Talk:Representation theory of finite groups
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Needs introduction
editThis page could do with an introduction. -- Fropuff 16:07, 2004 Aug 9 (UTC)
Well, it could do with taking in hand.
(Actually ANY finite dimensional rep of a finite group can be turned into a unitary rep. To see this, note that any finite dimensional space can be turned into a Hilbert space with a positive definite sesquilinear form <.,.>. The same thing too with the rep, but then it needn't necessarily be unitary. But we can construct a new positive definite sesquilinear form which makes it unitary).
![{\displaystyle <x,y>'=\sum _{g\in G}<g[x],g[y]>}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1795423d543007438956c4992d53515d8ff9cc0)
I've just written this in different words at unitary representation. It's an argument only available over the complex numbers, so probably belongs there.
Charles Matthews 18:43, 20 Sep 2004 (UTC)
Too technical?
editToo technical notice: actually group representation is the general introduction. Charles Matthews 21:17, 28 October 2005 (UTC)
- Also, I have added a simple example, near the top in non-technical language. So I've taken out the technical notice. The page still needs some work lower down though. Paul Matthews 17:32, 22 November 2006 (UTC)
Under Constructing New Representations from Old, I think there is a mistake. Surely the direct sum is (p_1(g)v, p_2(g)w) rather than (p_1(g)v, p_1(g)w). Also, I do not see why the definition of subrepresentations is only over the complex numbers.
Existence
editI came to this page to find out whether there exists a linear representation for every finite group. Shouldn't something like that be addressed here, or am I just too dense to find it? -GTBacchus(talk) 18:08, 13 December 2005 (UTC)
- The left regular representation is a faithful linear representation for any finite group. - Gauge 05:39, 9 January 2006 (UTC)
Application of Schur's lemma
editThis section requires cleanup:
- Needs an introduction
- Incredibly informal language
- Contradictory use of notation on at least two occasions
- The final theorem is false for every finite nontrivial group when the tensor product is taken to mean the standard tensor product of G-modules, so the original poster probably intends some other tensor product and that should be mentioned.
- Needs a source, since it is quite possibly original research, though it sounds vaguely familiar to an idea from modular representation theory
Most of the articles in this area need quite a bit of cleanup, so if this section cannot be sourced, it might be better to delete it. I just didn't want to delete it after cleaning it for 30 minutes. JackSchmidt (talk) 00:03, 5 December 2007 (UTC)