζ is an eigenvalue of some element of G if and only if d|di for some i.

For any positive integer d, we define a(d) to be the number of degrees which are divisible by d.

Example 4.2. Let G = Sn. Then, the degrees are {2,3,...,n}, so for any d ∈ N, we have  n − 1 if d = 1  a(d) =  n  d if 2 ≤ d ≤ n  0 if d > n.

35 36 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

For any w ∈ G, write the cycle type of w as (λ1,...,λr). Then, for any primitive dth root of unity ζ, we have,  λi ∑d|λ if 2 ≤ d ≤ n dimV(w,ζ) = i d r − 1 if d = 1.

 n  This is maximised when λ = (d,d,...,n − d d), in which case, we have dimV(w,ζ) = a(d). Springer showed that this is a general phenomenon

Theorem 4.3 (Springer, [34, Theorem 3.4]). Let G be any complex reflection group, d any natural number, and ζ any primitive dth root of unity. Then,

(i). maxdimV(w,ζ) = a(d), w∈G (ii). for any g ∈ G there exists w ∈ G with dimV(w,ζ) = a(d) and V(w,ζ) ⊇ V(g,ζ),

(iii). If w1,w2 ∈ G with dimV(w1,ζ) = a(d) = dimV(w2,ζ), then there exists g ∈ G such that
g V(w1,ζ) = V(w2,ζ).

4.2 Regular elements

Recall from Chapter 3 that a vector is said to be regular if it does not lie on any reflecting hyperplane, or equivalently, if its stabiliser is trivial.

Definition 4.4. We call an element w ∈ G regular if it has a regular eigenvector. For d ∈ N, w is said to be d-regular if it has a regular eigenvector with eigenvalue a primitive dth root of unity.

Theorem 2.29 implies that the Coxeter elements in a real reflection group are precisely the h-regular elements. Clearly, no such statement can be expected for arbitrary reflection group elements—for example, diagonal elements of G(m,1,n) with distinct eigenvalues do not have any regular eigenvectors—but Springer nevertheless showed that some properties of Coxeter elements generalise to regular elements.

Theorem 4.5 (Springer, [34, Thm 4.2]). Let w ∈ G, and suppose that w has a regular eigenvector with eigenvalue ζ a primitive dth root of unity. Then,

(i). w has order d,

(ii). dimV(w,ζ) = a(d). Moreover, the elements g ∈ G with dimV(g,ζ) = a(d) are conjugate in G. 4.3. WELL-GENERATED REFLECTION GROUPS AND COXETER ELEMENTS 37

(iii). The centraliser of w is a complex reflection group on V(w,ζ) with degrees those di that are divisible by d.

−m (iv). The eigenvalues of w are ζ i , where mi are the exponents of G.

Note that (i) implies that if an element of order d has a regular eigenvector, then the correspond- ing eigenvalue is a primitive dth root of unity.

4.3 Well-generated reflection groups and Coxeter elements

We now turn to the question of whether there is an analogue of Coxeter elements for complex reflection groups. Throughout this section, we assume that G is irreducible. First of all, note that we have the following analogue of (2.7.1), which was discovered via a case-by-case analysis by Orlik and Solomon [30]: ∗ N + N ≥ ndn. (4.3.1) Hence, it makes sense, following [20], to define the Coxeter number to be N + N∗ h = . (4.3.2) n The following proposition is mentioned without a full proof in [20].

Proposition 4.6. The Coxeter number h is an integer.

Proof. Consider the central element, z = ∑r∈R(1 − r) ∈ Z[G]. By [32, §6.5], z acts on V by an integer scalar. On the other hand, we have z = ∑H∈A eHprH⊥ , where prH⊥ denotes the projection ⊥ ∗ onto H with kernel H. Using again the fact that z is a scalar, we find that nz = ∑H∈A eH = N +N , since by composing the n copies of z with unitary transformations, we can arrange that for each H, the n copies of prH⊥ become projections onto an orthonormal basis.

Inequality (4.3.1) can be restated as dn ≤ h. However, equality does not always hold: for example, if G = G(4,2,2), then by Proposition 3.7 and Example 3.14, we have dn = 4 and h = 6.

Furthermore, when h > dn there cannot be a h-regular element, by Theorem 4.3.

Definition 4.7. A complex reflection group of rank n is well-generated if it can be generated by n reflections. Otherwise it is badly-generated.

For example, every real reflection group is well-generated. On the other hand, every two reflec- tions in G(4,2,2) generate a dihedral group (by the Shephard–Todd classification), so G(4,2,2) is badly-generated. 38 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

It was shown by Orlik and Solomon [30] that the irreducible well-generated reflection groups are the cyclic groups, the symmetric groups, the groups G(m, p,n) for m,n ≥ 2, (m, p,n) 6= (2,2,2) and p ∈ {1,m}, and all of the exceptional groups except G7, G11, G12, G13, G15, G19, G22 and

G31 [30]. Furthermore, they showed that an irreducible badly-generated reflection group of rank n can always be generated by n + 1 reflections. A more conceptual classification of the well-generated groups was proven by Orlik and Solomon ∗ [30], and uniformly by Bessis [1]. Recall from Section 3.5.1 that the codegrees di are defined to be m∗ ∗ i mi − 1, where ∑i t i = ∑i[(SG)i : V ]t .

Theorem 4.8 (Orlik-Solomon, Bessis). Let G be an irreducible complex reflection group with highest degree dn. Then, the following conditions are equivalent:

(i). G is well-generated;

∗ (ii).d i + di = dn;

(iii).d n = h

(iv). There exist generating reflections s1,...,sn for G such that s1 ···sn is dn-regular.

Notice the parallel between condition ((ii)) and the duality between the degrees for real reflection groups in Proposition 2.27.

4.4 Statement of the main theorem

∗ Recall from the introduction that if x ∈ V then we denote be Nx (respectively Nx ) the number of reflections (respectively, reflecting hyperplanes) in Gx. In this thesis we prove the following theorem:

Theorem 4.9. Let G be an irreducible complex reflection group, and x an eigenvector of w ∈ G with eigenvalue a primitive dth root of unity. Then,

(i) ∗ ∗ N + N − Nx − Nx ≥ dn. (4.4.1)

(ii) Equality holds if and only if G is well-generated and d = h. 4.5. NECESSITY OF CONDITIONS 39

4.5 Necessity of conditions

n If G is a real reflection group with degrees (di)i=1, w ∈ G is any element and x is an eigenvector of w, then the corresponding eigenvalue is a primitive dnth root of unity if and only if w is a Coxeter element [25, Thm 32-2 C], so in particular, it is regular by [26]. However, this does not generalise to complex reflection groups. Example 4.10 illustrates that for G a badly-generated reflection group,

(4.4.1) can be a strict inequality even when d = dn.

Example 4.10. Let G = G(m, p,n) with p ∈/ {1,m} and gcd(m,n−1) = 1. This is a badly-generated reflection group with largest degree m(n−1). Let ζ be a primitive mth root of unity and ω a primitive (n − 1)st root of unity. Let   0 0 ··· 0 ζ 0   ζ 0 ··· 0 0 0    .. . . .  0 ζ . . . .  w =   ∈ G,  ......   ......    0 0 ··· ζ 0 0   0 0 ··· 0 0 ζ

T and x = ωn−2,ωn−1,...,ω,1,0
. Then, x is an eigenvector of w with eigenvalue ζω of order p m(n − 1) = dn. However, Gx is the cyclic group generated by the matrix diag(1,1,...,1,ζ ), so ∗ ∗ m we have N + N − Nx − Nx = mn(n − 1) + (n − 1) p , which is strictly larger than dn = mn(n − 1). On the other hand, in the case of well-generated reflection groups, Proposition 5.12 below shows that for well-generated reflection groups, if d is equal to the Coxeter number, then x is regular, and in particular (4.4.1) is an equality. To see that the irreducibility condition is necessary, consider the following example.

