Total positivity on the Grassmannian
Alexander Postnikov

Algebraic Geometry Seminar
Brown University

November 22, 2004


     We investigate the totally nonnegative part of the Grassmannian and
its intimate relationship  with the inverse boundary  problem for planar
networks.   We  study  the  decomposition  of  the  totally  nonnegative
Grassmannian into cells, which is  a finer subdivision than the standard
Schubert decomposition.   The cells  generalize the double  Bruhat cells
of  Fomin-Zelevinsky (for  type  A).  We  present several  combinatorial
constructions for the cells and give their explicit parametrization.  We
introduce certain planar  3-valent graphs (or webs) that  can be thought
of as generalized reduced decompositions  in the Weyl group.  These webs
and their mutations play a crucial role in the construction.