Remarks on Hilbert identities, isometric embeddings, and invariant cubature

2012/04/09 by Nozaki, Hiroshi, Sawa, Masanori · 1 citation
#11E76 #52A21 #65D32 #Combinatorics (math.CO) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1204.1779

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.

Cited by

• Algebro-combinatorial generalizations of the Victoir method for constructing weighted designs

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