Talk:Character group

More work needed for this article?

This article, although tersely written, is not advanced material. It covers a topic that is commonly taught to second and third year undergraduates, and represents maybe one or two lectures worth of material. This article, as written, should be easy to understand by anyone who has been exposed to introductory group theory. linas 05:14, 26 Jan 2005 (UTC)

Thank you. I did not mean the contents. Complicated math is fine (in proper context). I meant the style. "Theorem, definition, theorem, theorem, definition..." looks like some text from Abramowitz and Stegun. It is good if you relax the style a bit, add some motivation, some more prose, etc. Take a look at Wikipedia: How to write a Wikipedia article on Mathematics.

By the way, you better use section headings rather than subsection headings in some places. I mean ==Section==. Oleg Alexandrov | talk 05:26, 26 Jan 2005 (UTC)

It might be useful to mention that the first definition of group characters is essentially the classical case of linear group representations in GL(n, C) when n = 1. This is really the way to think about it. You aren't representing the group by complex-valued functions, so much as you are representing them by 1 × 1 complex matrices, i.e. by linear maps from C to C. I understand where Oleg is coming from...the style is a bit too textbook-ish. What is already here is good, but it needs a lot more, e.g. why are characters important? Who came up with them? What problems led to their invention? etc. Also, the section on residue classes is out of place. That is not the topic of the article. If someone needs to know about them, they can be told where the appropriate article is. Revolver 06:04, 12 Apr 2005 (UTC)

Not commutative

Quick question: why can we not define the character group when G is not commutative? What fails? 219.117.195.84 (talk) 15:18, 7 July 2009 (UTC)

Well the problem depends on how you define it. The set of group homomorphisms from any group to the multiplicative group of the field of complex numbers forms a group. However, both the group G and the abelian group G/[G,G], define the same group of homomorphisms, so there is no generality gained in doing it this way. A more general method is to consider the set of group homomorphisms from a group to the general linear group over the field of complex numbers. Assuming you let the dimension of the matrices grow to any finite number, then you get a monoid under the tensor product, but no homomorphism to a group of dimension more than 1 will have an inverse. Since this method also allows adding the homomorphisms using the tensor product, this actually turns the set into a ring, called the representation ring, which is fairly useful. JackSchmidt (talk) 15:28, 7 July 2009 (UTC)

Conjugacy classes: In the preliminaries section, G is introduced as an abelian group, and then it is observed that a character is constant on the conjugacy classes. Since there is only a single element in each conjugacy class of an abelian group, the statement is empty. I don't know the subject, so I can't correct it: I don't know what was intended. Yasmar (talk) 10:14, 25 February 2012 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Character group/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Suffers from the identity crisis. Is this about finite groups? Abelian groups? Dirichlet characters? Connections with related notions have to be explained. Arcfrk 07:12, 25 May 2007 (UTC)

Last edited at 22:44, 28 May 2007 (UTC). Substituted at 01:52, 5 May 2016 (UTC)

Add characters for U(1)

There is another definition of characters used in the literature which is slightly different that the one used in this article. These are homomorphisms

$\phi :G\to U(1)$

which is a subgroup of $\mathbb {C} ^{*}$. Algebraically, they contain the same information for many cases, but having $U(1)$ as the target is useful while studying tori. For example, the dual torus of a torus $V/\Lambda$ where $\Lambda$ is a lattice can be found by looking at the character group

${\text{Hom}}(\Lambda ,U(1))$

One reference is Birkenhake's Complex Abelian Varieties. Kaptain-k-theory (talk) 20:46, 25 June 2021 (UTC)