Orbits of the hyperoctahedral group as Euclidean designs

Bajnok, Béla

Journal of Algebraic Combinatorics, Tome 25 (2007) no. 4, pp. 375-397.

Résumé

Summary: 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 }, \dots $, e $_{ n }$ denoting the standard basis vectors of $\sf R$ sfR $^{ n }$ and letting x $_{ k } = e _{ 1 } + \cdot \cdot \cdot + e _{ k } ( k = 1, 2, \dots , n)$, the set $ {\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}{\cal I}^n_k={\bf x}$_k^H=$ x$\_{\bf k}^g \mbox{} | \mbox{} g $inH$ \} is the vertex set of a generalized regular hyperoctahedron in $sfR$ \sf{R} $^ n $. A finite set $X$ \`I $sfR ^ n calX subset$\sf{R}^n with a weight function $w: X textregisteredsfR ^+$ w: $calX rightarrow$\sf{R}^+ is called a Euclidean $t$-design, if $ sum_r inR W_r barf_S_r = sum_x incalX w(x) f(x)$ \sum\_{r $inR$} W\_r \bar{f}\_{S\_{r}} = \sum\_$x incalX w(x) f(x)$ holds for every polynomial $f$ of total degree at most $t$; here $R$ is the set of norms of the points in $X calX, W _ r $ is the total weight of all elements of $X calX$ with norm $r, S _ r $ is the $n$-dimensional sphere of radius $r$ centered at the origin, and [ `$( f)] _ S $\_{ r}} \bar{f}\_{S\_{r}} is the average of $f$ over $S _ r $.$

Détail

Keywords: keywords Euclidean design, spherical design, tight design, harmonic polynomial, hyperoctahedral group