Tuesday and Thursday, 1-2:30
2-147
Overview: Root systems, Weyl groups, and more generally Coxeter groups are fundamental combinatorial objects which govern much of the geometry and representation theory of Lie groups. Our mission is to study the wealth of information one can obtain from these objects. The basic tools to be discussed included signed permutations groups, Bruhat orders, Kazhdan-Lusztig polynomials, Hecke rings, nil-Hecke rings, invariants, Poincare polynomials. The geometric thread that runs throughout this subject is Schubert varieties. Schubert varieties are the primary examples of non-trivial projective varieties. Therefore, we will examine these varieties carefully with respect to each new topic.
Requirements: The requirements for the course will be biweekly homework plus one lecture on a research paper related to Coxeter groups and Lie Theory (to be chosen jointly by the student and professor.) Some background in Algebra is necessary and familiarity with combinatorics and Lie theory would be beneficial. The lecture should be about 45 minutes. More time can be scheduled if needed.
Text books: The main text will be ``Reflection Groups and Coxeter Groups'' by Humphreys. The text has been ordered through Quantum Books. I will also be handing out chapters from ``Singular Loci of Schubert Varieties'' by Billey and Lakshmibai as they become available. A good source of research related papers in this area can be found in ``Algebraic Groups and their Generalizations'' edited by Haboush and Parshall.
Thursday February 4, 1999: Weyl groups vs. Coxeter groups or finite reflection groups.
Tuesday February 9, 1999: More on Weyl groups
Thursday February 11, 1999: Classification reduced irreducible root systems. Coxeter diagrams. Dynkin diagrams.
Tuesday February 16, 1999: \\ ** No Lecture: Monday schedule of classes
Thursday February 18, 1999: Schubert Varieties and connections to Lie theory.
Tuesday February 23, 1999: Invariant Theory: Chevalley's theorem
Thursday February 25, 1999: Invariant Theory cont: degrees and expontents
Tuesday March 2, 1999: Bruhat order. Mobius function. Computational tests. Subexpressions. Intervals. Poincare series. Weak order.
Thursday March 4, 1999: Chevalley's proof that Bruhat order is equivalent to subexpressions.
Tuesday March 9, 1999: Richard Stanley guest lecturing on unimodality, symmetry and Sperner properties of Bruhat orders.
Thursday March 11, 1999: Bruhat order continued.
Tuesday March 16, 1999: Hecke algebras, R-polynomials and Kazhdan-Lusztig polynomials.
Thursday March 18, 1999: Kazhdan-Lusztig polynomials continued.
Tuesday March 23, 1999: \\ ** No Lecture: Spring Break
Thursday March 25, 1999: \\ ** No Lecture: Spring Break
Tuesday March 30, 1999: Combinatorial methods for determining rational smoothness of a Schubert variety. Carrell-Peterson methods.
Thursday April 1, 1999: Rational smoothness cont. Billey's methods. Factoring Poincare polynomials.
Tuesday April 6, 1999: Nil Hecke ring. Kumar's tests for smoothness and rational smoothness of Schubert Varieties. Computing the singular locus.
Thursday April 8, 1999: Nil Hecke ring continued.
Tuesday April 13, 1999: Weyl Modules and Demazure Modules. Verma modules. Kostant partition formula.
Thursday April 15, 1999: Prof. Rota will give guest lecture.
Tuesday April 20, 1999: \\ ** No Lecture: Patriot's Day
Thursday April 22, 1999: Bases of Tangent spaces for Schubert varieties.
Tuesday April 27, 1999: To Be Announced.
Thursday April 29, 1999: Student presentations: Greg Warrington and Jennifer Galovich
Tuesday May 4, 1999: Student presentations: Pramod Achar and Kevin McGerty
Thursday May 6, 1999: Student presentations: Peter Clifford and Federico Ardila
Tuesday May 11, 1999: Student presentations: Tara Holm and Daniel Biss
Thursday May 13, 1999: (Last day of Class) Summary of open problems.