On the space of orthogonal multiplications in three and four dimensions and Cayley’s nodal cubic

Abstract

The study of orthogonal multiplications is more than 100 years old and goes back to the works of Hurwitz and Radon. Yet, apart from the extensive literature on admissibility of domain and range dimensions near the Hurwitz-Radon range (in codimension $\le 8$), only sporadic and fragmentary results are known about full classification (in large codimension), more specifically, about the moduli space $\mathfrak {M}_m$ of orthogonal multiplications $F:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}^n$ (for various n), even for $m\le 4$. In this paper we give an insight to the subtle geometries of $\mathfrak {M}_3$ and $\mathfrak {M}_4$. Orthogonal multiplications are intimately connected to quadratic eigenmaps between spheres via the Hopf-Whitehead construction. The 9-dimensional moduli space $\mathfrak {M}_3$ lies on the boundary of the 84-dimensional moduli of quadratic eigenmaps of $S^5$ into spheres. Similarly, the 36-dimensional moduli space $\mathfrak {M}_4$ is on the boundary of the 300-dimensional moduli of quadratic eigenmaps of $S^7$ into spheres. We will show that $\mathfrak {M}_3$ is the $SO(3)\times SO(3)$-orbit of a 3-dimensional convex body bounded by Cayley’s nodal cubic surface with vertices in a real projective space $\mathbb {R}P^3$, the latter imbedded equivariantly and minimally in an 8-sphere of the space of quadratic spherical harmonics on $S^3$. For $\mathfrak {M}_4$, we show that it possesses two orthogonal 18-dimensional slices each of which is an $SO(4)\times SO(4)$-orbit of a 6-dimensional polytope $\mathcal {P}\subset \mathbb {R}^6$. This polytope itself is the convex hull of two orthogonal regular tetrahedra. The corresponding orthogonal multiplications are explicitly constructible. Finally, we give an algebraic description of the 24-dimensional space of diagonalizable elements in $\mathfrak {M}_4$. The crucial fact in $\mathbb {R}^4$ is the splitting of the exterior product $\Lambda ^2(\mathbb {R}^4)$ into self-dual and anti-self-dual components. The techniques employed here can be traced back to the work of Ziller and the author (Toth and Ziller 1999) in describing the 18-dimensional moduli for quartic spherical minimal immersions of $S^3$ into spheres. As a new feature, we point out the importance of multiplicity of the zeros of the polynomials that define the boundary $\partial \,\mathfrak {M}_m$ as a determinantal variety.