Talk:Group representation

message: 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)
thread_title:  Orthogonality 

message: 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)
thread_title:  Permutation representations 

message: 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)
replies: text: 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)
thread_title:  Linear action 

message: :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)
thread_title:  Reducible representations 

message: A major omission here is the representation of groups as quotients of free groups. -- Dominus 18:03, 11 May 2004 (UTC)
thread_title:  Presentation of groups 

message: The statement: 
: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)
replies: text: ...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?
Sendhil (talk) 04:02, 3 August 2008 (UTC)

text: 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)
thread_title:  Complete reducibility 

message: 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)
replies: text: May I ask you? What are eigenvectors?--HydrogenSu 15:00, 22 January 2006 (UTC)
thread_title:  Relationship to physics? 

message: 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)
thread_title:  Induced Representations 

message: 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 128.95.141.35 (talk) 19:00, 9 May 2008 (UTC)
replies: text: This is not presently an issue; the first sentence of the lead paragraph says what group representations do, and introduces the commonest example of representations, namely matrices and their multiplication.  yoyo (talk) 14:21, 28 March 2018 (UTC)
thread_title: What is a representation?

message: 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)
replies: text: 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)
replies: text: @Geometry guy 9.5 years later, what's "the current state of the entire" :Category:Representation theory  - is it still as dire?  And what needs doing?  yoyo (talk) 14:26, 28 March 2018 (UTC)
thread_title:  Split 

message: 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)
replies: text: Logically speaking, you are correct. Unfortunately, in practice, the abuse of calling  the representation is very common. YohanN7 (talk) 06:01, 23 August 2014 (UTC)
replies: text: Do both abuses occur? —Quondum 06:05, 23 August 2014 (UTC)
replies: text: Yes. In terms of notation, if  is the homomorphism, then the pair  is also often referred to as the representation. YohanN7 (talk) 06:19, 23 August 2014 (UTC)
replies: text: This still includes a reference to the homomorphism, so it does not seem strange. Intuitively,  means that  is isomorphically embedded in . To refer to  on its own as the representation seems strange to be, as no element in  has much to do with the original group or the transformations in .  I can also picture the pair  being referred to as the representation.  I suppose the latter could be implied when one speaks of  only in this sense, much like one refers to many algebraic structures by the name of the set: e.g. by  one might mean either a set of elements, or the ring , so by "" 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)
thread_title:  The "common practice" use of the term "representation" 

message: The claim in the article, , "… just like the number 1 is considered to be neither composite nor prime" was flagged "dubious".  Doing so automatically generates a link to this talk page, under heading "Dubious".  But there was nothing here!
So I've created this section for any editor to explain, if they can, why they think this claim is dubious.  
Here's why I think this claim is correct:
# Checking the article Prime number, I found that it's been a long time since anybody persisted in calling 1 a prime.  Also, the clearest modern definition of a prime number is "a positive integer with exactly two distinct divisors (namely 1 and itself)".  Clearly 1 fails the test, since it has only one distinct divisor (namely 1, which also happens to be itself).
# Composite numbers, meanwhile, are usually defined as positive integers with more than two distinct divisors; again, 1 fails.
Also, from the perspective of algebra, it's usually more beneficial to think of 1 as a "unit" or "multiplicative identity".
yoyo (talk) 12:25, 29 March 2018 (UTC)
replies: text: Well, the issue is that I can’t find the claim on the zero-dimensional representation in the references; at least it’s not in Fulton-Harris. Also, “dubious” may not have been the best word but is it really useful to say something like “just like the number 1 ...”? What dubious is a line of thought: a representation is just like a number; highly misleading, I think, especially for infinite-dimensional representations. —- Taku (talk) 17:40, 29 March 2018 (UTC)
replies: text:  The concept of using prime numbers as an analogy for much more complex things than numbers is hardly unprecedented- see prime manifold or prime knot. In that sense, irreducible representations are quite similar to the concept of a 'prime' object. Plus, I see nothing wrong with the sentence as it is now, regardless - it isn't saying that irreducible representations are exactly like prime numbers, or even much like them, but rather using the primes as a an example of a similar phenomenon where one particularly trivial element isn't cleanly categorized into an otherwise dyadic group. If anything, there should be a citation needed tag next to the claim that the zero dimensional representation is not considered irreducible - Ramzuiv (talk) 03:33, 28 August 2019 (UTC)
thread_title:  Dubious 

message: The section Definitions begins as follows:
"A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V.  That is, a representation is a map 
:\rho \colon G \to \mathrm{GL}\left(V \right) 
such that
:\rho(g_1 g_2) = \rho(g_1) \rho(g_2) \qquad  \text{for all }g_1,g_2 \in G."
My question is this: Do representations of a Lie group need to be defined as continuous homomorphisms from G to GL(V)? 
Or does continuity follow from the assumption that the mapping is an algebraic homomorphism? 2601:200:C000:1A0:E0B5:CA14:5879:A6C9 (talk) 14:44, 18 June 2022 (UTC)
thread_title:  Question about the definition of a group representation