论文标题
Grigoriev-laurent假单单的范围
The spectrum of the Grigoriev-Laurent pseudomoments
论文作者
论文摘要
Grigoriev（2001）和Laurent（2003）独立地表明，半决赛程序的平方和层次结构并不能完全代表HyperCube $ \ {\ pm 1 \}^n $，直到层次结构至少$ n $。劳伦（Laurent）还观察到，随着$ n $的增加，她的证明构造的假单位矩阵似乎具有令人惊讶的简单和递归结构化的光谱。虽然此后出现了几个新的证明莫里耶氏王的下限，但洛朗的观察结果仍然没有得到证实。我们提供了下限的另一个表示理论的证明，这也为Grigoriev-laurent假单单的特征值提供了精确的公式。使用这些，我们证明并详细说明了洛朗的观察结果。 我们的论点具有两个可能具有独立兴趣的特征。首先，我们表明，格里戈里耶氏王的假单单是Bandeira和Kunisky（2020）提出的假单胞菌的克矩阵结构的特殊情况。其次，我们发现对应于两行的年轻图对应的对称组的不可约说明的新实现，是多元多项式的空间，这些空间相对于等边性是多峰值的。
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.
