Bruhat Order in Full Symmetric Toda System

TL;DR

This paper explores the geometric and topological structure of the full symmetric Toda system, revealing its phase transition diagram aligns with the Bruhat order of symmetric groups, and establishing it as a Morse-Smale system.

Contribution

It demonstrates the correspondence between Toda flow singular points and Bruhat order elements, extending the analysis to general n and linking trajectories to Bruhat comparability.

Findings

Phase transition diagram matches Bruhat order for n=3,4.

Singular points correspond to permutations in S_n.

The system is a Morse-Smale system.

Abstract

In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions $n=3,4$ coincides with the Hasse diagram of the Bruhat order of symmetric groups $S_3$ and $S_4$. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general $n$. The resulting theorem identifies the set of singular points of $\mathrm{dim}=n$ Toda flow with the elements of the permutation group $S_n$, so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagramm. This is equivalent to the fact, that the full symmetric Toda system is in fact a Morse-Smale system.