Note on cubature formulae and designs obtained from group orbits

TL;DR

This paper explores invariant cubature formulas and Euclidean designs linked to finite reflection groups, providing new proofs, extending classifications, and connecting these concepts through Sobolev's theorem.

Contribution

It offers a shorter, simpler proof of Xu's conditions for symmetric cubature formulas and extends Euclidean design classifications to invariant cases across various reflection groups.

Findings

Provided an alternative proof of Xu's theorems on cubature formulas.

Extended classification of tight Euclidean designs to invariant designs from reflection group orbits.

Generalized Bajnok's classification to other finite reflection groups.

Abstract

In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type B to other finite reflection groups.