New sets of equi-isoclinic n-planes from old

Abstract

Two n-planes Γ and Δ in real Euclidean r-space Rr are called isoclinic with parameter λ if the angle θ between any x in Γ and its orthogonal projection Px on Δ is unique, with cos2 θ = λ. Let vλ(n, r) denote the maximum number of equi-isoclinic (i.e. pairwise isoclinic) n-planes in Rr with parameter λ.

Equi-isoclinic planes in Euclidean even dimensional spaces

“…This result is new in comparison with [4], and enables us to prove here that if λ > 1/4 then v λ (2, 2r, R) ≤ r 2 (Corollary 1). These considerations combined with the above list solve the case r = 3.…”

“…Conversely every set of equiangular lines in C r with angle arccos √ λ yields trivially a set of equi-isoclinic planes in R 2r with parameter λ ([1], [4]). …”

“…Hoggar [4] proved that the last formula applies to vector spaces over F = C or H. In fact he gave the following bound for v(n, r,…”

“…He also extended the determination of v(n, 2n, F) to F = C or H. In comparison with [1], [2], [4], [5] and [6] where all the values of v λ (n, r, R) are determined only for some even integers r, the author proved in [3] that v(2, 5, R) = 5. The central problem of this paper is to determine v(2, 6, R).…”

Sous-espaces équi-isoclins de l'espace euclidien

“…[8]. Enfin, mentionnons [5] pour le problème plus général de remplissage avec des n-uplets non nécessairement équi-isoclins, [17] pour l'étude des n-uplets de d-plans de l'espace euclidien de dimension 2d, isoclins deux à deux non nécessairement équi-isoclins et [11] pour des généralisations de certains résultats de P.W.H. Lemmens et J.J. Seidel aux cas de l'espace complexe et de l'espace quaternionien.…”