Example 4.11. Consider any sequence of integers d1,...,dn ≥ 2 with n ≥ 2, and let

G = µd1 × µd2 × ··· × µdn .

Let w be a generator of the last factor, and x = (0,0,...,0,1)T . Then, in the notation of Theorem 4.9, we have d = dn, but n n−1 ∗ ∗ N + N − Nx − Nx = ∑ di − ∑ di i=1 i=1

= dn < nd, 40 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

which demonstrates that Theorem 4.9 fails to hold for reducible groups, regardless of their degrees.

Notice that the special case where all of the di are equal to 2 shows that Theorem 4.9 also fails to hold in the case of reducible real reflection groups.

4.6 Braid group formulation

reg We now reformulate Theorem 4.9 in terms of the pure braid group P = π1(V ,x0). Recall that the generator π of Z(P) introduced in Section 3.6.4 has length N +N∗. On the other hand, the generator ∗ π x of the parabolic subgroup Px in the braid group has length Nx + Nx . Thus, we can reformulate our main theorem as follows.

Theorem 4.12. Let G be an irreducible complex reflection group, and x an eigenvector of w ∈ G with eigenvalue a primitive dth root of unity. Then,

(i)

`(π) − `(π x) ≥ dn. (4.6.1)

(ii) Equality holds if and only if G is well-generated and d is the Coxeter number of G.

4.7 The case of real reflection groups

When G is a real reflection group, Theorem 4.9 was first proven case-by-case by Kamgarpour in [24]. Here we instead present a uniform proof communicated to us by Jean Michel. For I a subset of the positive roots, we denote by GI the corresponding parabolic, and π I the positively-oriented generator of the centre of the corresponding pure braid group, which we will henceforth identify with its image in P. The following theorem is a special case of [17, Thm 8.1].

Theorem 4.13 (Digne-Michel). Let G be a real reflection group, and w, x and d be above. Then, there exists a Coxeter system (G,S) and some I ⊆ S such that

(i).H = GI, and

0 0 (ii). if w is the element of smallest length in wGI, then there is a lift w ∈ B of w such that d w = π/π I.

Proof. We follow the proof in [17]. We first show that I can be chosen such that that for all d 1 ≤ i ≤ b 2 c, we have i `(w0i) = |Φ \ Φ |. d I 4.7. THE CASE OF REAL REFLECTION GROUPS 41

(Notice that this already establishes (4.4.1), but not the remaining assertions of Theorem 4.9). We then use this to show that the reduced word lift of w0 satisfies the required condition.

Let (·,·) denote the G-invariant hermitian form on V ⊗R C. Without loss of generality, we may assume that for α ∈ Φ we have Re(α,x) = 0 if and only if (α,x) = 0, since otherwise we can ensure this by multiplying x by a complex number of norm 1. Choose the Coxeter system S such that Wx = hrα | Re(α,x) = 0i is standard and

Φ+ ⊆ {α ∈ φ | Re(α,x) ≥ 0}.

Without loss of generality, assume that w is of minimal length in WIw = wWI. Notice for later, −1 that this means that the reduced word lift w of w is both a left-divisor of w0w0,I and a right-divisor −1 + of w0,I w0 in B . To see the first assertion, notice that w is a left-divisor of w0, say w0 = wwvv. By the + cancellativity of B , it is then enough to prove that w0,I is a right-divisor of v. But w0,I is a right 0 00 divisor of w0 = wwvv, so if it does not divide v then we can use the braid relations to write w = w w 00 where w ∈ BI, contradicting the minimality assumption. The other assertion is similar. d + d + We now show that w = 1, by showing that for all α ∈ Φ , we have w α ∈ Φ . First, if α ∈/ ΦI, then (wdα,x) = (α,w−dx) = (α,x) > 0, as required. On the other hand, the minimality assumption implies that w preserves the sign of every α ∈ I since otherwise rα w will have smaller length. By additivity, this means that w preserves the sign of every root in ΦI, so w stabilises ΦI.

By the previous paragraph, w acts on Φ \ ΦI with orbits of size exactly d: an orbit cannot be i i smaller than that, since for any α ∈ Φ \ ΦI, the complex numbers (w α,x) = ζ (α,x) are distinct d for all 0 ≤ i ≤ d − 1. Hence, we see that for 1 ≤ i ≤ b 2 c, the number positive roots in each orbit made negative by wi is exactly one more than the number made negative by wi−1 (since a root

α ∈ Φ \ ΦI is positive if and only if Re(α,x) > 0). Adding up these numbers for each orbit, we get

|Φ \ Φ | `(wi) = i I . d

d −1 d/2 Finally, to see that w = ππππ I , we consider two cases. First, if d is even, we get `(w ) = −1 −1 `(w0w0,I ) = `(w0,I w0). Since w is either a left- or a right-divisor of each of these elements, it d −1 2 −1 −1 follows that all three must be equal. Hence, w = w0,I w0w0,I = ππππ I , as required. The odd case is similar.

Remark 4.14. (i). Theorem 4.13 implies that in B, we have

|Φ| − |Φ | `(w) = I . (4.7.1) d
In particular, d | |Φ| − |Φx| . 42 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

(ii). On the other hand, we do not always have d | `(π) − `(π x) for complex reflection groups. 2πi
T ∼ For example, let G = G(3,1,2), w = exp 3 Id, and x = (1,1) . Then, Gx = S2, so `(π) − `(π x) = 10, but d = 3.

2 Example 4.15. If w is the identity element, then w = π = w0, since d = 1 and Gx is trivial. In particular, as explained in Remark 4.14, it is not an element of any parabolic subgroup of B.

Lemma 4.16. Let G be an irreducible complex reflection group, and let H, H0 be proper parabolic subgroups of G. Then, G contains a reflection which is contained in neither H nor H0.

Proof. If G is of rank 1, then it contains only one proper parabolic, namely the trivial subgroup. On the other hand, if the rank of G is at least 2, then by the assumption that G is irreducible, there must 0 exist reflections s1 ∈ H and s2 ∈ H which do not commute, and therefore generate an irreducible reflection subgroup H00 ≤ G of rank 2. Hence, the reflection arrangement of H00 is not contained in the union of the reflection arrangements of H and H0, so in particular, H00 contains a reflection which is contained in neither H nor H0.

We now outline Michel’s proof of our main theorem for real reflection groups. By (4.7.1), it suffices to prove that `(w) ≥ n, and that equality holds if and only if d = h. To prove the first assertion, we will show that w is not contained in any proper standard parabolic subgroup of the braid group, and therefore any word for w in the Artin generators σ i must contain every σ i. 2 2 From Proposition 4.13 and the fact that π = w0 and π I = w0,I (where w0,I is the reduced word lift of the longest element in the standard parabolic GI), it follows that:  wd/2 if d is even w0/w0,I = bd/2c w w1 if d is odd, where w1 is the first half of a word for w. Therefore, it suffices to show that w0/w0,I is not contained in any proper standard parabolic.

