The dimension of an orbitope based on a solution to the Legendre pair problem
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The Legendre pair problem is a particular case of a rank-1 semidefinite description problem that seeks to find a pair of vectors (𝐮,𝐯) each of length ℓ such that the vector (𝐮^⊤,𝐯^⊤)^⊤ satisfies the rank-1 semidefinite description. The group (ℤ_ℓ×ℤ_ℓ)⋊ℤ^×_ℓ acts on the solutions satisfying the rank-1 semidefinite description by ((i,j),k)(𝐮,𝐯)=((i,k)𝐮,(j,k)𝐯) for each ((i,j),k) ∈ (ℤ_ℓ×ℤ_ℓ)⋊ℤ^×_ℓ. 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 (𝐮^⊤,𝐯^⊤)^⊤ satisfying the rank-1 semidefinite description, we show that the dimension of the convex hull of the orbit of 𝐮 under the action of ℤ_ℓ or ℤ_ℓ⋊ℤ^×_ℓ is ℓ-1 provided that ℓ=p^n or ℓ=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 ℓ-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 ℓ is 2ℓ-2 provided that it has at least one feasible point.
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