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 F and Fx denote the root systems of G and StabG(x) respectively, then jFj − jFxj ≥ 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 `(p) − `(p x) ≥ dn; with equality if and only if G is well-generated and d = h. Here, p is the generator of the centre of the pure braid group of G, p 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 `(p) − `(p 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. 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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(mm), as a path in V (left), and as a loop in V reg=G (right). 25 3.3 The generator p = b 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 2 G any element, and x 2 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.