Remarks on Hilbert identities, isometric embeddings, and invariant cubature

Victoir (2004) developed a method to construct cubature formulas with various combinatorial objects. Motivated by this, the authors generalize Victoir's method with one more combinatorial object, called regular $t$-wise balanced designs. Many cubature formulas of small indices with few points are provided, which are used to update Shatalov's table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type $B$ is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.