> < ^ Date: Thu, 09 Sep 1999 11:01:34 +0200 (MET DST)
> < ^ From: Jan Draisma <Jan.Draisma@unibas.ch >
> ^ Subject: Testing modules for irreducibility

Dear GAP-Forum,

My question in short: do the generators in a MeatAxe-module have to be
_invertible_ matrices?

Explanation:
My question concerns MeatAxe, or more generally: modules for
associative algebras. I understand that MeatAxe is intended for
computations of modules over a group algebra (for example, in the
description of MeatAxe, it sais that the matrices that generate the
algebra must be invertible). However, from what I know about MeatAxe,
I think it should be usable for any associative algebra. Is that the
case?

Perhaps other methods have already been implemented for the application
I have in mind, which is the following. Given a Lie algebra G with a
subalgebra H, I want to test whether G/H is an irreducible H-module. One
can compute matrices for this action, simply by choosing a basis of G
of which a subset forms a basis of H, then computing the adjoint
matrices of the basis elements of H, and deleting rows and columns
that correspong to H. The resulting matrices are usually not
invertible. Is it still possible to use MeatAxe with them as input?

I tried a few small examples, which MeatAxe treated ok:
1. G=a simple lie algebra of type B2 with H=a maximal parabolic subalgebra
(of co-dimension 3, irreducible).
2. G=a simple lie algebra of type A2 with H=a maximal parabolic subalgebra
(of co-dimension 2, irreducible).
3. G=3*3-matrices with zeroes on positions (3,1) and (3,2), and
H=3*3-matrices with zero third row and zero third column.
(reducible).