Remarks on Hilbert identities, isometric embeddings, and invariant cubature
For researchers in combinatorial design, cubature, and Banach space embeddings, the work offers incremental improvements to existing tables and extends known results to broader group families.
The paper generalizes Victoir's method for constructing cubature formulae using regular t-wise balanced designs, providing new cubature with few points. This updates Shatalov's table of isometric embeddings and improves classical Hilbert identities, while extending a theorem of Bajnok to all finite irreducible reflection groups.
Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoir's method with one more combinatorial object, called regular t-wise balanced designs. Many cubature 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.