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Dear Gap-forum, and Philippe,

I was mistaken: it is handy to view so(4,C) as sl(2,C)+sl(2,C). Then,

the 8-dim. module

(V_1 x V_1)+(V_0 x V_1)+(V_1 x V_0)

where x denotes the tensor product, and V_i the sl(2,C)-module with

highest weight i, should work. (Hence, V_1 is the standard module, and

V_0 the trivial one; in fact it turns out to be easier to take V_1 x

V_1^* (2 by 2 matrices) for the former direct sum.) Indeed, the maximal

subalgebras are the diagonal one, which leaves a line in V_1 x V_1

invariant, and sl(2,C)+B, and B+sl(2,C), which lines in V_0 x V_1 and

V_1 x V_0 invariant, respectively.

Note that here is a subtlety which seems to have be overlooked in the

paper you quote: there is an automorphism that interchanges the latter

two subalgebras, but this automorphism takes one module to one which

is not isomorphic to it; therefore we need two modules. Of course, as

there is only a finite number of irreducible modules of a given

dimension, their proof remains correct.

In GAP, you can construct the above modules using the function

HighestWeightModule.

Hope this helps,

Jan

Dear gap-forum, and Jan,

> I'm far to be an expert in representation theory and in GAP, but I'm

>looking for a Chevalley module V for G=SO(4,C), taht is to say a faithful

>finite dimensional SO(4,C)-module such that:

> 1. V contains no one-dimensional G-modules

> 2. any proper connected closed subgroup H $\in$ G leaves a one-dimensional

> subspace W $\in$ V invariant.

I don't understand your question. Looking at the Lie algebra level,

you need a faithful module V for g=sl_2+sl_2 such that any proper subalgebra

leaves a 1-dim subspace invariant, contrary to g itself. Any g-module is

a direct sum of submodules V_i tensor V_j, where i,j>0 denote the highest

weights; i.e. the first sl_2 acts trivially on V_j and the second acts

trivially on V_i. Now h=the first sl_2-factor is a proper subalgebra

of g. Suppose that there exists a 1-d W<=V such that hW<=W. W has a

non-zero projection to some V_i tensor V_j, and we find a 1-d. subspace

W' of the latter with the same property. However, the sl_2 in h will

not leave W' invariant. So, no such module exists.

I look actually for a Chevalley module in order to use a theorem of C.Mitschi

and M.F.Singer to find a realization over C(x) of SO(4,C) by a system Y'=AY.

In their article, they prove that every connected semisimple linear group

defined over C has a Chevalley module. The references of their article are:

Connected Linear Groups as Differential Galois Groups; J.of Algebra,184(1996),

p.333-361 (p.344 for the lemma which interests us).

Hoping this may help you,

Best regards,

>Philippe Gaillard

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