Norton's trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry

Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module V-d, with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c = 1/2, the W-3 algebra at c = 4/5 or the W-4 algebra at c = 1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster.