Lie powers of modules for GL ( 2 , p )

In a previous paper [12] with almost the same title, two of the authors considered the Lie powers L(V ) of the natural module V for GL(2,p) (where p is prime). For the case when n is not divisible by p, it was shown there that the indecomposable direct summands of L(V ) are either simple or projective, and two methods were described for calculating the relevant Krull–Schmidt multiplicities. The hard part of the paper dealt with the case of n divisible by p, but only under the assumption that p 3: for p 5, this case was left open. The purpose of the present paper is to complete this investigation. For background and motivation, see [12]. The main qualitative result (see Corollary 2.2) is rather more general: if V is any finite direct sum of simple modules and projective indecomposable modules for GL(2,p) over a field of characteristic p, then the nonsimple, nonprojective indecomposable direct summands in any Lie power of V can only be of dimension p − 1 and composition length 2. The proof is constructive, and most of it is of even wider application: it amounts to a method for calculating Krull–Schmidt multiplicities in Lie powers of arbitrary finite-dimensional GL(2,p)-modules in characteristic p. (We do not think that this method improves computation in the cases already covered in [12].) One may well question how far it is practical to perform the actual calculations, even if one only wants Lie powers of the natural module, but the qualitative result paraphrased above demonstrates the penetrating power of this approach. Remarkably, the claim about the composition length of the nonsimple, nonprojective indecomposables which occur is equally true for the Lie powers of the natural module