Ultradiscrete solitary waves and the RSK bijection

Speaker: Travis Scrimshaw 
Affiliation: Hokkaido University

Abstract

The Korteweg-de Vries equation models water moving in a thin channel. It is the prototypical example of an integrable nonlinear PDE as it admits soliton solutions, waves that travel with speed according to their size and separate after interaction. In this talk, we will consider an ultradiscrete version of this due to Takahashi and Satsuma called the box-ball system, and the integrability survives by having the dynamics given by combinatorial data encoding quantum group representations. Here, we reinterpret this dynamic using a completely different combinatorial bijection: a variation of the famed Robinson-Schensted-Knuth  (RSK) bijection. We will conclude with an application of this reformulation, albeit at its natural generalization, to prove a symmetric function Cauchy-type identity involving transformed Hall-Littlewood functions and skew Schur functions. This is based on joint work with Takashi Imamura, Matteo Mucciconi, and Tomohiro Sasamoto.