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

The Legendre pair problem is a particular case of a rank-1 semidefinite description problem that seeks to find a pair of vectors \((\textbf{u},\textbf{v})\) $(\mathbf{u},\mathbf{v})$ each of length \(\ell \) $\ell$ such that the vector \((\textbf{u}^{\top },\textbf{v}^{\top })^{\top }\) ${({\mathbf{u}}^{\top },{\mathbf{v}}^{\top })}^{\top }$ satisfies the rank-1 semidefinite description. The group \((\mathbb {Z}_\ell \times \mathbb {Z}_\ell )\rtimes \mathbb {Z}^{\times }_\ell \) $({\mathbb{Z}}_{\ell }\times {\mathbb{Z}}_{\ell })⋊{\mathbb{Z}}_{\ell }^{\times }$ acts on the solutions satisfying the rank-1 semidefinite description by \( ((i,j),k)(\textbf{u},\textbf{v})=((i,k)\textbf{u},(j,k)\textbf{v}) \) $((i,j),k)(\mathbf{u},\mathbf{v})=((i,k)\mathbf{u},(j,k)\mathbf{v})$ for each \(((i,j),k) \in (\mathbb {Z}_\ell \times \mathbb {Z}_\ell )\rtimes \mathbb {Z}^{\times }_\ell \) $((i,j),k)\in ({\mathbb{Z}}_{\ell }\times {\mathbb{Z}}_{\ell })⋊{\mathbb{Z}}_{\ell }^{\times }$ . By applying the methods based on representation theory in Bulutoglu [Discrete Optim 45 (2022)], and results in Ingleton [J Lond Math Soc 1(4): 445–460, 1956] and Lam and Leung [J Algebra 224:91–109, 2000], for a given solution \((\textbf{u}^{\top },\textbf{v}^{\top })^{\top }\) ${({\mathbf{u}}^{\top },{\mathbf{v}}^{\top })}^{\top }$ satisfying the rank-1 semidefinite description, we show that the dimension of the convex hull of the orbit of \(\textbf{u}\) $\mathbf{u}$ under the action of \(\mathbb {Z}_{\ell }\) ${\mathbb{Z}}_{\ell }$ or \(\mathbb {Z}_\ell \rtimes \mathbb {Z}^{\times }_\ell \) ${\mathbb{Z}}_{\ell }⋊{\mathbb{Z}}_{\ell }^{\times }$ is \(\ell -1\) $\ell -1$ provided that \(\ell =p^n\) $\ell =p^n$ or \(\ell =pq^i\) $\ell =p{q}^i$ for \(i=1,2\) $i=1,2$ , any positive integer n, and any two odd primes p, q. Our results lead to the conjecture that this dimension is \(\ell -1\) $\ell -1$ in both cases. We also show that the dimension of the convex hull of all feasible points of the Legendre pair problem of length \(\ell \) $\ell$ is \(2\ell -2\) $2\ell -2$ provided that it has at least one feasible point.