Seminar talk, 28 October 2026

Speaker: Georgy Sharygin

Title: The full symmetric Toda system, Lie-Bianchi integrability and the "stochastic" group

Abstract:
In my talk I will continue to tell the story of the full symmetric Toda system. It is a rather straightforward generalisation of the usual "tridiagonal" open Toda chain: the corresponding equation has the form $\dot{L}=[M,L]$, where $L$ is a real symmetric matrix and $M=M(L)$ is the "naive" antisymmetrisation of $L$. This system is Hamiltonian and Liouville integrable. In a recent paper with D.Talalaev and Yu.Chernyakov we proved that this system is also integrable in the sense of the Lie-Bianchi criterion, its symmetries forming the Lie algebra of the so-called "stochastic" group: the group of invertible matrices that preserve the vector (or convector) (1,1,...,1). In this talk (also based on a joint work with D.Talalaev and Yu.Chernyakov) we describe how the action of the stochastic Lie algebra integrates to an action of the corresponding group itself, an action that preserves the trajectories of the Toda system. It turns out that every orbit of this action contains a tridiagonal symmetric matrix, so that solving the full symmetric Toda system reduces to solving the open Toda lattice. Moreover, the passage from a full symmetric matrix to a tridiagonal one can be carried out by means of the Lanczos-Krylov transformation, a rather explicit and simple numerical procedure.

References:
arXiv:2506.07113