|Representation theory has been listed as a level-4 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.|
|WikiProject Mathematics||(Rated B-class, Top-importance)|
Still to be done
I've created an article on representation theory, mostly based on what I could find on-wiki, with the back-up of on-line references. It is far from complete, and much remains to be done.
- A history section is needed, preferably as summary style for an article on the history of representation theory.
- Facts need to be checked and citations added. I've only really done the lead.
- Summary style spinout articles need to be expanded and this article updated accordingly. This whole category is in a mess, although I have tried to tidy it substantially as I wrote this.
On the plus side, improving this cat should be a lot of fun! Geometry guy 23:21, 20 September 2008 (UTC)
- I've partially done #2: there are now a sprinkling of relevant references to a mix of primary and secondary sources. Geometry guy 21:13, 22 September 2008 (UTC)
Another classical todo:
- Representations of finite dimensional associative algebras: Artin-Wedderburn on semisimple, then Osima and basic algebras, then Gabriel-Roiter on finite type including bound quivers with finitely many points, maybe general approximation theory
- The associative algebra stuff is really weak right now, so improving it is a priority. Geometry guy 21:55, 22 September 2008 (UTC)
"Equivalent representation" redirects here, but the phrase is nowhere to be found in the article. If someone wants to learn what this phrase means, wikipedia is of no help.22.214.171.124 (talk) 06:30, 18 May 2009 (UTC)
- If someone wants to fix this problem then your message is not of much help. "Equivalent representation" is not a common or unambiguous term: most likely, it means "Isomorphic representation". However, it would be helpful if you can supply the context in which you encountered the term. Geometry guy 09:16, 18 May 2009 (UTC)
- I agree Geometry guy. I encountered it in a text that used isomorphic for (what most would call) a homomorphism (operation preserving map) and "equivalent representaion" for isomorphism. As this is not the language Wikipedia uses, and would only clutter a rather meager article, I'll just leave this comment here. and do nothing. Fermionicstew (talk) 01:33, 23 June 2012 (UTC)
- The ip has a valid point. I can find several introductory graduate level references that use either "Equivalent representations" or "Isomorphism of representations" for the concept. "Intertwining maps of representations" and "G-maps" are used too, but can't find a single reference that uses the term "Equivariant maps" in the context of representation theory of groups or Lie algebras. But never mind.
- Besides, "...then an equivariant map from V to W is a linear map..." clashes with the definition of "equivariant map", which is a broader concept than is needed here. Those maps need not be linear. (Rephrasing to "...then a a linear equivariant map from V to W is ..." would fix the minor problem.) YohanN7 (talk) 20:36, 28 September 2012 (UTC)
"...makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication"
Hmm, matrices are one hundred percent abstract if you ask me... No, not their representation (in writing), but the actual concept behind the matrix notation; the coefficients for a system of equations that may represent many possible things having isomorph structures. This is, per definition, an abstraction! Not only the abstract coefficient matrix itself, but also the equations they represent, the basic number system, and so on.
We are dealing with abstractions of abstractions (of abstractions) here!
The basic flaw (in many people's mind) is trying to categorize concepts as being either abstract or concrete. In reality, many different abstraction levels are typically used. However, some of them are so familiar that we regard them as concrete, or perhaps confuse them with a particular written representation! Of course, there are no philosophically perfect views on reality or mathematics, but, keeping this in mind would lead us somewhat closer to "the truth" and create less self-contradictions. It is therefore better ;)
- Thanks for your comments. I've added the word "more". Geometry guy 21:03, 18 November 2009 (UTC)
- I'm glad you see my point, thanks. /HenkeB
Representation (mathematics) no longer redirects
Because there are significant usages of the term representation in mathematics that are unrelated to representation theory, and I needed a place to put information about some of those other uses, I've turned representation (mathematics) into a full-blown article. The first entry on the new page after the introductory paragraph does point to representation theory.
I understand that for many people, representation means the kind of object that is studied in representation theory. And I'm certainly not disputing their right to use the term in that way. My intent, far from being prescriptive, was to merely be descriptive of the undeniable fact that other mathematicians do use the word to mean something different.
I believe that I've also tied up any loose ends created by removing the redirect: I checked all the articles that had linked to the redirect and edited the ones that did seem to mean the representation-theoretic sense of representation. Another minor detail is that the template:otheruses widget for this article now points to the new page instead of merely to representation theorem.
And I don't think my change should cause any inconvenience for authors of articles that are about representation theory. The only modification to habit that will be needed is to type [[representation theory|representation]] instead of [[representation (mathematics)|representation]].—PaulTanenbaum (talk) 21:31, 11 February 2010 (UTC)
I think the section for the definition could be made much clearer by making separate definition for groups, associative algebras, and Lie Algebras. Trying to do them all at once is to confusing. Njerseyguy (talk) 17:10, 26 May 2010 (UTC)
Isn't representation theory more broad than group theory?
As a person who needs to look at the roots before analyzing the stems and the branches, it would seem easier to learn representation theory first before going into detail any of the subjects it deals with. It represents a giant outline of topics (pun intended) which are otherwise nearly impossible to navigate in an autodidact fashion.siNkarma86—Expert Sectioneer of Wikipedia
86 = 19+9+14 + karma = 19+9+14 + talk 15:43, 21 July 2011 (UTC)
From the irreducible representations (π,V) one can build other representations (sometimes all other). These constructs are
- Direct sum
- Tensor products
- Complex conjugate
- (Subrepresentations if π is not irreducible.)
I doubt that the list is exhaustive. Perhaps also they don't all generalize across all objects under consideration (groups, algebras etc). The article talks a little about subrepresentations, quotients and direct sums, a subset of the above. I'd like to include descriptions of those constructs that are missing in the article. The reason is partly for completeness, but mostly because I need to be able to link to these generalities from other articles.
The only problem I can see is whether a construct should be put in this article, or in the more specialized articles group representation, algebra representation and Lie algebra representation. The constructions (the action) can differ like, say, for dual representations of a group and its Lie algebra.
Many words here are close to being peacock wording. This is unusual for a mathematical article. See "...striking...vastly...deeply...profound...". I will leave them in the text, as they might be seen as true by those in the field. — Preceding unsigned comment added by 126.96.36.199 (talk) 11:21, 4 June 2014 (UTC)
Use of adjectives Comment
This  edit toned down a couple of statements a bit. Good. The problem is that the references probably use adjectives in the way (profound, etc) that they were used in the previous version. Not every branch of mathematics is equally important (useful, "profound", etc).
The comment(s) below were originally left at 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., and are posted here for posterity. Following
|(Self-assessment: please replace, update.) Needs coherent history (currently limited to a few remarks in various sections), and more coverage of Lie theory and associative algebras. "Generalizations" section needs more sources. Geometry guy 21:15, 22 September 2008 (UTC)|
Last edited at 21:15, 22 September 2008 (UTC). Substituted at 02:33, 5 May 2016 (UTC)