Representing measures on idempotent varieties

Determine whether every exceptional Hsiang algebra admits a positive representing measure supported on its idempotent variety, and determine whether a suitably weighted normalized Riemannian volume measure on that variety always provides such a representation.

Background

For a Hsiang cubic, a representing measure on the idempotent variety reproduces the cubic form as a positive average of cubes of normalized idempotents. Equivalently, the multiplication tensor is the third moment of a positive measure supported on the idempotents. Such a measure implies stationarity for the Chebyshev ratio.

The paper constructs representing measures for all polar Hsiang algebras and for every Hsiang algebra known to exist through symmetry, homogeneous-variety arguments, or Steiner decompositions. The unresolved case concerns hypothetical exceptional algebras with admissible Peirce profiles that have neither been realized nor excluded; the question about using normalized Riemannian volume remains open even for known algebras.

References

The three central open questions suggested by our results are formulated in \Cref{sec:open}: whether every Hsiang algebra, including the hypothetical exceptional ones, admits a representing measure on its idempotent variety (\Cref{prob:waring}), whether the global minimizers of kappa0 are Hsiang cubics (\Cref{prob:global}), and whether Hsiang cubics are characterized by equioscillation or stationarity (\Cref{prob:converse}).

kappa0:

$\kappa_n:=\min\bigl\{R(p):\ 0\ne p\inH(\R{n})\bigr\}. $

— Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature  (2610.13532 - Tkachev, 9 Oct 2026) in Section 7, Problem \ref{prob:waring} (Concluding remarks)