$SL(2,\mathbb{C})$ 稳定子群序列下 4-正则图上的 Holant 问题的计算复杂性
The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$
AI总结：
本文针对复值4元签名的Holant问题建立二分定理，引入Schur定理、SL(2,C)有限子群分类和稳定子群序列等新方法，为完整分类奠定关键基础。
AI中文摘要：
Holant 框架为研究计数问题提供了一个通用设置，并将图同态（\\#GH）和计数约束满足问题（\\#CSP）作为特例包含在内。在过去的二十年中，针对 Holant 问题已建立了一系列计算复杂性二分定理，但复值签名的分类仍然是开放的。主要障碍在于所有签名具有偶数元数的情况。在本文中，我们针对具有复值 4 元签名的 Holant 问题建立了一个二分定理，这是 Holant 问题完整分类的一个关键基础情形。我们提出了一种新策略，将 Schur 定理、$\mathrm{SL}(2,\mathbb{C})$ 的有限子群分类以及稳定子群序列引入证明中。这些新技术具有独立的研究价值。
英文摘要：
The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.
