1078:. 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 -
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1013:, 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
246:"? 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.
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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
407:, 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
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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
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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
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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.
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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
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Well, the issue is that I canāt find the claim on the zero-dimensional representation in the references; at least itās not in Fulton-Harris. Also, ādubiousā may not have been the best word but is it really useful to say something like ājust like the number 1 ...ā? What
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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 ...
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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?
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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.
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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.
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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.
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This may sound incredibly picky or pedantic, but to be perfectly precise, don't you have to say "a representation of a group is an action of that group on some vector space,
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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
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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. --
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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.
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671:(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.
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A set S is said to be a set-theoretic representation of a group G if there is a function, Ļ from G to S^S, the set of functions from S to S such that...
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962:" 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. ā
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is a line of thought: a representation is just like a number; highly misleading, I think, especially for infinite-dimensional representations. ā-
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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
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Thanks. I misunderstood the meaning of "representation"; I thought it referred to any interpretation of group elements as concrete objects. --
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Anyway, just a suggestion... (There are numerous textbooks on such consequences of group theory for physics. One I like is Inui et al.,
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was flagged "dubious". Doing so automatically generates a link to this talk page, under heading "Dubious". But there was nothing here!
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There doesn't seem to be much discussion of the deep consequences of representation theory for physical problems. (Or did I miss it?)
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Every positive integer divides 0 ( 0 = n*0, so n divides 0), but 0 does not divide any positive integer ( nĀ != 0*m for any m ).
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The concept of using prime numbers as an analogy for much more complex things than numbers is hardly unprecedented- see
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Representations of finite groups can always be decomposed into a direct sum of irreducible subrepresentations (see
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of optics follow from just such a conservation law (following from the translational symmetry of the interface).
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It is common practice to refer to V itself as the representation when the homomorphism is clear from the context
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Doesn't the standard definition of divisibility in the integers have zero as a divisor of any positive integer?
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documents a huge subject, and needs a main article which surveys its breadth and depth. So I plan to split off
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then there is only a single condition given, but the condition doesn't guarantee that the image of an element
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Perhaps I miss your point. It seems to me that presentations are an example of group representations. --
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This is not presently an issue; the first sentence of the lead paragraph says what group representations
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only in this sense, much like one refers to many algebraic structures by the name of the set: e.g. by
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being referred to as the representation. I suppose the latter could be implied when one speaks of
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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This still includes a reference to the homomorphism, so it does not seem strange. Intuitively,
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Or does continuity follow from the assumption that the mapping is an algebraic homomorphism?
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unless anyone has a better idea. I assume it would be best to keep the edit history with
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No - that is not the normal technical usage here. Like this: a presentation is more like
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Also, from the perspective of algebra, it's usually more beneficial to think of 1 as a "
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Logically speaking, you are correct. Unfortunately, in practice, the abuse of calling
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follows). A related phenomenon is that the representation generally corresponds to a
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A major omission here is the representation of groups as quotients of free groups. --
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My question is this: Do representations of a Lie group need to be defined as
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Composite numbers, meanwhile, are usually defined as positive integers with
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as a representation, though, since no element (vector) of the vector space
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on its own as the representation seems strange to be, as no element in
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So I've created this section for any editor to explain, if they can,
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specific meaning, and presentations don't fit under that definition.
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correct? The representation is, formally speaking, the homomorphism
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has much to do with the original group or the transformations in
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Well, it should then make clear the distinction representation
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and a link would be handy explaining how subgroups work etc.
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is not quite correct; it requires a restriction on the field
1199:{\displaystyle \rho \colon G\to \mathrm {GL} \left(V\right)}
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There's a limit to the amount that can go into one article.
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9.5 years later, what's "the current state of the entire"
779:) acts represents much of anything of the original group
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This article is serving a dual role: an introduction to
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Question about the definition of a group representation
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The "common practice" use of the term "representation"
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one might mean either a set of elements, or the ring
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For example, if one has a linear operator (such as a
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presentation; these are not the same concept at all.
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411:. Anyway, there is a genuine gap there to bridge.
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1405:Knowledge level-4 vital articles in Mathematics
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399:. I went, a long time ago it seems now, to a
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