Bruhat Order in Full Symmetric Toda 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 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.

Manifolds of Isospectral Matrices and Hessenberg Varieties

“…Since Bruhat and dual Bruhat cells are transversal, the generalized Toda flow is a Morse-Smale system on M R λ . All the ideas of the work [6] work in the complex case as well.…”

Bruhat order in the Toda system on so(2,4): An example of non-split real form

“…This case is well studied and a vast family of invariant subvarieties in SO(n, R) called the minor surfaces is known (see [10,9,1] for details): they are given by the solution of polynomial equations of the form…”

New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{s}\mathfrak{l}_n $ Toda lattice

“…Our results in this paper are essential for establishing the Bruhat order in the full symmetric sl n Toda lattice [21]. A short version of our paper was presented in [22].…”