Geometric lifting of the integrable cellular automata with periodic boundary conditions

Abstract

Inspired by G Frieden’s recent work on the geometric R-matrix for affine type A crystal associated with rectangular shaped Young tableaux, we propose a method to construct a novel family of discrete integrable systems which can be regarded as a geometric lifting of the generalized periodic box–ball systems. By converting the conventional usage of the matrices for defining the Lax representation of the discrete periodic Toda chain, together with a clever use of the Perron–Frobenious theorem, we give a definition of our systems. It is carried out on the space of real positive dependent variables, without regarding them to be written by subtraction-free rational functions of independent variables but nevertheless with the conserved quantities which can be tropicalized. We prove that, in this setup an equation of an analogue of the ‘carrier’ of the box–ball system for assuring its periodic boundary condition always has a unique solution. As a result, any states in our systems admit a commuting family of time evolutions associated with any rectangular shaped tableaux, in contrast to the case of corresponding generalized periodic box–ball systems where some states did not admit some of such time evolutions.

Differential equations for the closed geometric crystal chains

“…It also seems that the derivation is not restricted to the totally one-row tableaux case but can be generalized to the rectangular tableaux cases [13]. We hope that we can report a result for such cases in the near future.…”

“…It seems that the above derivations can also be applied to the case of general n, but where giving a remark on the largest eigenvalue of the monodromy matrix may be in order. In this generalization, the matrices corresponding to (34) and ( 35) are connected by the relation in which any element of the matrix g * (x, s; λ) is so defined as to be an order n − 1 minor of the matrix g(x, s; (−1) n λ) (See §3.1.1 of reference [13]). Consider the type I case with L = nκ + 1.…”

“…We briefly review on the closed geometric crystal chains for the totally one-row tableaux case ( [13], §3. 1).…”

“…In a recent paper [13], the author and T. Yoshikawa constructed a new class of discrete integrable systems that can be viewed as a geometric lifting of a class of integrable cellular automata known as the periodic box-ball systems [7,8,15]. Since they are related to a realization of type A n−1 geometric crystals and geometric R-matrices by G. Frieden [2,3], we called the new integrable systems closed geometric crystal chains.…”

“…Therefore, a closed geometric crystal chain is not equivalent to the discrete periodic Toda chain. As a result, while the discrete Toda chain yields the Toda equation in a continuous limit, it is unclear what kind of differential equations will be derived from the closed geometric crystal chains in such a limit, except for the case of n = 2 that was studied in §2.3.2 of reference [13].…”