Lie powers of the natural module for GL(2,K)

TL;DR

This paper investigates the structure of Lie powers of the natural module for GL(2,K), specifically identifying when certain indecomposable summands appear in the Lie powers based on the characteristic of the field and the power structure of r.

Contribution

It provides a complete characterization for GL(2,K) of when the indecomposable summand with highest weight (r-1,1) appears in the rth Lie power, filling gaps left by previous work.

Findings

The summand with highest weight (r-1,1) appears in the rth Lie power iff r is not a power of p.

The paper extends previous results by analyzing the cases r = p^m and r = 2p^m.

It establishes a precise criterion linking the appearance of summands to the power structure of r.

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 $\rm{GL}(n,K)$ was presented. There is was shown that nearly all of the indecomposable summands of the $r$th tensor power also occur up to isomorphism as summands of the $r$th Lie power provided that $r\neq p^m$ and $r \neq 2p^m$, where $p$ is the characteristic of $K$. In the current paper we restrict attention to ${\rm GL}(2,K)$ and consider the missing cases where $r = p^m$ and $r = 2p^m$. In particular, we prove that the indecomposable summand of the $r$th tensor power of the natural module with highest weight $(r-1,1)$ is a summand of the $r$th Lie power if and only if $r$ is a not power of $p$.