Talk:Group representation
What is said about the orthogonality of characters might mislead. For non-abelian groups the degrees of representations, sizes of conjugacy classes enter the inner product used. Suggest this goes on its own page, as this one is already long, and (rightly) aims to give an overview first.
Charles Matthews 11:39, 9 Feb 2004 (UTC)
A set S is said to be a set-theoretic representation of a group G if there is a function, ρ from G to S^S, the set of functions from S to S such that...
then there is only a single condition given, but the condition doesn't guarantee that the image of an element g of G under ρ will go to a permutation of S. As it stands, it seems like you could fix a in S, and then define ρ(g) to be the constant function a for all g in G, and this would be a representation, which it clearly isn't. Revolver 03:33, 23 Feb 2004 (UTC)
One may in fact define a representation of a group as an action of that group on some vector space, thereby avoiding the need to choose a basis and the restriction to finite-dimensional vector spaces.
This may sound incredibly picky or pedantic, but to be perfectly precise, don't you have to say "a [linear] representation of a group is an action of that group on some vector space, which respects the vector space [linear] structure"? I mean, a group action (as I understand the term) is nothing more than a group homomorphism into a permutation group. But this is nothing more than a set-theoretic representation, it seem like it doesn't take into account that the permutation group has to preserve the vector space structure as well. Revolver 03:42, 23 Feb 2004 (UTC)
- I see this is addressed in the group action page by describing different kinds of actions, based on looking at monoids of endomorphisms in different categories. But I still think it's not clear the way it's worded above. Revolver 03:51, 23 Feb 2004 (UTC)
- If V has a non-trivial proper subspace W such that W is contained in V, then the representation is said to be reducible.
Shouldn't this be more like: such that ρ(W) is contained in W?
Rvollmert 13:52, 29 Mar 2004 (UTC)