Remarks on Hilbert identities, isometric embeddings, and invariant cubature

TL;DR

This paper extends Victoir's method for constructing cubature formulas using regular t-wise balanced designs, providing new small-index cubatures, updating isometric embeddings tables, and generalizing a theorem on Euclidean designs invariant under reflection groups.

Contribution

It introduces a generalized method for cubature construction with an additional combinatorial object, expanding applications to isometric embeddings and invariant Euclidean designs.

Findings

Constructed many small-index cubatures with few points.

Updated Shatalov's table of isometric embeddings.

Extended Bajnok's theorem to all finite irreducible reflection groups.

Abstract

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.