Lie Powers of the Natural Module for GL 2
نویسندگان
چکیده
Let L be a free Lie algebra of finite rank r over a field . For each positive integer n, denote the degree n homogeneous component of L by L. The group of graded algebra automorphisms of L may be identified with GLr; in such a way that L1 becomes the natural module, and then the L are referred to as the Lie powers of this module. Understanding the GLr; -module structure of the L may be thought of as an essential part of understanding L. For the case when the characteristic of is 0, the L are semisimple and the multiplicities of the various simple modules in the L are given by a formula of Wever [24]. We are concerned here with the case when has prime characteristic, p; then very few of the L are semisimple, and the problem has a different complexion. If n is not divisible by p, then L is a direct summand of the n-fold tensor power of the natural module L1, so the indecomposable direct summands of L may be taken as known from that context. This has been exploited in Erdmann [12] and in Donkin and
منابع مشابه
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...
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