The discrete Toda equation revisited, dual $\beta$-Grothendieck polynomial, ultradiscretization and static soliton

TL;DR

This paper explores the algebraic solutions of the discrete Toda equation, linking them to dual Grothendieck polynomials, and introduces ultradiscretization, static soliton representations, and a cellular automaton model.

Contribution

It establishes a connection between the discrete Toda equation solutions and dual Grothendieck polynomials, and proposes new ultradiscrete and cellular automaton models.

Findings

Algebraic solutions relate to dual Grothendieck polynomials.

Tropical permanent solutions for ultradiscrete Toda equation.

New cellular automaton model for ultradiscrete Toda equation.

Abstract

This paper presents a study of the discrete Toda equation $(\tau_n^t)^2+\tau_{n-1}^t\tau_{n+1}^t=\tau_n^{t-1}\tau_n^{t+1}$, that was introduced in 1977. In this paper, it has been proved that the algebraic solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomial, which is a $K$-theoretic generalization of the Schur polynomial. A tropical permanent solution to the ultradiscrete Toda equation has also been derived. The proposed method gives a tropical algebraic representation of static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation has been proposed.