Grigoriev-Laurent赝动量的谱
The spectrum of the Grigoriev-Laurent pseudomoments
摘要：
Grigoriev（2001）和Laurent（2003）独立地证明了半定规划的平方和层次并不精确地表示超立方体$\{\pm 1\}^n$，直到度至少为层次的$n$。Laurent还观察到，随着$n$的增加，她的证明构造的伪动量矩阵似乎具有令人惊讶的简单和递归结构的光谱。虽然格里戈里耶夫-洛朗下界的几个新的证明已经出现，洛朗的意见仍然没有得到证明。我们给另一个，表示理论证明的下界，这也产生确切的公式的特征值的格里戈里耶夫-洛朗pseudoments。使用这些，我们证明和阐述劳伦特的意见。我们的论点有两个特点，可能是独立的利益。首先，我们证明了Grigoriev-Laurent伪动量是Bandeira和Kunisky（2020）提出的伪动量的Gram矩阵构造的特殊情况。其次，我们找到了对称群的不可约表示的一个新的实现，对应于两行的Young图，作为空间的多元多项式是多调和的等边单形。
英文摘要：
Grigoriev (2001) and Laurent (2003) independently showed that the sum-of-squares hierarchy of semidefinite programs does not exactly represent the hypercube $\{\pm 1\}^n$ until degree at least $n$ of the hierarchy. Laurent also observed that the pseudomoment matrices her proof constructs appear to have surprisingly simple and recursively structured spectra as $n$ increases. While several new proofs of the Grigoriev-Laurent lower bound have since appeared, Laurent's observations have remained unproved. We give yet another, representation-theoretic proof of the lower bound, which also yields exact formulae for the eigenvalues of the Grigoriev-Laurent pseudomoments. Using these, we prove and elaborate on Laurent's observations. Our arguments have two features that may be of independent interest. First, we show that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). Second, we find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex.
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