The dimension of an orbitope based on a solution to the Legendre pair problem

TL;DR

This paper investigates the geometric structure of solutions to the Legendre pair problem, showing that the convex hull of the orbit of solutions has dimension 1, and conjectures this holds generally, with implications for understanding solution spaces.

Contribution

The paper introduces a novel analysis of the orbit dimensions of solutions to the Legendre pair problem using representation theory, providing new insights into the convex hulls of feasible solutions.

Findings

Dimension of the convex hull of the orbit is 1 for certain  ext{ell} values.

Dimension of the convex hull of all feasible points is 2 ext{ell}-2.

Conjecture that the orbit dimension is 1 in general cases.

Abstract

The Legendre pair problem is a particular case of a rank-$1$ semidefinite description problem that seeks to find a pair of vectors $(\mathbf{u},\mathbf{v})$ each of length $\ell$ such that the vector $(\mathbf{u}^{\top},\mathbf{v}^{\top})^{\top}$ satisfies the rank-$1$ semidefinite description. The group $(\mathbb{Z}_\ell\times\mathbb{Z}_\ell)\rtimes \mathbb{Z}^{\times}_\ell$ acts on the solutions satisfying the rank-$1$ semidefinite description by $((i,j),k)(\mathbf{u},\mathbf{v})=((i,k)\mathbf{u},(j,k)\mathbf{v})$ for each $((i,j),k) \in (\mathbb{Z}_\ell\times\mathbb{Z}_\ell)\rtimes \mathbb{Z}^{\times}_\ell$. By applying the methods based on representation theory in Bulutoglu [Discrete Optim. 45 (2022)], and results in Ingleton [Journal of the London Mathematical Society s(1-31) (1956), 445-460] and Lam and Leung [Journal of Algebra 224 (2000), 91-109], for a given solution $(\mathbf{u}^{\top},\mathbf{v}^{\top})^{\top}$ satisfying the rank-$1$ semidefinite description, we show that the dimension of the convex hull of the orbit of $\mathbf{u}$ under the action of $\mathbb{Z}_{\ell}$ or $\mathbb{Z}_\ell\rtimes\mathbb{Z}^{\times}_\ell$ is $\ell-1$ provided that $\ell=p^n$ or $\ell=pq^i$ for $i=1,2$, any positive integer $n$, and any two odd primes $p,q$. Our results lead to the conjecture that this dimension is $\ell-1$ in both cases. We also show that the dimension of the convex hull of all feasible points of the Legendre pair problem of length $\ell$ is $2\ell-2$ provided that it has at least one feasible point.