Bruhat Order in the Full Symmetric $\mathfrak{sl}_n$ Toda Lattice on Partial Flag Space

TL;DR

This paper extends the understanding of the asymptotic behavior of the full symmetric $rak{sl}_n$ Toda lattice to cases with coinciding eigenvalues, linking trajectory limits to Bruhat order on partial flag spaces.

Contribution

It generalizes previous results by incorporating eigenvalue degeneracies and relates the asymptotic trajectories to Bruhat order on multiset permutations.

Findings

Trajectories converge to points indexed by Schubert cells in partial flag space.

Asymptotic behavior is governed by Bruhat order on multiset permutations.

Results connect dynamical systems with algebraic combinatorics in flag varieties.

Abstract

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations.