Bruhat Order in Full Symmetric Toda System

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.

💡 Research Summary

The paper investigates the full symmetric Toda system—a completely integrable Hamiltonian flow defined on the space of real symmetric matrices—through the lens of algebraic combinatorics, specifically the Bruhat order on the symmetric group. The authors begin by recalling the Lax formulation of the Toda lattice: for a symmetric matrix (L(t)) the evolution is given by (\dot L=