经典仿射型中高阶完美晶体张量平方上能量函数的封闭公式
Closed formulas for energy functions on tensor squares of higher-level perfect crystals in classical affine types
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中文总结 AI 辅助
该研究为经典仿射型中\(l\)级完美晶体张量积\(B_l\otimes B_l\)的局部能量函数构造封闭公式，通过验证递归得到等价形式，代入公式得模特征路径和，经主特化推出Rogers-Ramanujan型恒等式，并给出低秩情况示例。
中文摘要 AI 辅助
对于每个\(l\geq1\)，我们为经典仿射型中\(l\)级完美晶体的张量积\(B_l\otimes B_l\)上的局部能量函数构造了明确的封闭形式坐标公式。一个单一的有限级最大线性公式涵盖所有七种类型：在\(A_n^{(1)}\)型中其两个分支重合，产生部分和的循环最大值，而在其余六种类型中，每个分支是带杠坐标中有限多个明确分段线性表达式的最大值，具有依赖类型的边界数据。我们直接在有限晶体上验证定义局部能量的递归，并推导等价递归形式，无需应用组合\(R\)矩阵即可计算能量。代入KMN路径特征公式可得到所有\(l\)级可积最高权模特征的明确正坐标路径和。经过主特化后，路径指数可重写为位置无关的相邻对统计量的加权和；与特化的Weyl-Kac特征公式比较产生一族统一的\(l\)级Rogers-Ramanujan型恒等式，将这些和与明确的无穷乘积相等。对于每个族中二级的一个代表性低秩情况，我们展示完整的相邻对度矩阵并列出所得恒等式。
英文摘要
For every $l\geq1$, we construct explicit closed-form coordinate formulas for the local energy functions on the tensor products $B_l\otimes B_l$ of level-$l$ perfect crystals in classical affine types. A single finite-level max-linear formula covers all seven types: its two branches coincide in type $A_n^{(1)}$, yielding a cyclic maximum of partial sums, whereas in the remaining six types each branch is the maximum of finitely many explicit piecewise-linear expressions in barred coordinates, with type-dependent boundary data. We verify the defining local-energy recursion directly on the finite crystals and derive equivalent recursive forms, allowing the energy to be evaluated without applying the combinatorial $R$-matrix. Substitution into the KMN path character formula gives explicit positive coordinate path sums for the characters of all level-$l$ integrable highest weight modules. After principal specialization, the path exponent can be rewritten as a weighted sum of a position-independent adjacent-pair statistic; comparison with the specialized Weyl--Kac character formula yields a uniform family of level-$l$ Rogers--Ramanujan-type identities equating these sums with explicit infinite products. For a representative low-rank case at level two in each family, we display the complete adjacent-pair degree matrix and list the resulting identities.
