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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)
Hmmm...I think it's correct as stated, although the whole issue could be explained better. There are two different ways of "looking at" representations, like putting on different glasses; one as an actual linear action, or homomorphism into Aut(something), the other way puts all the information together into a single algebraic object, if it's a rep of a group G, e.g. it would called a "FG-module". The "nontrivial proper subspace" above isn't a subspace in the vector space sense, it's a subspace in the FG-module sense, which is different. I'm afraid my knowledge of this isn't very much to be good at explaining it, but I think the basic wording is correct, just unclear. Revolver 02:31, 1 Apr 2004 (UTC)
That interpretation is possible, though in that case it should be an FG-submodule, and the whole representation-as-module-concept would have to be introduced before. I'll try to improve it. Rvollmert 13:58, 26 Jul 2004 (UTC)
Presentation of groups
A major omission here is the representation of groups as quotients of free groups. -- Dominus 18:03, 11 May 2004 (UTC)
Yes. This page should at least link there, and should probably have a capsule summary of presentations. I will fix this later if someone else doesn't first. -- Dominus 18:48, 11 May 2004 (UTC)
Well, it should then make clear the distinction representation versus presentation; these are not the same concept at all. Charles Matthews 19:43, 11 May 2004 (UTC)
Perhaps I miss your point. It seems to me that presentations are an example of group representations. -- Dominus 03:09, 12 May 2004 (UTC)
No - that is not the normal technical usage here. Like this: a presentation is more like the way a group is handed to us; while a representation is how we represent or configure it for ourselves. I went, a long time ago it seems now, to a computational group theory course by John Conway, which started off on the problem of constructing a permutation representation for a group, for which we have already a presentation. That is, we know generators and relations; what we want is to find concrete permutations that give an isomorphic group (when it is finite). Solved in principle by coset enumeration. Anyway, there is a genuine gap there to bridge.
Charles Matthews 06:09, 12 May 2004 (UTC)
Thanks. I misunderstood the meaning of "representation"; I thought it referred to any interpretation of group elements as concrete objects. -- Dominus 17:11, 13 May 2004 (UTC)
Presentations are usually very difficult to work with directly, whereas representations are concrete and allow us to "get our hands dirty". "Representation" isn't used here with the ordinary English meaning, as in "a representation is just another way of representing it", it's a math term with specific meaning, and presentations don't fit under that definition.
- Representations of finite groups can always be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem.)
is not quite correct; it requires a restriction on the field K (see Maschke's theorem).
Crust 14:49, 29 Dec 2004 (UTC)
- ...the characteristic of the complex numbers is zero, which never divides the size of a group.
Doesn't the standard definition of divisibility in the integers have zero as a divisor of any positive integer?
- Every positive integer divides 0 ( 0 = n*0, so n divides 0), but 0 does not divide any positive integer ( n != 0*m for any m ). JackSchmidt (talk) 04:33, 3 August 2008 (UTC)
Relationship to physics?
There doesn't seem to be much discussion of the deep consequences of representation theory for physical problems. (Or did I miss it?)
For example, if one has a linear operator (such as a Hamiltonian, etc., or Maxwell's equations, or the wave equation, assuming linear media), then the eigenvectors of that operator can generally be chosen to transform as an irreducible representation of the symmetry group of the system (the group of symmetry operators that commute with the operator). This has far-reaching consequences (especially in my field of solid-state physics...for example Bloch's theorem follows). A related phenomenon is that the representation generally corresponds to a conserved quantity (i.e. if you start out in a state that transforms as a representation of the symmetry group, then in a linear system you will still have the same representation at any later time). For example, Snell's law of optics follow from just such a conservation law (following from the translational symmetry of the interface).
Anyway, just a suggestion... (There are numerous textbooks on such consequences of group theory for physics. One I like is Inui et al., Group Theory and Its Applications in Physics.)
—Steven G. Johnson 03:06, 1 October 2005 (UTC)
- May I ask you? What are eigenvectors?--HydrogenSu 15:00, 22 January 2006 (UTC)
Yes, you are right Steven, the entry has clearly been written by mathematicians (fair enough, it is maths and I'm a mathematician myself!) and gives no indication of what a representation really is physically, or why anyone should care. I might add this sometime ... Paul Matthews
Possibly a brief mention of induced representations and a link would be handy explaining how subgroups work etc.
- There's a limit to the amount that can go into one article. Charles Matthews 11:12, 10 November 2005 (UTC)
What is a representation?
The article doesn't actually say what a representation is until you get down to the definitions. The first sentence says what representation theory is and the next two say why its important. Perhaps make it a little more specific? —Preceding unsigned comment added by 126.96.36.199 (talk) 19:00, 9 May 2008 (UTC)
This article is serving a dual role: an introduction to representation theory (which redirects here) and as the article on one of the main types of representation, a group representation. This is too much for one article to bear. Category:Representation theory documents a huge subject, and needs a main article which surveys its breadth and depth. So I plan to split off group representation from representation theory unless anyone has a better idea. I assume it would be best to keep the edit history with group representation. Geometry guy 20:48, 14 September 2008 (UTC)
- Okay done, and I've made some minor tweaks to this article as a consequence, but the whole article really needs to be rewritten. In doing this split, I've gathered a feeling for the current state of the entire representation theory category, and there is much more to be done to bring it into some kind of reasonable shape. Please help! Geometry guy 23:53, 20 September 2008 (UTC)
The "common practice" use of the term "representation"
Is the statement
- It is common practice to refer to V itself as the representation when the homomorphism is clear from the context
correct? The representation is, formally speaking, the homomorphism G → GL(V). However, it might be common to refer informally to an element of the image of this mapping in GL(V) as a a representation. It does not seem to make sense to refer to an element of V as a representation, though, since no element (vector) of the vector space V upon which GL(V) acts represents much of anything of the original group G. As such, should this statement not read
- It is common practice to refer to the image of the mapping in GL(V) itself as the representation when the homomorphism is clear from the context
instead? —Quondum 05:43, 23 August 2014 (UTC)
- Logically speaking, you are correct. Unfortunately, in practice, the abuse of calling V the representation is very common. YohanN7 (talk) 06:01, 23 August 2014 (UTC)
- Do both abuses occur? —Quondum 06:05, 23 August 2014 (UTC)
- This still includes a reference to the homomorphism, so it does not seem strange. Intuitively, Π : G → GL(V) means that G is isomorphically embedded in GL(V). To refer to V on its own as the representation seems strange to be, as no element in V has much to do with the original group or the transformations in GL(V). I can also picture the pair (Im(Π), V) being referred to as the representation. I suppose the latter could be implied when one speaks of V only in this sense, much like one refers to many algebraic structures by the name of the set: e.g. by Z one might mean either a set of elements, or the ring (Z,+,×), so by "V" one might really mean the set plus the group action on it. If this is the case, it would make sense to highlight this "meaning" in the article. —Quondum 15:22, 23 August 2014 (UTC)