Math 262A
Spring 2013
Reflection Groups

Instructor: Brendon Rhoades
Email: bprhoades (at)
Instructor's Office: 7250 APM
Instructor Office Hours: By appointment
Location: APM 7421
Lecture Time: 2:00-2:50 pm MWF

Course Description: Let V be a Euclidean space. A *reflection group* W is a finite group generated by reflections in GL(V). Many interesting families of groups (for example symmetric groups, Weyl groups of Lie algebras, and symmetry groups of regular polytopes) arise as reflection groups. In this course we'll study reflection groups (and some generalizations thereof) from a combinatorial point of view. We'll start off by covering the basic aspects of the theory (Bruhat order, the Cartan-Killing classification of finite reflection groups, the invariant theory of reflection groups, and the theory of Coxeter group-style presentations) and will end the term talking about newer topics (absolute order, W-Catalan combinatorics). Many definitions and examples will be given throughout. I will also present a variety of open problems.

Prerequisites: Not all that much. A decent background in group theory and linear algebra will be necessary. No experience whatsoever with reflection groups will be assumed.

Textbook: Reflection Groups and Coxeter Groups by James E. Humphreys.
(A good first book to read about reflection groups. Concise and well written.)


4/1/2013: Introductions, three ways of viewing the symmetric group (combinatorial: one-line notation and wiring diagrams, algebraic: words and presentation, geometric: reflecting hyperplanes).
4/3/2013: Section 1.1. Hyperplanes, reflections, reflection groups. Examples: Z_2, dihedral groups, the symmetric group, the group of signed permutation matrices. Irreducible and essential reflection groups. The rank of a reflection group.
4/5/2013: Sections 1.1, 1.2. The Type D reflection group, wiring diagrams of types B and D. Root systems. The correspondence between root systems and reflection groups.
4/8/2013: Section 1.3. Orders on real vector spaces, positive systems, simple systems. Every simple system is contained in a unique positive system. Every positive system contains at most one simple system.
4/10/2013: Sections 1.3, 1.4. Every positive system contains a unique simple system (so simple systems exist!), positive and simple systems in types ABI, observation that the W-action sends simple (positive) systems to simple (positive) systems.
4/12/2013: Sections 1.4, 1.5, 1.6. Any simple reflection swaps the corresponding simple root with its negative, but permutes all other positive roots. Any two simple (positive) systems are W-conjugate. W is generated by simple reflections and has no smaller generating set consisting of reflections. Definition of length.
4/15/2013: Sections 1.6, 1.7. Definition of Inv(w) and inv(w) for w in W. Proof that inv(w) = l(w). The Deletion and Exchange Properties for words in S. Description of Inv(w) in terms of reduced words. Characterization of when a subset G of a positive system arises as Inv(w) for some w.
4/17/2013: Sections 1.8, 1.9. The action of W on simple systems (or on positive systems) is simply transitive. The long element (examples: ABC). Matsumoto's Theorem.
4/19/2013: Class to meet in Porter's Pub due to the Graduate Student Combinatorics Conference at UMinnesota.
4/22/2013: Section 1.10. Parabolic subgroups W_I, parabolic factorization. Example: The symmetric group.
4/24/2013: Sections 1.11, 1.12. Poincare polynomials and parabolic recursion. The Coxeter arrangement of W and a fundamental domain for the action of W on V. The structure of stabilizers W_X for subsets X of V: they're reflection groups, generated by reflections in W which stabilize X (and in fact parabolic subgroups!).
4/26/2013: Section 1.13+. Definition of posets and lattices. The lattice of standard parabolic subgroups is the Boolean algebera on S. The lattice of parabolic subgroups is the intersection lattice of the reflection arrangement. So, parabolic subgroups of W "are" flats in the reflection arrangement lattice "are" W-set partitions. Examples: Types ABC.
5/1/2013: Sections 1.14, 1.15, 2.1. The reflections in W are precisely the root reflections. The Coxeter Complex. Coxeter graphs. A reflection group is ``determined" by its Coxeter graph.
5/3/2013: Sections 2.2, 2.3, 2.4, 2.5. Disjoint unions of Coxeter graphs correspond to orthogonal direct products of Coxeter groups. Positive (semi)definite matrices and graphs. Examples of positive(semi)definite graphs.
