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I'm not exactly sure what the scope of this idea is. For example, taking a representation and its contragredient and all tensor products, and decomposing, may not be exactly the same thing in some cases; though I expect it is in many cases of interest.
The Tannakian category thing is a sledgehammer, and I'm aware of that.
Charles Matthews 11:18, 12 May 2004 (UTC)
Dear Charles, how can I edit these pages in a more automatic way, so that the date and time is here immediately?
- You need to type ~ four times for sig+time stamp, three times for signature only.Charles Matthews
Yes, I would not claim that the definition is mathematically rigorous; it is rather describing the rough flavor how this word is used in particle physics - and I guess that particle physicists use the word "fundamental representation" more often than the mathematicians. For example, I am not sure whether the physicists would agree that one must include the tensor products with the dual (anti-fundamental) representations, too. In most cases it does not matter - for example Nbar of SU(N) can be obtained from N-1-th power of N.
- Well, there is going to be some theory about this, for a given class of groups. For example for compact groups, it has to come somehow out of the Peter-Weyl theorem.
- One approach is this: starting with a representation you say, OK, create all the representations that come as direct summands out of tensor powers. Then you should get a collection of representations that is closed under tensor power, and direct summands (in the good case of complete reducibility, under subquotients, which says really 'anything you can get ypur hands on'); what is this? With a theorem saying "it's representations of something", you can probably say, either it's the group we had first, or some quotient.
- So the other way is, take all the matrix coefficients like <v,g.w> for fixed v and w in a representation, and generate a ring with them. Then if this is good at 'separating points' in your group, one expects that matrix coefficients of big tensor powers will 'find' functions that also separate out points. From there, getting to having enough functions that are constant on conjugacy classes, i.e. group characters, is sort of expected. (This is vague - I know the other way somewhat better.)
- In the end for the kind of example cited here, there are probably more direct ways. Charles Matthews 14:22, 12 May 2004 (UTC)
In light of the fact that fundamental representations, at least, according to Lumidek's definition are highly nonunique, in what sense is it right to speak of THE fundamental rep instead of A fundamental rep? Phys 20:44, 2 Aug 2004 (UTC)
Why do you need compactness and simply connectedness? The highest weight theory works well rather for semisimple Lie groups (that is, in my opinion, the proper context). The following sentence is very confusing for me: "The corresponding irreducible representations are the fundamental representations of the Lie group. In particular, from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding tensor product of the fundamental representations..." So, what is the fundamental representation? Any representation with an integral dominant highest weight? As far as I know, fundamental representations are representations with highest weight that is some fundamental weight -- so easy it is. Franp9am 07:25, 16 September 2006 (UTC)