投影子的嵌套序列（2）：多参数多态统计模型，哈密顿量，S-矩阵
A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices
摘要：
我们的出发点是一类辫子矩阵，在以前的论文中，构造的基础上的嵌套序列的投影仪。本文研究了奇$N$的这种$N^2\times N^2$矩阵的统计模型。$\frac 12(N+3)(N-1)$自由参数的存在是我们的模型的关键特征，将它们与其他知名模型区分开来。每个位点都有$N$可能的状态。传递矩阵的迹线取决于$\frac 12(N-1)$参数。对于$r$阶，$N$特征值构成迹，并且涉及整个参数范围的其余$(N^r-N)$特征值以由单位的$r$次根形成的零和多重态或当$r$不是素数时对应于$r$阶因子的低维多重态出现。任何本征值的模的形式为$e^{μθ}$，其中$μ$是自由参数的线性组合，$θ$是谱参数。对素数$r$，指出了多重态个数与Fermat定理的有趣关系。从辫子矩阵出发，构造了对应于可分解$S$ -矩阵的链哈密顿算子和势函数.前景进行了讨论。
英文摘要：
Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{μθ}$, where $μ$ is a linear combination of the free parameters, $θ$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.
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