  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.
