Talk:Unitarian trick

Opening sentence

"the unitarian trick (occasionally unitarian trick)" looks like a repetition to me. maybe "Weyl's unitarian trick" was intended to be in parenthesis?

since Unitary trick redirects here, I think it was intended to be the alternative. I've edited accordingly. Hv 02:17, 15 April 2007 (UTC)[reply]

Weyl's theorem

The page on semisimple Lie algebras states

   *  ${\mathfrak {g}}$  is semisimple if and only if every representation is completely reducible, 
     that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).

directing here. It's hard to find a clear definition of Weyl's theorem here on the "unitarian trick"-page, and I'm not able to derive it from what's here. Does anyone perhaps have a better reference that we could cite? Especially good would be one where the theorem is explicitly stated. 192.38.109.188 16:32, 28 August 2007 (UTC)[reply]

Complex general linear group

The complex general linear group is not semisimple, so contrary to the statement in the article, the unitarian trick does not apply to it. Not every finite-dimensional representation of it is completely reducible! In fact, this property is equivalent to semisimplicity - see Fulton & Harris, Representation Theory, Ex. 9.21. 129.175.94.201 (talk) 16:41, 10 April 2024 (UTC)[reply]

So something may be mis-said. A reference is [1] for approaches to showing that the complex general linear group is a reductive group. Charles Matthews (talk) 17:43, 10 April 2024 (UTC)[reply]

By the way, the problematic statement can be made correct by limiting the conclusion to holomorphic representations. But more explanation and context is needed anyway. Charles Matthews (talk) 07:26, 17 April 2024 (UTC)[reply]

I have removed the following sentence for now:

An important example is that in which G is the complex general linear group GLn(C), and K the unitary group U(n) acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for those of G, at least in finite dimensions.

I am confused. On reductive group it is stated

For a reductive group G over a field of characteristic zero, all finite-dimensional representations of G (as an algebraic group) are completely reducible, that is, they are direct sums of irreducible representations.

with a reference to Milne's book https://www.jmilne.org/math/CourseNotes/iAG200.pdf. If this is true, and using the fact stated above that GL(n,C) is reductive and the above fact from Fulton & Harris, we would have to conclude that GL(n,C) is semisimple, which I believe is false. What gives? Is there a difference between "representations of G" and "representations of G (as an algebraic group)"? AxelBoldt (talk) 05:37, 11 April 2024 (UTC)[reply]

The point here is about non-algebraic representations. There are 1-dimensional representations of the complex general linear group that are trivial on the complex special linear group, i.e. those coming from representations of the multiplicative group of complex numbers. Using polar coordinates we see representations of the circle group, combined with representations of the multiplicative group of real numbers. The "algebraic" ones of those come from taking the n-power of a complex number, so clearly there are many non-algebraic ones.

A correct statement would be that the unitary group representations determine the complex general linear group, up to twisting with one-dimensional representations. I think Weyl's context is starting from the natural representation of the complex general linear group and then looking at all the tensor representations that can be formed, because this is the way to recover classical invariant theory, as the theory of the trivial representation occurring in symmetric tensors. In that case the 1-dimensional representations are to do with "weight" and so are easy to account for.

Probably this old article needs to be rewritten. Charles Matthews (talk) 07:10, 14 April 2024 (UTC)[reply]