UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 8680: Topics in Combinatorics
Combinatorics of reflection groups
and invariant theory

Fall 2022

Location, time: Mon-Wed-Fri, 11:15am - 12:05pm in VinH 364
Instructor: Victor Reiner (he/him/his; you can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): (612) 625-6682
E-mail is better: reiner@math.umn.edu
Office hours: Wed 5:00-6:00pm in-person at Vinh 256,
Tues 10:10-11:00am via Zoom at this Zoom link
Discord server: Here is the link to our class Discord server.
Study group? Organizing some sort of study group would be wonderful, too!
Prerequisites: We will assume knowledge of basic abstract algebra (groups, rings, fields, modules), such as in Math 5285/86 or Math 8201-02,
and representation theory of finite groups over the complex numbers, e.g., as in
  • Chapters 1 and 2 of Sagan's book
  • Chapters 1-4 and 10 of B. Steinberg's book
Familiarity with simplicial homology would help, but is less important.
Grading: Students registered for the class wanting to get an A should attend regularly and hand in by December 1, 2022 five homework problems from the list of homework sources below.
Course content: Reflection groups have long played an important role in Lie and representation theory, and later became pervasive in modern combinatorics. Much amazing combinatorial numerology stems from the invariant theory of reflection groups, which studies how these groups act on polynomial rings and exterior algebras. We hope to explain this, and discuss some of the topics below.
  • Coxeter groups
  • Classifications (irreducible complex, real, crystallographic, A-D-E)
  • Weak and strong Bruhat orders
  • Invariant theory (theorems of Shephard-Todd/Chevalley, Solomon; never got to Springer)
  • Coxeter elements (never got there)
  • Coxeter-Catalan combinatorics (never got there)
  • Analogies with GLn(Fq) (never got there)
Notes:
Topic/notes Dates
Overview Wed Sept 7, Fri Sept 9, Mon. Sept 12
Roots for reflection groups Wed Sept 14, Fri Sept 16, Mon Sept 18
Geometric representation for Coxeter groups Mon Sept 19, Wed Sept 21, Fri Sept 23, Mon Sept 26
Consequences: roots, length, exchange and deletion conditions Mon Sept 26, Wed Sept 28, Fri Sept 30
More Consequences: parabolic factorization and longest element Mon Oct 3, Wed Oct 5 Fri Oct 7, Mon Oct 10
Length generating function recurrence,
and some extra non-lecture notes		 Wed Oct 12, Fri Oct 14
Irreducibility, nondegeneracy, chamber geometry and finiteness Mon Oct 17, Wed Oct 19, Fri Oct 21
Classifications: finite, affine Mon Oct 24, Wed Oct 26, Fri Oct 28, Mon. Oct 31
Strong Bruhat order Wed Nov. 2, Fri Nov. 4, Mon Nov 7, Wed Nov. 9
Bruhat order on quotients and tableau criterion Fri Nov. 11, Mon Nov. 14
Topology of Bruhat intervals Wed Nov. 16, Fri Nov. 18, Mon. Nov. 21
Weak order Wed Nov. 23
Invariant Theory: Shephard-Todd, Chevalley, Molien
(Tonny Springer notes)
 Mon Nov. 28, Wed Nov. 30, Fri Dec. 2, Mon. Dec. 5
Solomon's Theorem and consequences Mon Dec. 5, Wed Dec. 7, Fri Dec. 9
Factoring the length generating function Mon Dec. 12, Wed Dec. 14
Sources:
Homework sources:
  • Bjorner-Brenti
    • Chapter 1 #1,2,3,5,6,10,12,13,15,16,17
    • Chapter 2 #1,2,3,4,5,10,11,14,15,20
    • Chapter 3 #3,5,10
  • Humphreys
    • Chapter 1: 1.10 #3, 1.11 #1,2, 1.12 #3, 1.13 #1
    • Chapter 3: 3.10 #1, 3.19 #1,2, 3.29 #3
    • Chapter 5: 5.1 #2, 5.8 #2,4, 5.12 #1
  • Exercises from a summer school in 2012 in Portugal
  • Exercises from a series of two lectures, 1, 2 in a 2017 CRM-LaCIM-UQAM spring school
Here are some selected solutions to a few exercies that had some subtleties.
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