Syllabus 280: Coxeter groups and Schubert calculus
Winter 2009

Lectures: MWF 11:00am-11:50pm in MSB 2112
Instructor: Anne Schilling, MSB 3222, phone: 754-9371, anne@math.ucdavis.edu
Text: The course will not strictly follow a particular texts. Some useful references are:
  • A. Bjorner, F. Brenti, Combinatorics Of Coxeter Groups, Springer 2005
  • J. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29
  • L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMFAMS vol. 6, American Math. Soc.
  • I.G. Macdonald, Notes on Schubert polynomials, Publications du Laboratoire de Combinatoire et d' Informatique Mathematique, Volume 6
  • W. Fulton, Young Tableaux, London Mathematical Society, Student Texts 35
  • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press
We will also use recent papers which I will announce or hand out in class.
Grading: Every registered student is required to take notes for at least two classes and hand them in in latex-format. A pdf version of the notes will be posted on the class web-site. You are allowed to work on this in pairs. The notes should be handed in no later than 6 days after the lecture. Please use the following template.
Web: http://www.math.ucdavis.edu/~anne/WQ2009/280.html

Course description

This course will provide an introduction to the combinatorial aspects of Coxeter groups and Schubert calculus (background on the symmetric group and symmetric functions will be helpful, but not required). Topics include:

Lecture 1: Jan 05 Schubert polynomials written by Steven Pon and Alexander Waagen
Lecture 2: Jan 07 Coxeter systems and examples written by Joshua Clement and Brandon Crain
Lecture 3: Jan 09 Permutation representation written by Jeff Ferreira
Lecture 4: Jan 12 Strong exchange property written by Brandon Crain and Steven Pon
Lecture 5: Jan 14 Characterization theorem and example written by Mihaela Ifrim and Brandon Barrette
Lecture 6: Jan 16 Proof of Characterization theorem written by Brandon Barrette and Mihaela Ifrim
Lecture 7: Jan 21 Bruhat order written by Carlos Barrera-Rodriguez and Mohamed Omar
Lecture 8: Jan 23 Geometric interpretation and Subword Property written by Qiang Wang
Lecture 9: Jan 26 Lifting property and poset structure of finite Coxeter groups written by Tom Denton
Lecture 10: Jan 28 Weak order written by Steve Pon
Lecture 11: Jan 30 Parabolic subgroups written by Euna Chong
Lecture 12: Feb 02 Divided difference operators written by Qiang Wang
Lecture 13: Feb 04 Yang-Baxter equation written by Igor Rumanov
Lecture 14: Feb 06 Yang-Baxter equation and double Schubert polynomials written by Carlos Barrera-Rodriguez
Lecture 15: Feb 09 Symmetry and stability of double Schubert polynomials written by Mihaela Ifrim
Lecture 16: Feb 11 Combinatorial formula for Schubert polynomials and rc-graphs written by Joshua Clement
Lecture 17: Feb 13 Properties of rc-graphs written by Tom Denton
Lecture 18: Feb 18 Monk's rule written by Mohamed Omar
Lecture 19: Feb 20 Stanley symmetric functions and the affine symmetric group written by Jeff Ferreira
Lecture 20: Feb 23 The affine symmetric group written by Brandon Barrette
Lecture 21: Feb 25 (k+1)-cores and k-bounded partitions written by Euna Chong
Lecture 22: Feb 27 Weak k-tableaux written by Qiang Wang
Lecture 23: Mar 02 k-Schur functions (see papers by Lapointe and Morse A k-tableau characterization of k-Schur functions; Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions)
Lecture 24: Mar 04 Properties of k-Schur functions and Sage Tutorial/Examples (see Sage and Sage-Combinat)
Lecture 25: Mar 06 Affine Stanley symmetric functions written by Alexander Waagen and Igor Rumanov
Lecture 26: Mar 16 Sorting monoids on Coxeter groups (presented by Nicolas Thiery)