Indeed, w0/w0,I is not contained in Gx, otherwise we would have w0 ∈ Gx, which is impossible for length reasons. On the other hand, let H be any other proper parabolic. Then, by Lemma 4.16, there exists a reflection sα ∈ G which is not contained in Gx or H. We then see that α is negated by w0, but it is not negated by w0,I (because, since Gx is a standard parabolic, the lengths of w0,I in Gx and G coincide, so w0,I cannot negate any roots other than those in Φx). Therefore, w0/w0,I negates α, so by similar reasoning, it cannot be an element of H. For the second assertion, suppose that equality holds in (4.6.1), which by (4.7.1) means that `(w) = n. Then, since the word w contains every simple braid reflection, it is a Coxeter element. 4.8. CASES WHERE EQUALITY ALMOST HOLDS IN (4.6.1) 43

Therefore, in B we have: d 2 2 h 2 w = w0/w0,I = w /w0,I, 2 h−d which implies that w0,I = w . But since w is not in any proper parabolic, this can only happen if d = h and I = /0. Conversely, if d = h, then in B we have:

2 h 2 2 w0 = w = w0/w0,I.

Therefore, w0,I = 1 and Gx is trivial, so equality holds in (4.6.1) by Corollary 2.28.

4.8 Cases where equality almost holds in (4.6.1)

The proof in Section 4.7 allows us to make a stronger statement, namely that if d < h, then

|Φ| − |Φx| ≥ (n + 1)d. (4.8.1)

To see this, recall that w ∈ B+, and that w does not lie in any proper parabolic subgroup. Write w as a product of the simple braid reflections σ i (to positive powers, by positivity). By the pigeonhole principle, we see that if `(w) = n, then either all of the σ i occur exactly once, or at least one σ j does not occur. The first case cannot happen because then w would be a Coxeter element, contradicting our assumption that d < h. But the second case also cannot occur, because then w would lie in the standard parabolic generated by the σ i with i 6= j. Therefore, `(w) ≥ n + 1, so (4.8.1) follows by taking the lengths of both sides in (4.7.1). However, it is not clear in which cases (4.8.1) is attained. In one direction, we have the following proposition.

Proposition 4.17. Let G be an irreducible real reflection group of rank n on the vector space V, w ∈ G, and x ∈ V an eigenvector of w with eigenvalue a primitive dth root of unity. Suppose in addition that Gx is a standard parabolic, and that w has minimal length in wGx.

(i). If equality holds in (4.8.1), then w is a Coxeter element in a maximal parabolic.

(ii). On the other hand, if (4.8.1) is strict, then we have,

|Φ| − |Φx| ≥ (n + 2)d. (4.8.2)

Proof. If |Φ| − |Φx| = (n + 1)d, then by the above discussion, the braid group element w would need to have length n + 1. Now consider any word for w in the σ i. Since w is not an element of any 44 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

proper parabolic, a pigeonhole principle argument similar to the above discussion shows that there is some σ j that occurs twice in this word, and every other σ i occurs exactly once. Passing to the reflection group, we can write,

w = s(1) ···s(k)s js(k+1) ···s(l)s js(l+1) ···s(n−1), (4.8.3) where 0 ≤ k ≤ l ≤ n − 1, and the s(i) are distinct simple reflections different from s j. Denoting the 0 reflections s js(i)s j by s(i), we then see that

0 0 w = s(1) ···s(k)s(k+1) ···s(l)s(l+1) ···s(n−1), which shows that `T (w) ≤ n − 1. On the other hand, we can apply the Steinberg relations to (4.8.3) to write w = sc, where s is some reflection, and c = s(1) ···s(k)s(k+1) ···s(l)s js(l+1) ···s(n−1) is a Coxeter element; equivalently, c = sw. Since `T (c) = n and `T (s) = 1, this implies that c >T w. Hence, by Proposition 3.44, we conclude that w is a parabolic Coxeter element. On the other hand, we must have `T (w) = n − 1, so any such parabolic must be maximal.

Finally, if |Φ|−|Φx| > (n+1)d, then by Remark 4.14, we must have |Φ|−|Φx| ≥ (n+2)d.

Remark 4.18. The converse of Proposition 4.17 is false. For example, there are no elements of

I2(6) for which equality holds in (4.8.1).

⊥ Example 4.19. Let W = W(B2), w = s1s2s1s2, which acts by −1, and take x ∈ α for any root α. Then, one can easily check that equality holds in (4.8.1), but w is not in any proper parabolic, since it has no fixed points. This shows that the minimality assumption in Proposition 4.17 is necessary.

Remark 4.20. One might wonder whether the above pattern continues: that is, whether under the conditions of Proposition 4.17, if |Φ| − |Φx| = (n + 2)d then w is a parabolic Coxeter element. This turns out to be false, as the following example illustrates. Let G = W(B2), w = s1s2s1s2, and take x to be any regular vector. Then, w does not belong to a proper parabolic, but

|Φ| − |Φx| = 8 = d(n + 2).

4.9 Towards a uniform proof

In order to generalise the proof in Section 4.7 to complex reflection groups, Jean Michel [29] has privately suggested the following conjectural generalisation of Theorem 4.13. However, a proof of this conjecture seems much more elusive, because we no longer have access to either the 4.9. TOWARDS A UNIFORM PROOF 45

root-theoretic description of length, nor the reduced word lifting of elements of G to B, both of which are relied heavily upon in the proof of Theorem 4.13 in [17].

Conjecture 4.21 (Michel, private communication). In the setting of Theorem 4.9, there exists a lift d −e w of some element of Gxw to B such that w = ππππ x for some e ≥ 0.

Notice that we cannot expect the above conjecture to always hold with e = 1, since that
would imply d | `(π) − `(π x) , contradicting Remark 4.14 (ii). We were unable to prove that Conjecture 4.21 implies Theorem 4.12. However, we have the following partial result.

Proposition 4.22. Let G be an irreducible complex reflection group, w ∈ G, and x ∈ V an eigenvec- tor of w with eigenvalue a primitive dth root of unity. Suppose that either V w = 0, or dimV w = 1 w and the base point x0 is chosen such that the angle between x0 and V with respect to the Euclidean inner product Re(·,·) is smaller than the angle between V w and any reflecting hyperplane not con- + d −e taining it. Finally, suppose that there exists a lift w of w to B such that w ∈ B (x0) and w = ππππ x for some e ≥ 1. Then,

`(π) − e`(π x) ≥ dn.

d −e Proof. By the assumption w = ππππ x , we have,

d `(w ) = `(π) − e`(π x), so it remains only to show that `(w) ≥ n. If not, then by positivity of w we see that w is a product of at most n − 1 braid reflections. The intersection U of the corresponding hyperplanes has positive + dimension, and w is a product of braid reflections σ 1,...,σ k ∈ B (x0) about hyperplanes containing U. w If V = 0, then this is an immediate contradiction, because it means that w = s1 ···sk ∈ GU , w w w so U ⊆ V . From now on, assume that dimV = 1, so V = U. Hence, if x0 is sufficiently close to U, then the real line segments γ and γ are homotopic1 where the H are the reflecting γ x0,U γ x0,Hi i hyperplanes for the σ , and for W a linear subspace of V, γ is the line segment from x to W σ i γ x0,W 0 orthogonal to W with respect to Re(·,·). So in fact, the σ i belong to BU , and in particular, w ∈ BU . d −e Since w = ππππ x , we then have

d e π = (σ 1 ···σ k) π x. (4.9.1)

But we claim that this would mean that A = Ax ∪ AU , contradicting Lemma 4.16. To see this, note −e −e that ππππ x ∈ P, and it is also in BU since it is a power of w. This means that ππππ x ∈ PU . 1The difficulty is that without the assumption on dimV w, the interior of the triangle defined by these line segments w could have intersected some other reflecting hyperplane, since even if we assume that x0 is near V , we cannot ensure that it is near U ⊆ V w. 46 CHAPTER 4. EIGENVECTORS OF REFLECTION GROUP ELEMENTS

Hence, by Proposition 3.26, we can write

r  e  π = σ Hi π e, (4.9.2) ∏ Hi,γ i x i=1 where the Hi belong to AU . reg × Let H ∈ A, and consider αH : V → C . Since all r + e terms on the right hand side of (4.9.2) reg are in P = π1(V ,x0), we can write,

r eHi π1(αH)(π) = π1(αH)(σ ) + eπ1(αH)(π x). ∑ H1,γ i i=1

But the left hand side is 1, while the first term on the right-hand side can only be nonzero if H ∈ AU , and the second can only be nonzero if H ∈ Ax. So H ∈ AU ∪ Ax. Chapter 5

Proof of the main theorem

In order to prove (4.4.1), we proceed case-by-case via the Shephard–Todd classification. In doing so, we will also see that when G is well-generated and d < h, (4.4.1) is always a strict inequality. We then prove the remainder of statement (ii) in Section 5.6, using Springer’s theory of regular elements.

