AI 中文总结
本文用谱图理论将SU(3) Verlinde和表示为谱zeta函数值，通过谱多项式因式分解得到短递推与Binet公式，实现高效计算。
AI 中文摘要
我们将 $\mathrm{SU}(3)$ Verlinde 和 $V_n(m)$，在显式因子内，实现为三角离散环面（$m^2$ 个顶点）上高阶 Laplace 算子的谱 zeta 函数在 $n$ 处的值。对于固定的 $m$，我们用相关的偶谱多项式的对数导数表示其生成函数，并将该多项式表示为显式的迭代结式。利用置换对称性，我们证明该多项式在 $\mathbb{Q}$ 上是立方，除非当 $3\mid m$ 时有一个显式的二次因子。该因式分解产生 $V_n(m)$ 满足的具有常系数的较短线性递推关系。我们还推导出 Binet 型公式，将 $V_n(m)$ 表示为谱多项式根的缩放逆平方的幂的有限线性组合。这些结果提供了计算这些 Verlinde 和的高效算法。文中给出了几个充分发展的例子，展示了该方法的计算效率，包括用斐波那契数和卢卡斯数表示的表达式。
英文摘要
We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over $\mathbb{Q}$, apart from an explicit quadratic factor when $3\mid m$. This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constant coefficients. We also derive Binet-type formulas expressing $V_n(m)$ as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.
Comments31 pages, 1 figure
