Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature

Abstract

A Hsiang cubic is a cubic form on $\mathbb{R}^{n}$ whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on $S^{n-1}$: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial $T_3$. For the spherical analogue of the problem of least deviation, the minimization of $\|p\|_\infty^2/\|p\|_2^2$ over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.