Orbits of the hyperoctahedral group as Euclidean designs

• Authors: Béla Bajnok
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2007

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e 1 , ..., e n denoting the standard basis vectors of $$\sf{R}$$ n and letting x k = e 1 + ··· + e k (k = 1, 2, ..., n), the set $$ {\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}$$ is the vertex set of a generalized regular hyperoctahedron in $$\sf{R}$$ n .A finite set $${\cal X} \subset \sf{R}^n$$ with a weight function $$w: {\cal X} \rightarrow \sf{R}^+$$ is called a Euclidean t-design, if $$ \sum_{r \in R} W_r \bar{f}_{S_{r}} = \sum_{{\bf x} \in {\cal X}} w({\bf x}) f({\bf x})$$ holds for every polynomial f of total degree at most t; here R is the set of norms of the points in $${\cal X}$$ ,W r is the total weight of all elements of $${\cal X}$$ with norm r, S r is the n-dimensional sphere of radius r centered at the origin, and $$\bar{f}_{S_{r}}$$ is the average of f over S r .Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum t for which it is a Euclidean design) equal to 3, 5, or 7.We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7.In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for t = 5, a set of three equations for t = 7, and a set of seven equations for t = 9.Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fisher-type inequality $$|{\cal X}| \geq N(n,p,t)$$ for the minimum size of a Euclidean t-design in $$\sf{R}$$ n on p = |R| concentric spheres (assuming that the design is antipodal if t is odd).A Euclidean design with exactly N (n, p, t) points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for N(n, p, t) = (3, 2, 5), (3, 3, 7), and (4, 2, 7).