Regular simplices, symmetric polynomials and the mean value property

LetP be ann-dimensional regular simplex in ℝⁿ centered at the origin, and let P(k) be thek-skeleton ofP fork = 0, 1,…,n. Then the set of all continuous functions in ℝⁿ satisfying the mean value property with respect to P(k) forms a finite-dimensional linear space of harmonic polynomials. In this paper the function space is explicitly determined by group theoretic and combinatorial arguments for symmetric polynomials.