The University of Michigan Combinatorics Seminar
The aim of this talk is to discuss total positivity in Grassmann manifolds and
its links with the inverse boundary problem for planar oriented networks. This
problem emerged in an attempt to generalize and simplify several recent
combinatorial and algebraic constructions related to representation theory of
GL(n) and canonical bases. The combinatorial classes of networks
correspond to certain totally positive Grassmann cells.
The collection of these cells forms a (conjecturally regular) CW-complex.
They give a finer subdivision of the
Grassmannian than the decomposition into the Schubert cells. The totally
positive Grassmann cells extend the notion of double Bruhat cells (for type A)
that were recently studied by Fomin and Zelevinsky.