The full Kostant-Toda hierarchy on the positive flag variety
Some geometric and combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy are studied, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of \(\text{SL}_n(\mathbb{R})\). The f-KT flows on the tnn flag variety are complete, and it is shown that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch. These results represent the first results on the asymptotics of the f-KT hierarchy (and even the f-KT lattice); moreover, these results are not confined to the generic flow, but cover non-generic flows as well. The f-KT flow on the weight space via the moment map is defined, and it is shown that the closure of each f-KT flow forms an interesting convex polytope which is called a Bruhat interval polytope. In particular, the Bruhat interval polytope for the generic flow is the permutohedron of the symmetric group \(\mathfrak{G}_n\). An analogous results for the full symmetric Toda hierarchy, by mapping f-KT solutions to those of the full symmetric Toda hierarchy is also proved. In the appendix, it is shown that Bruhat interval polytopes are generalized permutohedra.
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