Lie Powers of the Natural Module for Gl(2,k)

Abstract

In recent work of R. M. Bryant and the second author a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural GL(n, K) was presented. There is was shown that nearly all of the indecomposable summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power provided that r = p m and r = 2p m , where p is the characteristic of K. In the current paper we restrict attention to GL(2, K) and consider the missing cases where r = p m and r = 2p m . In particular, we prove that the indecomposable summand of the rth tensor power of the natural module with highest weight (r − 1, 1) is a summand of the rth Lie power if and only if r is a not power of p.