5.1 The case G = G(m,1,1)

∗ In this case, every w is a scalar given by an mth root of unity, N + N = m, and Gx is trivial. Therefore, (4.4.1) holds, with equality if and only if w is a primitive mth root of unity, i.e., if d = m, which is the Coxeter number.

5.2 The case G = G(1,1,n + 1) = Sn

This case was first proven by Kamgarpour [24], and the proof by Michel in § 4.7 also applies. How- ever, in order to motivate our approach to the imprimitive cases, we briefly reproduce Kamgarpour’s proof here.

First, one considers the case d ≤ 2. Using the fact that the largest proper parabolic in Sn+1 is

Sn, we see that the left hand side of (4.4.1) is at least 2n. Hence, we only need to prove that we never have both d = 2 and rk(Gx) = n − 1. But the latter condition holds if and only if x is a scalar n multiple of the image of a standard basis vector ei ∈ C , and there is no permutation matrix which negates such a vector.

47 48 CHAPTER 5. PROOF OF THE MAIN THEOREM

For d ≥ 3, we identify V with the Cartan subalgebra in sln+1 (via the Killing form), and let n+1 n n−i f (T) = T + ∑i=1 aiT be the characteristic polynomial of x. The ai are then the evaluations at x of a set of fundamental invariants for G (up to a sign, they are the elementary symmetric polynomials), so by Theorem 4.1, ai = 0 if and only if d | di = i + 1, or in other words, the only terms appearing in f are of degree congruent to n+1 modulo d. Therefore, there exists a polynomial n+1−dk d Q ∈ C[T] of degree k such that f (T) = T Q(T ). Hence, the multiset of the xi consists of the elements i 1/d ζ γ j for 0 ≤ i ≤ d − 1 and 1 ≤ j ≤ k (where ζ is a primitive dth root of unity and γ j are the roots of Q), as well as n + 1 − dk copies of 0. Since Gx can only permute two xi if they are equal, we see that

Gx is no larger than d Sn+1−dk × Sk , where the first factor permutes the zeroes, and the remaining factors permute equal nonzero coordinates. Since |Φ| = n(n + 1), this implies that

|Φ| − |Φx| ≥ dk(2n + 2 − k − dk) ≥ dk(n + 1 − k). (5.2.1)

Combining this with the assumption that d ≥ 3 (so k ≤ n/3), it is easy to show that the right hand side of (5.2.1) is greater than or equal to dn, with equality if and only if k = 1 and d = n + 1 = h. This establishes (4.4.1).

5.3 The groups G(m, p,n): structure of the stabiliser

We now turn our attention to the groups G(m, p,n) for m,n ≥ 2 and (m, p,n) 6= (2,2,2). In order to generalise the method of Kamgarpour, we will need to determine the structure of the stabiliser. We will need the following lemma.

Lemma 5.1. Let G = G(m, p,n) with m,n ≥ 2, (m, p,n) 6= (2,2,2), and w, x, d, Gx be as in the statement of Theorem 4.9. Then, there exist integers 1 ≤ λi ≤ k for i ∈ {1,...,k} such that

d G ∼ (m, p,r) × S . x = G ∏ λi i=1 5.3. THE GROUPS G(m, p,n): STRUCTURE OF THE STABILISER 49

We will prove Lemma 5.1 later in this section. We first introduce some notation that will be needed for the remainder of this chapter. Reorder the di as follows:  im if i ≤ n − 1 di = mn  p if i = n.

Correspondingly, we have a set of fundamental invariants,  m m ei(x1 ,...,xn ) if i ≤ n − 1 fi = m en(x1,...,xn) p if i = n.

Let ω be a generator of µm, and ζ a generator of µd. Finally, let r denote the number of coordinates of x which vanish.

dk Lemma 5.2. There exists a positive integer k such that dk ≤ mn, m | dk, and r = n − m , and nonzero complex numbers γ1,...,γk such there is an equality of multisets

j rm j 1/d {ω xi | 1 ≤ i ≤ n,1 ≤ j ≤ m} = {0 } ∪ {ζ γi | 1 ≤ i ≤ k,1 ≤ j ≤ d}. (5.3.1)

Proof. Consider the polynomial

j f (T) = ∏ (T − ω xi) (5.3.2) 1≤i≤m 1≤ j≤n n mn mi m m m(n−i) = T + ∑(−1) ei(x1 ,...,xn )T i=1 n−1 mn mi m(n−i) mn p = T + ∑ (−1) fi(x1,...,xn)T + (−1) fn(x1,...,xn) . i=1 ∼ By construction, this polynomial is invariant under the natural action of G(m,1,n) on C[V][T] = Sym(V ∗)[T], and hence also under the action of its subgroup G. mn−mi By Theorem 4.1, we see that the coefficient of T is nonzero if and only if d|di. Therefore f can be written as f (T) = T mn−dkQ(T d), where Q(T) is a monic polynomial of degree k such that m|dk and dk ≤ mn. Since, by Theorem 4.3, d divides at least one di, we have k > 0. Let γ1,...,γk denote the roots of Q. Then the roots of f are

j 1/d ζ γi , 1 ≤ i ≤ k, 0 ≤ j ≤ d − 1, and 0 with multiplicity mn − dk. 50 CHAPTER 5. PROOF OF THE MAIN THEOREM

j By definition, G permutes the vectors ω ei. If g = (θ,σ) ∈ Gx, this implies that

n n ∑ xiθieσ(i) = ∑ x je j. i=1 i=1

j Thus, Gx permutes the ω xiei. This induces an action of Gx on the multiset in (5.3.1).

j1 j2 If ω xi1 and ω xi2 are distinct elements of the same Gx-orbit, we immediately see that xi1 /xi2 ∈ µm. Furthermore, i1 6= i2, since otherwise, the ith coordinate of x takes two different values.

Lemma 5.3. Let m, n be positive integers, and ∼ an equivalence relation on Z/mZ × Z/nZ such that for all a,b ∈ Z/mZ and c,d ∈ Z/nZ, we have:

(i). if c 6= d then (a,c) 6∼ (a,d), and

(ii). if (a,c) ∼ (b,d), then (a,c + 1) ∼ (a,d + 1).

Then, there exists a subset S ⊆ Z/mZ × Z/nZ of cardinality m, such that

(iii). S is a union of ∼-equivalence classes;

Sn
(iv). Z/mZ × Z/nZ = i=1 (0,i) + S

Proof. We proceed by induction. If m = 1, the subset S = (0,0) trivially satisfies properties (iii) and (iv). Now suppose m ≥ 2. By the inductive hypothesis, there exists a subset S0 ⊆ {0,...,m − 2}×Z/nZ of cardinality m−1 which is a union of ∼-equivalence classes in {0,...,m−2}×Z/nZ, and whose Z/nZ-translates partition {0,...,m − 2} × Z/nZ. If (m − 1,0) is not ∼ equivalent to any element of {0,...,m − 2} × Z/nZ, then let S = S0 ∪ {(m − 1,0)}. Otherwise, there is a unique r ∈ Z/nZ for which (m − 1,0) is ∼-equivalent to an element of (0,r) + S0; in this case, let S = S0 ∪ {(m − 1,−r)}. It is obvious that in both cases, S has cardinality m and satisfies (3) and (4).

