The University of Michigan Combinatorics Seminar
Fall 2001
September 21, 4:10-5:00, 3866 East Hall

Quantum cohomology of partial flag varieties and totally positive Toeplitz matrices

Konstanze Rietsch

Cambridge University


A matrix is called totally positive if all of its minors are positive. We show that the set of totally positive unipotent upper-triangular Toeplitz matrices in GL(n) form a real semi-algebraic cell of dimension n-1. Furthermore we prove a natural cell decomposition for its closure. The proof makes use of the beautiful connection between Toeplitz matrices and quantum cohomology rings of flag varieties due to Dale Peterson. And properties of these quantum cohomology rings, their Schubert bases and especially the positivity of the structure constants are applied. We also give a characterization of total positivity for Toeplitz matrices in terms of (quantum) Schubert bases.