Invariant cubature formulae for spheres and balls by combinatorial methods

Invariant cubature formulae for a class of weight functions on the simplex T-d are derived using combinatorial methods, extending the formulae in [Gru ndmann and Moller, SIAM J. Numer Anal., 15 (1978), pp. 282-290] for the uni t weight function on T-d. These formulae are used to derive cubature formul ae on the surface of the sphere S-d and on the unit ball B-d using connecti ons between cubature formulae on T-d, B-d and S-d.