Proof of Lemma 5.1. Consider G as a subgroup of the group Smn of permutations on the multiset j 1/d in (5.3.1). The rm copies of zero are permuted transitively, so we restrict our attention to the ζ γi . Define the relation ∼ on Z/kZ × Z/dZ by (i1, j1) ∼ (i2, j2) if and only if

1/d 1/d (ζ j1 γ )/(ζ j2 γ ) ∈ µ . i1 i2 m

Then, ∼ satisfies the conditions of Lemma 5.3, so let S be a subset satisfying the conclusions of Lemma 5.3. 1/d Since S is a union of ∼-equivalence classes, we see that G can only permute ζ j2 γ and x i2 1/d ζ j2 γ if (i , j ) and (i , j ) are in the same translate of S. i2 1 1 2 2 5.3. THE GROUPS G(m, p,n): STRUCTURE OF THE STABILISER 51

Since |S| = k and there are d such translates, we see that Gx is contained in the group d Srm × Sk ⊂ Smn, where the Srm factor consists of the permutations of the rm copies of zero, and the Sk are the d permutations on each translate of S. But the intersection of Srm × Sk with G is d G(m, p,r) × Sk , and since Gx is a parabolic which contains the G(m, p,r) component, we see from Theorem 3.46 that Gx is a product of G(m, p,r) with a product of one or more symmetric groups, and that this d product of symmetric groups is contained in Sk . Corollary 5.4. We have,  m ∗ ∗ dk(n + r − k) + p if p < m N + N − Nx − Nx ≥ (5.3.3) dk(n + r − k) if p = m. Proof. For groups of the form G(m, p,n) with p < m we have, nm N + N∗ = + mn(n − 1). (5.3.4) p By Lemma 5.1, we therefore have, rm N + N∗ ≤ + mr(r − 1) + k(k − 1), (5.3.5) x x p where k and r are as in the statement of Lemma 5.1. Taking the difference of (5.3.4) and (5.3.5), we get, m N + N∗ − N − N∗ ≥ (n − r) + mn(n − 1) − mr(r − 1) − dk(k − 1) x x p m = (n − r) + m(n − r)(n − r − 1) + 2mr(n − r) − dk(k − 1) p = dkp−1 + dk(n − r − 1) + 2dkr − dk(k − 1), dk(n + r − k + p−1), where the second-last line follows from n − r = dk/m. For groups of type G(m,m,n) we have, N + N∗ = mn(n − 1). Lemma 5.1 therefore gives, ∗ Nx + Nx ≤ mr(r − 1) + k(k − 1), where k and r are as in the statement of Lemma 5.1. By a similar calculation to the p < m case, we then get, ∗ ∗ N + N − Nx − Nx ≥ dk(n + r − k). 52 CHAPTER 5. PROOF OF THE MAIN THEOREM

5.4 The groups G(m, p,n): proof of the inequality

As with the case of the symmetric group, we prove (4.4.1) as follows. First, we show that the inequality holds for small values of d (Lemma 5.6), by calculating the largest value that the left hand side can take for any proper parabolic (Lemma 5.5). For large values of d, the condition dk ≤ mn then gives an upper bound on k. On the other hand, Corollary 5.4 gives a lower bound on the left hand side of (4.4.1) which is quadratic in k with negative leading coefficient. Combining these bounds allows us to deduce (4.4.1).

Lemma 5.5. Suppose m,n ≥ 2, and let Gx range over all proper parabolic subgroups of G(m, p,n). ∗ ∼ Then, Nx + Nx is maximised when Gx = G(m, p,n − 1), in which case,  m ∗ ∗ 2m(n − 1) + p if p < m N + N − Nx − Nx = (5.4.1) 2m(n − 1) if p = m.

α Proof. First, suppose that p < m. From Proposition 3.7, we see that if Gx is a parabolic GΛ of type (ii) corresponding to a partition n = n1 + ··· + nq, then q q ∗  2 Nx + Nx = ∑ ni(ni − 1) = ∑ ni − n. (5.4.2) i=1 i=1

Similarly, if Gx is a parabolic GΛ of type (i), where Λ consists of a thick triple (m, p,n0) and thin triples (1,1,n1),...,(1,1,nq), then q ∗  2 mn0 Nx + Nx = ∑ ni − n + + (m − 1)n0(n0 − 1). i=0 p 1 ∗ In both cases, one can easily check that Nx + Nx is maximised (subject to Gx 6= G) when the partition of n is (n − 1,1). Thus, the worst case is Gx = G(m, p,n − 1), in which case, mn m(n − 1) N + N∗ − N − N∗ = mn(n − 1) + − m(n − 1)(n − 2) − (5.4.3) x x p p m = 2m(n − 1) + . (5.4.4) p

Now suppose p = m. Then, again by Theorem 3.46, Gx must either be of the form GΛ where Λ α contains exactly one thick feasible triple (m,m,n0), or of the form GΛ, where Λ contains only thin ∗ feasible triples. In the latter case, Nx + Nx is again given by (5.4.2). In the former case, we have q ∗  2 Nx + Nx = ∑ ni − n + (m − 1)n0(n0 − 1). i=0 1 2 The function (n1,...,nq) 7→ ∑i ni is order-preserving with respect to the refinement partial order on partitions, and every partition other than (n) is a refinement of a 2-row partition, so it suffices to check the claim for 2-row partitions, which is elementary algebra. 5.4. THE GROUPS G(m, p,n): PROOF OF THE INEQUALITY 53

∗ Again, we see that Nx + Nx is maximised when Gx = G(m,m,n − 1), in which case,

∗ ∗ N + N − Nx − Nx = 2m(n − 1).

Lemma 5.6. Inequality (4.4.1) holds in each of the following cases:

(i).p < m and n = 2,

mn (ii).p < m, n ≥ 3 and d ≤ n−1 ,

(iii).p = m and n ≤ 3,

3m (iv).p = m, n ≥ 4 and d < 2 . If in addition, d < h, then (4.4.1) is a strict inequality.

