Designs related through projective and Hopf maps

TL;DR

The paper develops a unified construction for spherical $t$-designs by placing a spherical $t$-design on each fiber of a $\mathbb{K}$-projective map $\Pi_{\mathbb{K}}$ or a $\mathbb{K}$-Hopf map $\pi_{\mathbb{K}}$, given a weighted $\lfloor t/2\rfloor$-design on a quotient space $\mathbb{KP}^n$ or on a sphere, for $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$. This generalizes König–Kuperberg’s complex-projective construction and Okuda’s Hopf-based results, yielding new $t$-designs on spheres such as $S^{4n+3}$ from $\mathbb{HP}^n$ designs and $S^7$ from $S^8$ designs. The main theorem provides a two-way equivalence: a $t$-design on $S^d$ arises from a $\lfloor t/2\rfloor$-design on $\Sigma$ together with $t$-design fibers, and conversely, if the base is a $\lfloor t/2\rfloor$-design and each fiber is a $t$-design, the total is a $t$-design. The constructions are asymptotically optimal, with design sizes scaling as $|X_t|\asymp t^d$, and multiple explicit, near-optimal examples are discussed, highlighting the approach’s breadth across real, complex, quaternionic, and octonionic settings.

Abstract

We verify a construction which, for $\Bbb K$ the reals, complex numbers, quaternions, or octonions, builds a spherical $t$-design by placing a spherical $t$-design on each $\Bbb K$-projective or $\Bbb K$-Hopf fiber associated to the points of a $\lfloor t/2\rfloor$-design on a quotient projective space $\Bbb{KP}^n\neq\Bbb{OP}^2$ or sphere. This generalizes work of König and Kuperberg, who verified the $\Bbb K=\Bbb C$ case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the $\Bbb K=\Bbb C$ case of the generalized Hopf settings.

Figures (5)

• Figure 1: The Fano plane, which visualizes how to multiply octonions. We consider 7 lines in this picture: the 3 sides of the triangle, its 3 altitudes, and the circle inscribed in the triangle. If arrows point from $e_i$ to $e_j$ to $e_l$ along any one of these lines, we have $e_i e_j=-e_j e_i=e_l$.
• Figure 2: Left: vertices of an octahedron, a 3-design on $S^2$. Center: antipodal points, a 1-design on $S^2$. Right: vertices of a triangle, a 2-design on $S^1$.
• Figure 3: Taking coordinates $(a,b=b_1+ib_2)\in\Bbb C^2$ on $S^3\cong\Bbb{R}^3\cup\{\infty\}$ and $(\xi,\eta)\in\Bbb R\times\Bbb C$ on $\Bbb{CP}^1\cong S^2$, restrictions of $f=|a|^2-|b|^2+2|a|^2|b|^2-b_1$ to projective fibers are pictured in gray on the left and $(I_\Bbb Cf)(\xi,\eta)=\xi+\frac{1}{2}|\eta|^2$ is visualized in gray on the right, with points on the right denoted by the same color as their corresponding projective fibers on the left.
• Figure 4: A visualization of Example \ref{['ex:okudaex']}, where we have that $Y=\{(1,0),(-1,0)\}\subset S^2$ is a 1-design and each $Z_y$, both of which are sets of vertices of equilateral triangles, is a 2-design on $S^1$. Elements of $X$, a 2-design on $S^3\cong\Bbb R^3\cup\{\infty\}$, are shown in green on the left.
• Figure 5: The Hopf fibers in $S^3\cong\Bbb R^3\cup\{\infty\}$ corresponding to the vertices of an octahedron on $S^2$. To construct the family of 7-designs described in Example \ref{['ex:cohnex2']}, distribute points corresponding to the vertices of an octagon on each marked fiber.

Theorems & Definitions (19)

• Theorem 1.1: Main Theorem
• Definition 1.2
• proof : Proof of the projective settings of Theorem \ref{['thm:bigthm']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4