The dimension of an orbitope based on a solution to the Legendre pair problem
Document Type
Article
Publication Date
12-2024
Abstract
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 (Zℓ×Zℓ)⋊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)∈(Zℓ×Zℓ)⋊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 Zℓ or Zℓ⋊Z×ℓ is ℓ−1 provided that ℓ=pn or ℓ=pqi 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.
Source Publication
Journal of Algebraic Combinatorics
Recommended Citation
Kilpatrick, K., & Bulutoglu, D. (2025). The dimension of an orbitope based on a solution to the Legendre pair problem. Journal of Algebraic Combinatorics, 61(1), 11. https://doi.org/10.1007/s10801-024-01367-2
arXiv:2301.01839 [math.RT]
Comments
The "Link to Full Text" opens the e-print version of the article, as found in the arXiv e-print repository, at https://arxiv.org/abs/2301.01839v1
The published article is scheduled to appear in the February 2025 issue of the source journal, as cited, and is available to subscribers through the DOI link below.