# The full Kostant-Toda lattice on the positive flag variety and a flag matroid polytope

### Yuji Kodama (The Ohio State University)

Friday 23rd May, 2014 15:00-16:00 Maths 326

#### Abstract

The full Kostant-Toda lattice hierarchy is given by the Lax equation
$\frac{\partial L}{\partial t_j}=[(L^j)_{\ge 0}, L],\qquad j=1,...,n-1$
where $L$ is an $n\times n$ lower Hessenberg matrix with $1$'s in the super-diagonal, and $(L)_{\ge0}$ is the upper triangular part of $L$.

We study combinatorial aspects of the solution to the hierarchy when the initial matrix $L(0)$ is given by an arbitrary point of the totally non-negative flag variety of $\text{SL}_n(\mathbb{R})$.
We define the full Kostant-Toda flows on the weight space through the moment map, and show that the closure of the flows forms a convex polytope inside the permutohedron of the symmetric group $S_n$.
This polytope is a flag matroid polytope and is uniquely determined by a pair of permutations $(v,w)$ which is used to parametrize the component of the Deodhar decomposition of the flag variety.
This is a joint work with Lauren Williams.