Journal of Algebraic Combinatorics

, Volume 25, Issue 4, pp 375–397

Orbits of the hyperoctahedral group as Euclidean designs

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Article

DOI: 10.1007/s10801-006-0042-3

Cite this article as:
Bajnok, B. J Algebr Comb (2007) 25: 375. doi:10.1007/s10801-006-0042-3
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Abstract

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type Bn) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e1, ..., en denoting the standard basis vectors of \(\sf{R}\)n and letting xk = e1 + ··· + ek (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 \({\cal X}\),Wr is the total weight of all elements of \({\cal X}\) with norm r, Sr is the n-dimensional sphere of radius r centered at the origin, and \(\bar{f}_{S_{r}}\) is the average of f over Sr.

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).

Keywords

Euclidean designSpherical designTight designHarmonic polynomialHyperoctahedral group

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© Springer Science+Business Media, LLC 2006