关于几何复杂性理论：多重性障碍比发生性障碍更强
On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions
摘要：
Mulmuley和Sohoni在两篇论文（SIAM J Comput 2001，2008）中提出的几何复杂性理论旨在通过特定群簇的坐标环中的表示论多重性来分离代数复杂性类。这些论文还推测，这些多重性的消失行为将足以分离复杂性类别（所谓的发生障碍）。最近，在2016年连续发表的两篇论文中，Ikenmeyer-Panova（Adv. Math.）和Bürgisser-Ikenmeyer-Panova（J. AMS）。这就提出了一个问题，即通过表示论的多重性来分离群簇是否比通过出现来分离群簇更强。本文第一次提供了一个可以实现多重性分离的设置，而发生分离是证明不可能的。我们的设置是令人惊讶的简单和自然的：我们研究各种产品的齐次线性形式（所谓的周品种）和各种多项式的有界边界华林秩（即更高的割线品种的Veronese品种）。作为一个附带的结果，我们证明了一个轻微的推广厄米的互易定理，这证明了福克斯猜想的一个新的无限家庭的情况。
英文摘要：
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. The papers also conjecture that the vanishing behavior of these multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. This paper provides for the first time a setting where separating with multiplicities can be achieved, while the separation with occurrences is provably impossible. Our setting is surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite's reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.
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