Criterion for (non)decomposability of a representation?

1. Jul 25, 2013

JustMeDK

Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma.

My question is: Does there exists some similar criterion for deciding whether or not a given representation is decomposable, i.e., whether or not there exists some change of basis that brings the representation on (sub)block-diagonal form?

2. Jul 25, 2013

fzero

One way to do this is to determine the character of the given representation. Then, assuming one knows the characters of the irreducible representations, one uses orthogonality relationships to decompose the character in terms of the irreducible characters.

3. Jul 26, 2013

JustMeDK

Thanks for your reply, fzero. I am not quite certain, though, that the method you propose is of any use to me for the particular case I have at hand: the representation is not that of any group (and it seems that the Schur orthogonality relations concern representations of groups). Also, and this is perhaps being too demanding, I would appreciate something conceptually as easy as the criterion of irreducibility (but perhaps that is just not possible concerning nondecomposability?).

4. Jul 26, 2013

JustMeDK

To be a little more specific about the representation at hand: it consists of elements, NOT all of which are invertible, thus, of course, not generating any group.

5. Jul 26, 2013

fzero

If you exponentiate the matrices, you generate a Lie group, which would then let you use the character. I don't know if this is feasible for your application.

6. Jul 27, 2013

JustMeDK

It is perhaps an idea worth looking into, thanks.

I would prefer working at the linear level, though, for the following two reasons: 1.) I have no experience with the machinery of characters; 2.) I fear that the exponentiation and subsequent generation of a group (by obtaining closure under multiplication and taking the inverse) may turn out to be rather difficult to work with.

Generating the Lie group corresponds, I guess, at the linear level to generating from the matrices at hand a closed algebra under commutation, i.e., finding a Lie algebra. But then, of course, the machinery of characters is of no use; or am I mistaken?

Anyway, my original question remains open. Therefore,

If anyone knows of some criterion on (non)decomposability, analogous in simplicity to Schur's Lemma on (ir)reducibility, then please let me know.

Last edited: Jul 27, 2013
7. Jul 30, 2013