A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda equation). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda equation. The formula for the fundamental cycle of the pBBS is obtained as a corollary.

The K-homology ring of the affine Grassmannian of SLn(C) was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions.On the other hand, for the quantum K-theory of the flag variety F ln, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars's relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart-Maeno's quantum Grothendieck polynomial associated with a Grassmannian permutation.

A box-ball system with more than one kind of balls is obtained by the generalized periodic discrete Toda equation (pd Toda eq.). We present an algebraic geometric study of the periodic Toda equation. The time evolution of pd Toda eq. is linearized on the Picard group of an algebraic variety, and theta function solutions are obtained.

In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemann surfaces.

This paper presents a study of the discrete Toda equation that was introduced in 1977. In this paper, it is proved that the determinantal solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomials, a K-theoretic generalization of the Schur polynomials. A tropical permanent solution to the ultradiscrete Toda equation is also derived. The proposed method gives a tropical algebraic representation of the static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation is proposed.