Factorisation Dynamics and Total Positivity at Several Scales
When
In this talk, I will present a discrete dynamical system that is closely related to the full Kostant-Toda lattice originally introduced by Ercolani-Flaschka-Singer in 1993. By employing a particular matrix factorisation due to Fomin and Zelevinsky, using so-called Lusztig coordinates and the Loewner-Whitney characterisation of total positivity, I will describe a way of stratifying the overall full dynamics into simpler and more familiar instances of the usual discrete Toda lattice map. The same techniques can be used to provide a simple description of the (continuous-time) full Kostant-Toda lattice in terms of an iterated dynamic evolution on the associated Lusztig coordinates. Going in the other direction, one can perform a spatial discretisation known as ultradiscretisation to obtain a cellular automaton-like evolution which generalises the classical box-ball system of Takahashi and Satsuma (1996) for which the individual strata are comprised precisely of coupled box-ball iterations. If time permits, I will present a beautiful application of this new cellular automaton to the famous Robinson-Schensted-Knuth correspondence which is a fundamental combinatorial bijection with uses in the representation theory of semisimple Lie groups. This is joint work with Nick Ercolani.
Speaker will be in-person!
Place: Hybrid, Math, 402 and Zoom: https://arizona.zoom.us/j/81150211038 Password: “arizona” (all lower case)