The Full Kostant–Toda Hierarchy on the Positive Flag Variety

We study some geometric and combinatorial aspects of the solution to the full Kostant–Toda (f-KT) hierarchy, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of S L n ( R ) . The f-KT flows on the tnn flag variety are complete, and we show that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch (Total positivity and real flag varieties. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, 1998 ). Our results represent the first results on the asymptotics of the f-KT hierarchy (and even the f-KT lattice); moreover, our results are not confined to the generic flow, but cover non-generic flows as well. We define the f-KT flow on the weight space via the moment map, and show that the closure of each f-KT flow forms an interesting convex polytope which we call a Bruhat interval polytope . In particular, the Bruhat interval polytope for the generic flow is the permutohedron of the symmetric group S n . We also prove analogous results for the full symmetric Toda hierarchy, by mapping our f-KT solutions to those of the full symmetric Toda hierarchy. In the appendix we show that Bruhat interval polytopes are generalized permutohedra , in the sense of Postnikov (Int. Math. Res. Not. IMRN (6):1026–1106, 2009 ).