Proof. Consider first the case p < m. If n = 2, then (5.4.3) states that the left hand side of (4.4.1) is m 2m at least 2m + p . But the degrees in this case are m and p , so by Theorem 4.1, we have dn < 2m unless p = 1 and d = 2m = h. But since G(m,1,n) is well-generated, (4.4.1) also holds in this case by Proposition 5.12 below. m If n = 3, then (5.4.3) states that the left hand side of (4.4.1) is at least 4m + p . On the other 3m hand, the degrees are m, 2m and p . Hence, by Theorem 4.1, the only cases in which dn can be greater than or equal to the left hand side of (4.4.1) are:

• d = 2m;

3m • p = 1 and d = 3m or d = 2 (m even);

3m • p = 2 and d = 2 . The case p = 1 and d = 3m = h is again covered by Proposition 5.12. In all of the other cases, we mn 3m have d ≥ n−1 = 2 . Finally, if n ≥ 4, then we have, 2(n − 1)2 > n2. Therefore, mn2 2m(n − 1) > ≥ dn, n − 1 so the claim follows by (5.4.3). Now consider the case p = m. If n = 2 then (5.4.1) becomes 2m, whereas the fundamental degrees are {2,m}, so dn ≤ 2m. Therefore, by Theorem 4.1, (4.4.1) always holds. Similarly, if n = 3, the above expression becomes 4m, whereas the degrees are {3,m,2m}, so (4.4.1) strictly holds unless d = 2m = h. But in that case, we already know that d must be regular. 3 3 3 Finally, if n ≥ 4 and d < 4 m, then n − 1 ≥ 4 n, so 2m(n − 1) ≥ 4 mn > dn. 54 CHAPTER 5. PROOF OF THE MAIN THEOREM

Lemma 5.7. With notation as above, if r = 0, then inequality (4.4.1) holds. Equality holds if and only if p ∈ {1,m} and d = h.

m Proof. Since by assumption r = 0, we have fn(x) = (x1 ···xn) p 6= 0, so Theorem 4.3 implies that mn d|deg( f ) = . (5.4.5) n p

We will consider the three cases p = m, 1 < p < m and p = 1 separately. First, one can easily show that if 2 ≤ k ≤ n − 2, then k(n − k) ≥ n with equality only when n = 4 and k = 2; we will use this fact repeatedly. 3 In the case p = m, we may assume by Lemma 5.6 that n ≥ 4 and d ≥ 2 m. Since d|n (by (5.4.5)) 3 2 and dk = mn, we get k ≥ m ≥ 2. On the other hand, since d ≥ 2 m, k ≤ 3 n ≤ n − 2 (since n ≥ 3). ∗ ∗ Hence, N + N − Nx − Nx ≥ dk(n − k) ≥ dn. Furthermore, if equality holds, then n = 4, k = 2, so ∗ m = 2 and d = 4. In this case, we would need to have Nx + Nx = 8, but one can easily check that there is no parabolic in G = W(D4) satisfying this condition. In the cases 1 < p < m and p = 1, we may assume (by Lemma 5.6, (i) and (ii)) that n ≥ 3 and mn d ≥ n−1 . In the case 1 < p < m, (5.4.5) and the fact that dk = mn imply that p|k. Since p > 1, it follows mn that k ≥ 2. On the other hand, we have k ≤ n − 1, since by assumption, d ≥ n−1 . If k = n − 1, then, since n ≥ 3, we have k > n/2. Therefore, since dk = mn, we must have d = 1. But this contradicts r = 0. Therefore, k ≤ n − 2. By (5.3.3), we obtain

∗ ∗ −1 N + N − Nx − Nx ≥ dk(n − k) + dkp dk ≥ dn + . p

Therefore, the inequality (4.4.1) is strict in this case. Finally, if p = 1, then we again have that 1 ≤ k ≤ n − 2, and the same argument shows that if in addition k ≥ 2, then (4.4.1) holds and is a strict inequality. This leaves the case k = 1, for which, mn d = d = mn = h. On the other hand, the right hand side of (5.3.3) becomes dn, so equality holds as required.

3 Lemma 5.8. Let m,n,r,d,k be positive integers such that m(n − r) = dk, n ≥ 3 and d ≥ 2 m. Then,

k(n + r − k) ≥ n.

Furthermore, if either k ≥ 2 or r ≥ 2, then the inequality is strict. 5.5. THE EXCEPTIONAL GROUPS 55

3 2 Proof. Since dk < mn and d ≥ 2 m, we have k < 3 n ≤ n − 1. On the other hand, k(n + r − k) = n + (n − k)(k − 1) + k(r − 1) ≥ n. The second assertion follows, since if k ≥ 2 then the second term is positive and the third is nonnegative, and if r ≥ 2 then the third term is positive and the second is nonnegative.

Proposition 5.9. Inequality (4.4.1) holds whenever G = G(m, p,n) where m,n ≥ 2 and (m, p,n) 6= (2,2,2). Moreover, it is a strict inequality unless G is well-generated and d = h.

3m Proof. Lemma 5.6 shows that if either n < 3 or d < 2 , then the statement is true. Hence, from 3m now on, we assume that n ≥ 3 and d ≥ 2 . If r = 0, the statement is proven in Lemma 5.7, so we will also assume from now on that r > 0. By combining Corollary 5.4 and Lemma 5.8, we see that (4.4.1) holds, and that it is strict if in addition either k ≥ 2 or r ≥ 2. Finally, if k = 1 = r, then by definition we get d = m(n − 1). If p = m, this means that d = h. If p < m, we again get a strict inequality due to the extra term in (5.3.3).2

5.5 The exceptional groups

5.5.1 Exceptional groups of rank 2

We first consider the case where n = 2. Then, Gx will have rank 1, and will therefore be a cyclic ∼ group, G(m,1,1) = Z/mZ. Hence, Gx has m − 1 reflections (namely, all nontrivial elements), and one reflecting hyperplane (the trivial subspace). Therefore, we see that

∗ ∗ ∗ N + N − Nx − Nx = N − (m − 1) + N − 1 = 2h − m, where h is as in (4.3.2). So we need to prove that 2h − m ≥ 2d, or equivalently,

m ≤ 2(h − d). (5.5.1)

For the exceptional groups of rank 2 (G4 through G22 in the Shephard–Todd classification), this inequality is checked in Table B.1 (which is derived from the tables in [27]). The column maxd is the largest divisor, other than h, of a degree. As is evident from the table, if d < h, then (5.5.1) is always satisfied, and is a strict inequality.

2 Note that when 1 < p < m, we also have d = dm, but these groups are not well-generated. 56 CHAPTER 5. PROOF OF THE MAIN THEOREM

5.5.2 Exceptional groups of higher rank

We now turn our attention to the remaining exceptional groups, G24, G25, G26, G27, G29, G31, G32,

G33 and G34. (The Coxeter groups, G23,G28,G30,G35,G36 and G37 have already been considered by Kamgarpour.) We proceed by determining the isomorphism classes of maximal parabolics, as per the tables in [37, §6]. In some cases we can prove that the inequality holds for every maximal parabolic

Gx < G (on the left hand side) and the largest possible value of d (on the right hand side), which is clearly sufficient. We shall see that amongst the exceptional groups, there are only a handful of cases where this strategy does not work, and which therefore need to be considered separately. Table B.2 gives the isomorphism classes of maximal parabolics (column 2), the largest value ∗ ∗ of Nx + Nx amongst these parabolics (column 3), N + N (column 4), the rank (column 5), the fundamental degrees (column 6) and the largest possible value of d (column 7; note that for G31, h 6= dn, so maxd = dn). By inspection, we see that the inequality is immediate, except possibly in the following five cases (one can easily check, by similar methods, that for these groups, the smaller parabolics and smaller values of d do not add any further exceptions). Hence, it remains only to show that there is no pair (w,x) for which Gx and d take these values.

(i). G = G25, Gx = G4, d = 9;

(ii). G = G31, Gx = G(4,2,3), d = 24;

(iii). G = G32, Gx = G25, d = 24;

(iv). G = G33, Gx = G(3,3,4), d = 12;

(v). G = G34, Gx = G33, d = 30.

Cases (iv) and (v) do not occur, due to the following lemma:

Lemma 5.10. Let G be a complex reflection group, and suppose that w ∈ G has an eigenvector x with eigenvalue a primitive dth root of unity. Then,

(i). The element w normalises Gx.

d (ii). The positive integer d is minimal such that w ∈ Gx.

(iii). The order of the subgroup hw,Gxi ≤ G is d |Gx|. 5.6. THE CASE d = h 57

(iv). We have, (d |Gx|) |G|.

