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COXETER GROUPS Class Notes from Math 665, Taught at the University COXETER GROUPS Class notes from Math 665, taught at the University of Michigan, Fall 2017. Class taught by David E Speyer; class notes edited by David and written by the students: Elizabeth Collins-Wildman, Deshin Finlay, Haoyang Guo, Alana Huszar, Gracie Ingermanson, Zhi Jiang and Robert Walker. Comments and corrections are welcome! 1 Contents Introduction to the finite Coxeter groups September 6 { Introduction4 September 8 { Arrangements of hyperplanes5 September 11 { Classification of positive definite Cartan matrices6 Geometry of general Coxeter groups September 13 { Non-orthogonal reflections 10 September 15 { Roots, hyperplanes, length, reduced words 11 September 18 { Rank two computations 13 September 20 { A key lemma 15 September 25 { Consequences of the Key Lemma 17 September 27 { Adjacency of chambers, the set T , assorted homework topics 19 September 29 { Inversions and length 21 October 2 { David pays his debts 23 Special cases October 4 - Crystallographic groups 25 October 6 { The root poset and the highest root 28 October 9 - Affine groups 32 October 11 { Affine groups II 34 October 13 { Affine Groups III 35 October 18 { Hyperbolic groups 38 October 20 { Assorted topics about inversions and parabolic subgroups 42 Lie groups October 23 { Examples of Lie Groups 44 October 25 { Lie algebra, exponential, and Lie bracket 47 October 29 - Co-roots in Lie groups 50 Invariant Theory Begins October 30 { Examples and notation for invariants 51 November 1 { Commutative algebra background 53 November 3 { Commutative algebra of rings of invariants 54 Invariant Theory of Laurent Polynomial Rings November 6 { Invariants of Coxeter actions on Laurent polynomial rings 55 November 8 { Anti-symmetric functions 58 November 10 { Extensions and Variants of the Weyl Denominator Formula 60 Polynomiality of Rings of Invariants November 13 { Regular Rings 63 November 15 { Rings of Coxeter invariants are polynomial, first proof 64 November 17 { Molien's formula and consequences 66 November 20 { The ring of invariant differentials 67 November 22 { The formula of Shephard and Todd 69 Divided Difference Operators November 27 { Divided Difference Operators 70 November 29 { R is free over S, part I 72 December 1 { R is free over S, part II 74 December 4 { Rings of invariants are polynomial, second proof 75 Coxeter elements December 6 { Coxeter Elements 77 December 8 { The Coxeter Plane 80 December 11 { Coxeter eigenvalues and invariant theory 82 COXETER GROUPS 3 September 6 { Introduction. This course will study the combinatorics and geometry of Coxeter groups. We are going to spend the first few days discussing the finite reflection groups. We are doing this for three reasons: (1) The classification of finite reflection groups is the same as that of finite Coxeter groups, and the structure of finite reflection groups motivates the definition of a Coxeter group. (2) One of my favorite things in math is when someone describes a natural sort of object to study and turns out to be able to give a complete classification. Finite reflection groups is one of those success stories. (3) This will allow us to preview many of the most important examples of Coxeter groups. I'll also take the opportunity to tell you their standard names. Today we will, without proof, describe all the finite reflection groups. Let V be a vector space with a positive definite symmetric bilinear form · (dot product). Let α be a nonzero vector in V . The orthogonal reflection across α? is the linear map α · x x 7! x − 2 α: α · α This fixes the hyperplane α? and negates the normal line Rα. (The word \orthogonal" is there in anticipation of future weeks, when we'll have non-orthogonal reflections.) A orthogonal reflection group is a subgroup of GL(V ) generated by reflections. We will be classifying the finite orthogonal reflection groups. Let W1 ⊂ GL(V1) and W2 ⊂ GL(V2) be reflection groups. Then we can embed W1 × W2 w1 0 into GL(V1 ⊕ V2), sending (w1; w2) to 0 w2 . We call a reflection group irreducible if we can not write it as a product in this way. We will now begin describing all of the irreducible orthogonal reflection groups. The trivial group: Take V = R and W = f1g. The group of order 2: Take V = R and W = {±1g. This group can be called A1 or B1. The dihedral group: Let be a positive integer and consider