The spectrum of the Grigoriev-Laurent pseudomoments

TL;DR

This paper provides a new representation-theoretic proof of the Grigoriev-Laurent lower bound on the sum-of-squares hierarchy for the hypercube, revealing explicit eigenvalues and connecting pseudomoments to Gram matrix constructions.

Contribution

It offers a novel proof of the lower bound, derives exact eigenvalue formulas, and links pseudomoments to Gram matrices and multiharmonic polynomial representations.

Findings

Exact eigenvalues of Grigoriev-Laurent pseudomoments are derived.

Pseudomoments are shown to be a special case of a Gram matrix construction.

A new realization of symmetric group representations as multiharmonic polynomials is presented.

Abstract

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.