Combinatorics Seminar: Gradient Flows on Totally Nonnegative Flag Varieties

Speaker: Steven Karp, University of Notre Dame

Abstract: One can view a flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a flag variety from an algebraic, geometric, and dynamical perspective. We connect the tridiagonal totally nonnegative part to the classical Toda lattice flow, and show that it is homeomorphic to a permutohedron. We also classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. An important role is played by a twist map which preserves the totally nonnegative part. This is joint work with Anthony M. Bloch.

Host:  Martha Precup