the group of symmetries of regular m-gon in R2. This is generated by reflections across two hyperplanes with angle π=m between them. Calling these reflections σ and τ, we have σ2 = τ 2 = (στ)m = 1. This dihedral group has order 2m and can be called I2(m). We note a general rule of naming conventions { the subscript is always the dimension of V . n The symmetric group: Consider the group Sn acting on R by permutation matrices. ? The transposition (ij) is the reflection across (ei −ej) . This breaks down as R(1; 1;:::; 1)⊕ ? ? (1; 1;:::; 1) So the irreducible reflection group is Sn acting on (1; 1;:::; 1) . This is called An−1. n The group Sn n {±1g : Consider the subgroup of GLn(R) consisting of matrices which are like permutation matrices, but with a ±1 in the nonzero positions. This is a reflection ? ? ? group, generated by the reflections over (ei − ej) ,(ei + ej) and ei . We give example matrices for n = 3 below: 20 1 03 2 0 −1 03 2−1 0 03 41 0 05 4−1 0 05 4 0 1 05 0 0 1 0 0 1 0 0 1 ? ? ? reflection across (e1 − e2) (e1 + e2) e1 For reasons we will get to eventually, this is both called Bn and Cn. 4 COXETER GROUPS H1 H2 H3 β β ρ 2 1 D β3 Figure 1. Our running example of a hyperplane arrangement n−1 The group Snn{±1g : Consider the subgroup of the previous group where the product of the nonzero entries in the matrix is 1. This is an index two subgroup of Bn, generated by ? the reflections across (ei ± ej) . It is called Dn. ∼ ∼ ∼ ∼ ∼ Collisions of names: We have A1 = B1, D1 = f1g, A1 × D1 = I2(1), A1 × A1 = D2 = ∼ ∼ ∼ I2(2), A2 = I2(3), B2 = I2(4) and A3 = D3. (The last is not obvious.) Some authors will claim that some of these notations are not defined, but if you define them in the obvious ways, this is what you have. Also, I2(6) has another name, G2. We have now listed all but finitely many of the finite orthogonal reflection groups. The remaining cases are probably best understood after we start proving the classification, but we'll try to say something about them. Their names are E6, E7, E8, F4, H3 and H4. Sporadic regular solids: H3 is the symmetry group of the dodecahedron. H4 is the symmetry group of a regular 4-dimensional polytope called the 120-cell, which has 120 dodecahedral faces. 4 4 1 1 1 1 Symmetries of lattices F4 is the group of symmetries of the lattice Z [ Z + 2 ; 2 ; 2 ; 2 4 in R . E8 is the group of symmetries of the eight dimensional lattice n 8 8 1 1 1 X o (a1; a2; : : : ; a8) 2 Z [ Z + 2 ; 2 ;:::; 2 : ai 2 2Z : E7 is the subgroup of this stabilizing (1; 1; 1; 1; 1; 1; 1; 1); E6 is the subgroup of E7 stabilizing (0; 0; 0; 0; 0; 0; 1; 1). September 8 { Arrangements of hyperplanes. Let V be a finite dimensional R-vector ? space and let H1;H2; ··· ;HN be finitely many hyperplanes in V . Let Hi = βi for some _ vectors βi 2 V . The Hi divide V up into finitely many polyhedral cones. 0 Choose some ρ not in any Hi. Let D be the open polyhedral it lies in and D its closure. We'll choose our normal vectors βi such that hβi; ρi > 0. Observe: There can be no linear P relation between the βs of the form ciβi = 0 with ci ≥ 0 (except c1 = c2 = ··· = cN = 0) P since then cihβi; ρi = 0. P We will call βi simple if βi is not of the form cjβj with cj ≥ 0. So j6=i D = fσ : hβi; σi ≥ 0; 1 ≤ i ≤ ng = fσ : hβi; σi ≥ 0; βi simple g: In Figure1, β1 and β2 are simple and β3 is not. Now let Hi be the reflecting hyperplanes of some finite orthogonal reflection group. (Now 0 V has inner product) Let ρ, D ;D as before, βi as before. Call the βi \positive roots". Let α1; ··· ; αk be the simple roots. COXETER GROUPS 5 n Example. Consider An−1 (the symmetric group Sn) acting on R . The reflections are the transpositions (ij); the hyperplanes are xi = xj with normal vectors ei − ej. If we take ρ = (1; 2; ··· ; n), then the positive roots are ei − ej for (i > j). The simple roots are ei+1 − ei =: αi; for anything else, ej − ei = (ej − ej−1) + (ej−1 − ej−2) + ··· + (ei+1 − ei). So D = f(x1; x2; ··· ; xn): x1 ≤ x2 ≤ · · · ≤ xng. 1 What's the angle between α1 and α3? Since hα1; α3i = 0, it is 2 π. 2 2 p What's the angle between α1 and α2? We have hα1; α1i = (1) + (−1) = 2, so jα1j = 2 = jα j. α · α = (e − e ) · (e − e ) = −1, so cos θ = p−p1 = − 1 and θ = 2 π. 2 1 2 2 1 3 2 2 2 2 3 Lemma. In any finite reflection group with α1; ··· ; αk as before, the angle between αi and 1 ? αj is of the form π(1 − ), mij 2 f2; 3; 4; · · · g.
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