Lie powers and Witt vectors

In the study of Lie powers of a module V in prime characteristic p, a basic role is played by certain modules Bn introduced by Bryant and Schocker. The isomorphism types of the Bn are not fully understood, but these modules fall into infinite families {Bk,Bpk,Bp2k, . . .}, one family B(k) for each positive integer k not divisible by p, and there is a recursive formula for the modules within B(k). Here we use combinatorial methods and Witt vectors to show that each module in B(k) is isomorphic to a direct sum of tensor products of direct summands of the kth tensor power V⊗k .