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.

受 G Frieden 最近关于仿射A型几何R矩阵的工作的启发与矩形杨氏画面相关的晶体，我们提出了一种方法来构建一个新的离散可积系统族，可以将其视为广义周期盒球系统的几何提升。通过转换矩阵的传统用法来定义离散周期 Toda 链的 Lax 表示，并巧妙地使用 Perron-Frobenious 定理，我们给出了我们系统的定义。它是在实正因变量的空间上进行的，不认为它们是由自变量的无减法有理函数编写的，但仍然具有可以热带化的守恒量。我们证明，在此设置中，用于确保其周期性边界条件始终具有唯一解的盒球系统“载体”的模拟方程。因此，我们系统中的任何状态都允许与任何矩形画面相关的时间演化的通勤族，这与相应的广义周期盒球系统的情况形成对比，其中某些状态不允许某些此类时间演化。