A modular version of Klyachko's theorem on Lie representations of the general linear group

Abstract

Klyachko, in 1974, considered the tensor and Lie powers of the natural module for the general linear group over a field of characteristic 0 and showed that nearly all of the irreducible submodules of the rth tensor power also occur up to isomorphism as submodules of the rth Lie power. Here we prove an analogue for infinite fields of prime characteristic by showing, with some restrictions on r, that nearly all of the indecomposable direct summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power.

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

“…We note that Theorem A (i) can also be obtained as a special case of [1, Theorem 6.8], which is stated for GL(n, K) modules. However, the methods used in [1] do not apply when r = p m or r = 2p m . Thus, for n = 2 and p = 2, we see that Theorem A (ii) deals with the cases not covered by [1,Theorem 6.8].…”

“…However, the methods used in [1] do not apply when r = p m or r = 2p m . Thus, for n = 2 and p = 2, we see that Theorem A (ii) deals with the cases not covered by [1,Theorem 6.8]. In the following section we give some partial results for n = 2 and p > 2 which are not covered by [1,Theorem 6.8].…”

“…Part (i) of the theorem is a special case of [1,Theorem 6.8]. We give an alternative proof here, using a result of Stöhr [16,Corollary 9.2] on free Lie algebras of rank two in characteristic 2.…”

“…When K is an infinite field of prime characteristic p Klyachko's original argument can be modified to prove a similar result for Lie powers of certain degree, see [1]. Unfortunately the methods used there do not work well when the degree is a power of p or twice a power of p. Throughout this paper we shall restrict attention to the case where K is an infinite field of prime characteristic p and G = GL (2, K).…”

“…We give an alternative proof here, using a result of Stöhr [16,Corollary 9.2] on free Lie algebras of rank two in characteristic 2. Part (ii) of the theorem deals with the cases not covered by [1,Theorem 6.8] when p = n = 2. Note that part (iii) ca be used to give a fairly large lower bound for the multiplicity of a given indecomposable tilting module T (λ) as a direct summand of L r (E).…”