Spring 2013

Reflection Groups

**Instructor:** Brendon Rhoades

**Email:** bprhoades (at) math.ucsd.edu

**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.)

**Lectures:**

**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!