Note on cubature formulae and designs obtained from group orbits
For mathematicians working on cubature formulas and Euclidean designs, this work offers streamlined proofs and generalizations, though it is incremental in nature.
This paper provides a simpler proof of Xu's theorems on cubature formulas with strong symmetry and extends Neumaier and Seidel's theorem on Euclidean designs to invariant designs, classifying tight Euclidean designs from unions of orbits of corner vectors for general finite reflection groups.
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.