Lecture notes.
Problem difficulty: [2+] - normal, [3-] - harder, [3] - hard, [*] - open.
hw1 (due 9/22) hw2 (due 10/25) hw3 (due 11/19)
The course will concentrate on algebraic aspects of total positivity and related areas. Specific topics to be covered:
- Lindstrom lemma and relation to planar networks;
- positivity properties of immanants;
- Edrei-Thoma theorem and its relation to character theory of infinite symmetric group;
- Berenstein-Fomin-Zelevinsky theory of double Bruhat cells and Chamber Ansatz;
- Postnikov theory of plabic graphs and total positivity in Grassmanians;
- a gentle introduction to cluster algebras;
- total positivity in loop groups and beyond.
The grading will be based on problem sets and possibly a final project. The only prerequisite is good understanding of basic graduate algebra, although some knowledge of representation theory will be helpful.
There is no textbook. Each of the following references covers some part of the material.
Total positivity: tests and parametrizations (S. Fomin, A. Zelevinsky) - a good introductory reading.
Totally positive matrices (A. Pinkus).
Asymptotic representation theory of the symmetric group and its applications in analysis (S. Kerov).
Parametrizations of canonical bases and totally positive matrices (A. Berenstein, S. Fomin, A. Zelevinsky).
Total positivity, Grassmannians, and networks (A. Postnikov).
Total positivity in loop groups I: whirls and curls (Th. Lam, P. Pylyavskyy).