Authors

K. Erdmann and L. G. Kovacs

Title

Metabelian Lie powers of the natural module for a general linear group


Abstract

Consider a free metabelian Lie algebra $M$ of rank $r$
over an infinite field $K$ of prime characteristic $p$.
Given a free generating set, $M$ acquires a grading;
its group of graded automorphisms is the general linear group $GL_r(K)$,
so each homogeneous component $\met[d]$ is
a finite dimensional $GL_r(K)$-module.
The homogeneous component $M^1$ of degree 1 is the natural module,
and the other $M^d$ are the metabelian Lie powers of the title.
(Like symmetric powers, they are also known as dual Weyl modules.)

This paper investigates the submodule structure of the $M^d$.
In the main result, a composition series is constructed in each $M^d$
and the isomorphism types of the composition factors are identified
both in terms of highest weights
and in terms of Steinberg's twisted tensor product theorem;
their dimensions are also given.
It turns out that the composition factors are pairwise non-isomorphic,
from which it follows that the submodule lattice is finite and distributive.
By the Birkhoff representation theorem, any such lattice is
explicitly recognizable from the poset of its join-irreducible elements.
Of course a submodule is join-irreducible if an only if it has a unique maximal 
submodule, and here this gives a bijection between the poset of
join-irreducibles and the set of isomorphism types of composition factors.
Thus the information provided by the main result about the latter
gives natural labels for the former as well.
The relevant partial order in terms of these labels is then described
by exploiting a very close connection with
a 1975 paper of Yu.\,A.\,Bakhturin
on identical relations in metabelian Lie algebras.

In case $d$ is not larger than $r$ and is not divisible by $p$,
the description is relatively simple; for flavour, we sketch this.
Consider all possible ways of writing $d$ as a sum of powers of $p$,
not caring about the order of summands.
Call one such `$p$-partition' a refinement of another
if it breaks some parts into sums of smaller powers of $p$.
Results of S.\,R.\,Doty and Leonid Krop may be interpreted as saying that
the set of all $p$-partitions of $d$ partially ordered `by refinement'
is the poset relevant for the submodule structure
of the symmetric power of degree $d$.
For $\met[d]$ in the special case indicated, all one has to change is
to exclude the $p$-partitions
which have only one part $p^0$.


Status: Submitted.