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

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.