Summer School
on Algebraic and Enumerative Combinatorics 2012
S. Miguel de Seide, Portugal
Lectures by Vic Reiner
Topic: Reflection group counting and q-counting
Certain families of numbers,
such as in G.-C. Rota's "Twelve-fold way",
appear repeatedly as solution to enumeration problems,
and in other locations, e.g.
- 2n, triangular numbers, binomial coefficients, multinomial coefficients
- the partition number p(n)
- Stirling numbers of the 1st and 2nd kind
- (n+1)n-1, Catalan, Narayana, and Kirkman numbers
- tableaux numbers
Much effort has gone into understanding how to
view these numbers as coming from the
symmetric groups, or the finite reflection/Weyl
groups/Weyl group of type A, and
generalizing them to all finite reflection groups.
This viewpoint not only illuminates connections between them,
and other areas of mathematics, but also on how to define
useful q-analogues of these numbers. We hope to
illustrate this here.
- Things we count
- What is a finite reflection group?
- Taxonomy of reflection groups
- Back to the Twelvefold Way
- Transitive actions and CSPs
- Multinomials, flags, and parabolic subgroups
- Fake degrees
- The Catalan and parking function family
- Bibliography
- Exercises
Suggested reading
- S. Fomin and N. Reading,
"Root Systems and
Generalized Associahedra",
from IAS/Park City Summer Math Institute on Geometric Combinatorics, 2004.
- D. Armstrong,
"Generalized noncrossing partitions and combinatorics of Coxeter groups",
Mem. Amer. Math. Soc. 202 (2009), no. 949.
- A. Björner and F. Brenti, "The combinatorics of Coxeter groups",
Springer-Verlag, Graduate Texts in Mathematics 227.
- J.E. Humphreys "Reflection groups and Coxeter groups",
Cambridge Studies in Advanced Mathematics 29.
- R. Kane, "Reflection groups and invariant theory",
CMS Books in Math. 5, Springer-Verlag.
- Chapter 1 of
R.P. Stanley, "Enumerative Combinatorics, Vol. 1" (Chapter 1),
Cambridge Studies in Advanced Mathematics 29.