发表机构
Budapest University of Technology and Economics（布达佩斯技术与经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对矩阵张量幂线性组合的Schatten范数与行列式计算难题，提出基于虚拟表示的多项式时间方法，解决了三项及以上组合的高效计算问题，且验证了其高效性与高精度。
AI 中文摘要
设 $X_n=\sum_{i=1}^s t_i A_i^{\otimes n}$，其中 $A_1,\ldots,A_s\in M_d(\mathbb C)$，$t_1,\ldots,t_s\in\mathbb C$ 为固定值，$n$ 不断增长。直接计算 $X_n$ 的行列式或Schatten范数的计算复杂度随 $n$ 呈指数增长。对于单个张量幂，这些量是初等的；即使是一般的两项组合的行列式也可归约为多项式数量的标量因子；但对于三项及以上的组合，尚无类似的初等归约方法。我们提出一种精确的表示论方法，对于固定的 $d$ 和 $s$，可在 $n$ 的多项式时间内计算 $\\|X_n\\|_p$（$0<p<\infty$）及行列式。Schur-Weyl对偶给出了同时块分解，而Grothendieck环中的Jacobi-Trudi恒等式将Schur模替换为对称幂张量积的带符号组合。对于 $d=3$，每个不可约贡献可归约为两个可显式计算的对称幂项的差，进而可实现开源代码。在单线程CPU基准测试中，一个真实的三项 $3\times3$ 迹范数问题在 $n=18$ 时约需47秒完成，而仅存储未归约矩阵就需要约 $2.4\times10^{18}$ 字节；在共同范围 $n\leq9$ 内，直接计算与归约计算的相对误差均低于 $3.4\times10^{-15}$。
英文摘要
Let $$X_n=\sum_{i=1}^s t_i A_i^{\otimes n},$$ where $A_1,\ldots,A_s\in M_d(\mathbb C)$ and $t_1,\ldots,t_s\in\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\|X_n\|_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times10^{-15}$ throughout their common range $n\leq9$.
