关于 immanant 计算复杂性的一个开放问题的解答
Solution to an open problem on the computational complexity of immanant
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中文总结 AI 辅助
本文解决了宽度为 2 的矩形杨图对应的 immanant 族是否为 VNP 完全问题的开放问题，证明在零特征域上该类 immanant 及长度为 2 的同类 immanant 均为 VNP 完全问题。
中文摘要 AI 辅助
Immanant 是一类与对称群不可约特征标相关的广义矩阵函数。Bürgisser [SIAM J. Comput., 30 (2000), pp. 1023--1040] 证明，在 p-投影归约下，hook immanant 以及对应宽度多项式增长的矩形杨图的 immanant 计算是 VNP 完全问题，并提出一个开放问题：宽度为 2 的矩形杨图对应的 immanant 族在 p-投影归约下是否为 VNP 完全问题。本文给出该问题的解答，证明在任意特征为零的域上，宽度为 2 和长度为 2 的矩形杨图对应的 immanant 族在 p-投影归约下均为 VNP 完全问题。
英文摘要
Immanants are a class of generalized matrix functions associated with the irreducible characters of the symmetric group. Bürgisser [SIAM J. Comput., 30 (2000), pp. 1023--1040] proved that the computation of hook immanants and immanants corresponding to rectangular Young diagrams of polynomially growing width is VNP-complete under $p$-projections. And he posed an open problem: whether the family of immanants corresponding to rectangular Young diagrams of width $2$ is VNP-complete under $p$-projections. This paper gives a solution to this problem. We prove that, over any field of characteristic zero, the immanant families associated with rectangular Young diagrams of width $2$ and of length $2$ are both VNP-complete under $p$-projections.