−1 −1 Proof. Let g ∈ Gx. Then, since x is an eigenvalue of w, we have w gw(x) = x, so w gw ∈ Gx, a a whence the first assertion. The second assertion holds because w ∈ Gx is equivalent w (x) = x, which is equivalent to d|a. From the first two assertions, we deduce that any element of hw,Gxi i can be expressed uniquely in the form w g, where 0 ≤ i ≤ d − 1 and g ∈ Gx. There are d |Gx| such expressions, hence (iii) holds. The final assertion follows immediately.

5 5 In case (iv), we see from e.g. the tables in [27] that d |Gx| = 2 · 3 , which does not divide 7 4 8 5 2 9 7 |G| = 2 ·3 ·5. In case (v), we get d |Gx| = 2 ·3 ·5 , which again does not divide |G| = 2 ·3 ·5·7.

Thus, Theorem 4.9 holds for the groups G33 and G34.

In cases (i), (ii) and (iii), the index [G : hw,Gxi] is 3, 50 and 10 respectively, so we cannot apply Lemma 5.10. Instead, we checked these as follows, using the SageMath computer algebra system [?]. First, we note that in each of these cases, there is only one conjugacy class of parabolics isomorphic to Gx (see the tables in [37]). Therefore, we can, without loss of generality, fix a vector x which is stabilised by such a parabolic. We then ran an exhaustive search for elements w such that x is an eigenvector of w with eigenvalue a primitive dth root of unity3. The Sage code can be found in Appendix A. No counterexamples were found, thus proving (4.4.1) in these cases.

5.6 The case d = h

From now on, suppose that G is well-generated. We have already seen, case-by-case, that if d < h, then (4.4.1) is a strict inequality. Below, we prove that if d = h then (4.4.1) is an equality. This has been proven by Chapuy and Stump [11, Prop. 2.1] using a counting argument; however, we give a simpler proof, along the same lines as the proof in the case of real reflection groups in [24, §2.3].

Lemma 5.11. Let G be an irreducible complex reflection group.

(i). The codegree 0 occurs only once.

(ii). If G is in addition well-generated, then dn occurs only once as a degree.

Proof. Because the action of G on V has no fixed points, the contragredient action on V ∗ also has ∗ ∼ ∗ ∗ no fixed points, so m ⊂ Sym(V )≥2. Therefore, (SG)1 = V , and hence [(SG)1 : V ] = 1. Thus, there is a unique coexponent equal to 1 and hence a unique codegree equal to 0, proving (i). Part (ii)

3 We also recorded the values of w ∈ G and ζ ∈ µd primitive for which kwx − ζxk is minimised, however, in each case this gives w = 1. In each case, the second-nearest miss was a reflection 58 CHAPTER 5. PROOF OF THE MAIN THEOREM

then follows from part (i) and the duality between the degrees and codegrees of a well-generated reflection group.

Proposition 5.12. Let (G,V) be an irreducible well-generated complex reflection group with n degrees (di)i=1, and x an eigenvector of w ∈ G with eigenvalue a primitive dnth root of unity. Then x is regular.

Proof. Let ζ be the eigenvalue of w on x. By Theorem 4.8, dn is a regular number. Thus, there is an element c ∈ V and a regular eigenvector y ∈ V of c with eigenvalue a primitive dnth root of unity. By replacing c with a power of c if necessary, we can assume that the eigenvalue is ζ. But by Theorem 4.3 (i), for any element g ∈ G, the eigenspace V(g,ζ) has dimension at most a(dn), and Lemma 5.11 (ii) implies that a(dn) = 1. Since we have two eigenspaces V(w,ζ) = Cx and V(c,ζ) = Cy of maximal dimension, Theorem 4.3 (iii), implies that there is some g ∈ G such that g(Cx) = Cy. Therefore, x is regular, since elements of G send reflecting hyperplanes to reflecting hyperplanes. Bibliography

[1] D. Bessis. Zariski theorems and diagrams for braid groups. Invent. math., 145:487–507, 2001.

[2] D. Bessis. The dual braid monoid. Ann. Sci. Ecole´ Norm. Sup. (4), 36(5):647–683, 2003.

[3] D. Bessis. Finite complex reflection arrangements are K(π,1). Ann. of Math. (2), 181:809–904, 2015.

[4] J. Birman. Braids, links and mapping class groups, volume 82. Princeton Univ. Press, 1974.

[5] N. Bourbaki. Groupes et Algebres` de Lie, Chapitres 4 a` 6. Springer-Verlag, 1981.

[6] E. Brieskorn. Die Fundamentalgruppe des Raumes der regularen¨ Orbits einer endlichen komplexen Spiegenlungsgruppe. Invent. math., 12:57–61, 1971.

[7] E. Brieskorn and K. Saito. Artin-Gruppen und Coxeter-Gruppen. Invent. Math., 17:245–271, 1972.

[8] M. Broue.´ Introduction to Complex Reflection Groups and Their Braid Groups, volume 1988 of Lecture Notes in Math. Springer-Verlag, 2010.

[9] M. Broue,´ G. Malle, and R. Rouquier. Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math., 500:127–190, 1998.

[10] R.W. Carter. Conjugacy classes in the Weyl group. Compositio Math., 25(1):1–59, 1971.

[11] G. Chapuy and C. Stump. Counting factorizations of Coxeter elements into products of reflections. J. London Math. Soc., 90(3):919–939, 2014.

[12] C. Chevalley. The Betti numbers of the exceptional simple Lie groups, 1950. International Congress of Mathematicians talk.

59 60 BIBLIOGRAPHY

[13] C. Chevalley. Invariants of finite groups generated by reflections. Amer. J. Math., 77(4):778– 782, 1955.

[14] A.J. Coleman. The Betti numbers of the simple Lie groups. Canad. J. Math., 10:349–356, 1958.

[15] H.S.M. Coxeter. The product of the generators of a finite group generated by reflections. Duke Math. J., 18:765–782, 1951.

[16] P. Deligne. Les immeubles des groupes des tresses gen´ eralis´ es.´ Invent. Math., 17:273–302, 1972.

[17] F. Digne and J. Michel. Parabolic Deligne-Lusztig varieties. Adv. Math., 257:136–218, 2014.

[18] I. Digne, F. Marin and J. Michel. The center of pure complex braid groups. J. Algebra, 347(1):206–213, 2011.

[19] D. Goldsmith. Homotopy of braids: in answer to a question of Artin. In Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973), volume 375 of Lecture Notes in Mathematics, pages 91–96. Springer-Verlag, 1974.

[20] I. Gordon and S. Griffeth. Catalan numbers for complex reflection groups. Amer. J. Math., 134(6):1491–1502, 2012.

[21] A. Hatcher. Algebraic Topology. Cambridge Univ. Press, New York, 2001.

[22] J.E. Humphreys. Reflection groups and Coxeter groups. Number 29 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, 1990.

[23] V.G. Kac and D.H. Peterson. Generalized invariants of groups generated by reflections. In Geometry Today (Rome, 1984), volume 60 of Progress in Mathematics, pages 231–250. Birkhauser,¨ Boston, 1985.

[24] M. Kamgarpour. Stabilisers of eigenvectors of finite reflection groups. 2016. arXiv:1512.01591.

[25] R. Kane. Reflection Groups and Invariant Theory. Number 5 in CMS Books in Mathematics. Canadian Math. Soc., 2001.

[26] B. Kostant. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81(4):973–1032, 1959. BIBLIOGRAPHY 61

[27] G.I. Lehrer and D.E. Taylor. Unitary Reflection Groups, volume 20 of Australian Mathematical Society Lecture Series. Cambridge University Press, 2009.

