Grassmann Cells and Planar Networks

Lusztig defined {\it totally positive parts} in partial flag manifolds. The aim of this talk is to discuss total positivity in Grassmann manifolds and its links with the {\it 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 {\it 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 {\it double Bruhat cells} (for type A) that were recently studied by Fomin and Zelevinsky.

We present several combinatorial constructions for the totally positive Grassmann cells in terms of web diagrams, tableaux, chord diagrams, and matroids. The totally positive Grassmann cells can also be thought of as all possible degenerations of {\it cyclic polytopes}. We describe the partial order on the cells by containment of closures, which is a certain extension of the Bruhat order on the symmetric group.

This theory seems to have deep and intriguing relations with {\it honeycombs} of Knutson and Tao and with {\it cluster algebras} of Fomin and Zelevinsky.

This talk will be joint with the Lie Groups Seminar.