5/6/2013: Sections 2.6, 2.7. Classification Day! Some Perron-Frobenius theory. Application: Every proper subgraph of a positive semidefinite Coxeter graph is positive definite. Application of application: the graphs presented last time form a full list of the positive semidefinite graphs (and so the positive definite graphs correspond precisely to real reflection groups). A little cheating: We didn't construct the "exceptional" groups for the graphs of type EFH.
5/7/2013: Bonus Lecture: The Shi arrangement and the Ish arrangement.
5/8/2013: Sections 2.8, 2.9+: Crystallographic groups and root systems. The positive root poset and nonnesting partitions attached to positive root posets (in *crystallographic* type!).
5/10/2013: More on nonnesting partitions. Nonnesting partitions inject into the intersection lattice. Nonnesting partitions label dominant regions of the Shi arrangement. The definition of the W-Catalan number Cat(W) in terms of degrees, Coxeter number.
5/13/2013: Section 3.1. Coordinate rings, finite group actions, invariant rings. The fraction field of the invariant ring is the invariant field of the fraction field, so k[V]^G isn't ``too small".
5/15/2013 Sections 3.2, 3.5. k[V]^G is a finitely generated k-algebra (and not "too big"). Chevalley's Theorem: If W acting on V is a reflection group, then k[V]^W is generated by an algebraically independent set of n = dim(V) homogeneous polynomials. Examples: Z_2, S_n.
5/17/2013: Sections 3.5, 3.6, 3.7, 3.8. More examples of invariant rings: types B_n/C_n, D_n, I_2(m). Statement of the W-module structure of the coinvariant algebra. Uniqueness of degrees. Molien's Theorem.
5/20/2013: Sections 3.8, 3.9, 3.10, 3.11. Molien's Theorem continued. Corollaries: The product of the degrees is |W|. The sum of the degrees is (number of reflections) + dim(V). Shepard-Todd's Theorem: If a finite subgroup G of GL(V) has polynomial invariant ring, then G is a reflection group.
5/22/2013: Chevalley-Shepard-Todd over C. Complex reflection groups and their classification. Well-generated complex reflection groups. Additional facts about degrees: -1 is in W iff all degrees are even. Degrees keep track of the grading of V inside the coinvariant module for W. For W crystallographic, degrees control the generating functions for length and height. Section 3.13: Factoring the invariant Jacobian.
5/24/2013: Sections 3.16, 3.18, 3.19. Coxeter elements and standard Coxeter elements. Any two Coxeter elements are W-conjugate. The Coxeter number. Exponents. The degrees and exponenets are related by d_i = m_i + 1 for irreducible and essential reflection groups.
5/29/2013: Sections 5.1, 5.2, 5.3. Coxeter systems and Coxeter groups. Examples. The sign homomorphism and the length function. Intro to the geometric representation.
5/31/2013: Sections 5.3, 5.4, 5.5. The geometric representation is faithful, st has order m(s,t) in W for s, t in S. Root systems, positive roots, negative roots. Parabolic subgroups.
6/3/2013: Sections 5.5, 5.12, 5.6, 5.7, 5.8. Parabolic length and the lattice of standard parabolic subgroups. Poincare polynomials, parabolic factorization, and recursion. Inversion sets and l(w) = #Inv(w). Roots and reflections. The Strong Exchange Property.
6/5/2013: Sections 5.8, 5.9, 5.10, 5.11. Characterization of Coxeter groups in terms of deletion and exchange axioms, Bruhat order, Bruhat order on S_n via rank matrices, the subword characterization of Bruhat order.
6/7/2013: Noncrossing partitions in type A. Reflection length and absolute order. Noncrossing partitions in type W. W-noncrossing parking functions.

Other Resources: Combinatorics of Coxeter Groups by Anders Bjorner and Francesco Brenti.
(A great resource for the combinatorial aspects of Coxeter systems. Longer than Humphreys; a good `second book' to read in my opinion.)

My coauthor Drew Armstrong taught/is teaching a year-long course in reflection groups at the University of Miami. His course notes are available here and here. Drew does a really good job motivating why reflection groups are interesting objects of study. We won't have time to cover all of this in a quarter, but his notes are a great resource.

Federico Ardila taught a course at SFSU on Coxeter groups. His lectures are on youtube. Federico uses Coxeter systems as a starting point; we will use reflection groups. Check his lectures out!