求最高权向量的复杂性
On the complexity of evaluating highest weight vectors
摘要：
几何复杂性理论（Geometric Complexity Theory，GCT）是一种通过代数几何和表示理论来分离代数复杂性类的方法。最初Mulmuley和Sohoni提出（SIAM J COMPUT 2001，2008）使用出现障碍来证明Valiant的行列式与永久猜想，但最近Bürgisser，Ikenmeyer和Panova（AMS杂志2019）证明了这是不可能的。然而，代数几何和表示论的基本定理允许GCT中的每一个下界都可以通过使用所谓的最高权向量（HWVs）来证明。在设置感兴趣的GCT（即在设置多项式），我们证明了NP-硬度的评价HWVs一般，我们给出了有效的算法，如果树的宽度相应的杨图是小的，其中的评价点被简洁地编码为一个非交换代数分支程序！特别是，这给出了一个大的新类的分离功能，可以有效地评估在点与低（边界）华林秩。
英文摘要：
Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant's determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-diagram is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank.
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