[28] H. Matsumoto. Gen´ erateurs´ et relations des groupes de Weyl gen´ eralis´ es.´ C.R. Acad. Sci. Paris, 258:3419–3422, 1964.

[29] J. Michel. Personal communication.

[30] P. Orlik and L. Solomon. Unitary reflection groups and cohomology. Invent. Math., 59:77–94, 1980.

[31] P. Orlik and H. Terao. Arrangements of Hyperplanes, volume 300 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Heidelberg, 1992.

[32] J.-P. Serre. Linear representations of finite groups, volume 42 of Graduate Texts in Mathemat- ics. Springer-Verlag, New York, 1977.

[33] G.C. Shephard and J.A. Todd. Finite unitary reflection groups. Canad. J. Math., 6:274–304, 1954.

[34] T. A. Springer. Regular elements of finite reflection groups. Invent. Math., 25:159–198, 1974.

[35] R. Steinberg. Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc., 112(3):392–400, 1964.

[36] R. Steinberg. Lectures on Chevalley groups, 1967. Notes prepared by J. Faulkner and R. Wilson.

[37] D. E. Taylor. Reflection subgroups of finite complex reflection groups. J. Algebra, 366:218– 234, 2012.

[38] The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.1), 2017. http://www.sagemath.org.

Appendix A

Sage code

Here we give the Sage code that we used to check the main theorem in the cases G25, G31 and G32, by exhaustively determining that the triples (G,Gx,d) listed in Subsection 5.5.2 cannot occur for any values of w and x. def FixH(G, L): # vector stabilised by the standard parabolic ,→ given by list L of indices corresponding to simple ,→ reflections idty = G[0] X = idty.fix_space() # whole vector space V for i in L: r = G.simple_reflection(i) U = (r.to_matrix()-1).kernel() X = X.intersection(U) B = X.basis() assert (len(B) == 1) # if the parabolic has rank less than n ,→ -1, the search will not be exhaustive. return B[0] def RootsOfUnityList(d): # list of primitive dth roots of unity out = list() for k in range(1,d+1): # need k=d in case d=1 if gcd(k, d) == 1: out.append((2*pi*I*k/d).exp())

63 64 APPENDIX A. SAGE CODE

return out def CounterexamplesList(G, L, d, tol=0.01): # G is the group, L ,→ is the list of generators of the parabolic, d is the ,→ desired order of an eigenvalue. n = G.rank() # rank of G. out = list() # initialise list of counterexamples count = 0 # initialise number of counterexamples found x = FixH(G, L) # the vector x vector_space = VectorSpace(CC,n) # The vector space V xx = vector_space(x) # coerce x into V dth_roots_of_unity = RootsOfUnityList(d) zeta = dth_roots_of_unity[0] # wlog eigenvalue is exp(2pi i/d) min_w = G[0] # initialise nearest-miss to the identity min_diff = 100 # initialise size of nearest-miss to be larger ,→ than possible for w in G: # Test whether x is an eigenvector of w with ,→ eigenvalue of order d. wx = vector_space(w.action(x)) diff = vector_space(wx-zeta*xx) diff_norm = RR(diff.norm()).abs() # the norm of diff if diff_norm < tol: # test whether (zeta, x) is an ,→ eigenpair for w count = count+1 if count == 1: print "Counterexample found:", w min_w = w min_diff = diff_norm if count <= 100: out.append(w) elif count == 0 and diff_norm < min_diff and w != G[0]: # w ,→ is the new nearest miss min_w = w min_diff = diff_norm 65

return (G, count, [w.reduced_word() for w in out], min_w. ,→ reduced_word(), RR(min_diff))

# search for counterexamples in G25: P=CounterexamplesList(ReflectionGroup(25), [1,2], 9) print(P)

# search for counterexamples in G31: P=CounterexamplesList(ReflectionGroup(31), [1,2,3,4], 24) print(P)

# search for counterexamples in G32: P=CounterexamplesList(ReflectionGroup(32), [1,2,3], 24) print(P)

Appendix B

Tables

The following two tables are used in Section 5.5 to check the main theorem for the exceptional groups. Table B.1 deals with the exceptional groups of rank two, while Table B.2 deals with the exceptional groups of higher rank. See Subsections 5.5.1 and 5.5.2 respectively for an explanation of what the columns mean.

Table B.1: Groups of rank 2

Group N N∗ Degrees maxd h 2(h − maxd) Orders of the reflections

G4 8 4 4, 6 4 6 4 3 G5 16 8 6, 12 6 12 12 3 G6 14 10 4, 12 6 12 12 2, 3 G7 22 14 12, 12 12 18 12 2, 3 G8 18 6 8, 12 8 12 8 2, 4 G9 30 18 8, 24 12 24 24 2, 4 G10 34 14 12, 24 12 24 24 2, 3, 4 G11 46 30 24, 24 24 38 28 2, 3, 4 G12 12 12 6, 8 8 12 8 2 G13 18 18 8, 12 12 18 12 2 G14 28 20 6, 24 12 24 24 2, 3 G15 34 26 12, 24 24 30 12 2, 3 G16 48 12 20, 30 20 30 20 5 G17 78 42 20, 60 30 60 60 2, 5 G18 88 32 30, 60 30 60 60 3, 5 G19 118 62 60, 60 60 90 60 2, 3, 5 G20 40 20 12, 30 15 30 30 3 G21 70 50 12, 60 30 60 60 2, 3 G22 30 30 12, 20 20 30 20 2

67 68 APPENDIX B. TABLES

Table B.2: Groups of higher rank

Maximal parabolics G x |G | |G| N + N∗ N + N∗ n Degrees d h Group (up to isomorphism) x x x max

A2, 2 · 3, 4 6, G24 3 2 · 3 · 7 42 3 4, 6, 14 7 14 B2 2 8 2 2 (Z/3Z) , 3 , 3 4 6, G25 3 2 · 3 36 3 6, 9, 12 9 12 G4 2 · 3 12 Z/2Z × Z/3Z, 2 · 3, 5, 3 4 4 G26 G4, 2 · 3, 2 · 3 12, 54 3 6, 12, 18 12 18 G(3,1,2) 2 · 32 8 A2, 2 · 3, 6, 3 4 3 G27 B2, 2 , 2 · 3 · 5 8, 90 3 6, 12, 30 15 30 I2(5) 2 · 5 10 3 A3, 2 · 3, 12, 2 A1 × A2, 2 · 3, 9 4, 4, 8, 12, G29 4 2 · 3 · 5 80 4 12 20 B3, 2 · 3, 15, 20 G(4,4,3) 25 · 3 24 3 A3, 2 · 3, 12, 8, 12, 20, G A × A , 22 · 3, 210 · 32 · 5 8, 120 4 24 30 31 1 2 24 G(4,2,3) 26 · 3 30 3 2 Z/3Z × G4, 2 · 3 , 7 5 15, 12, 18, 24, G32 3 4 2 · 3 · 5 120 4 24 30 G25 2 · 3 36 30 3 A4, 2 · 3 · 5, 20, 4 A1 × A3, 2 · 3, 7 4 14, 4, 6, 10, G33 3 2 · 3 · 5 90 5 12 18 D4, 2 · 3, 24, 12, 18 G(3,3,4) 23 · 34 36 4 2 A5, 2 · 3 · 5, 30, 4 A1 × A4, 2 · 3 · 5, 22, 4 2 A2 × A3, 2 · 3 , 18, 6, 12, 18, G A × G(3,3,4), 24 · 34, 29 · 37 · 5 · 7 38, 252 6 30 42 34 1 24, 30, 42 G(3,3,5), 23 · 35 · 5, 60, 7 D5, 2 · 3 · 5, 40, 7 4 G33 2 · 3 · 5 90