Comments on: Representation Theory I http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/ Musings on art, philosophy, mathematics, and physics Thu, 31 Dec 2009 21:09:11 +0000 http://wordpress.com/ hourly 1 By: Basics of group representation theory « Delta Epsilons http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/#comment-420 Basics of group representation theory « Delta Epsilons Fri, 10 Jul 2009 21:58:35 +0000 http://hilbertthm90.wordpress.com/?p=480#comment-420 [...] back. But the basics are well known and have been discussed at length on other blogs (see, e.g. here, which is discussing the subject right now), so I am merely going to summarize these facts without [...] [...] back. But the basics are well known and have been discussed at length on other blogs (see, e.g. here, which is discussing the subject right now), so I am merely going to summarize these facts without [...]

]]> By: Akhil Mathew http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/#comment-418 Akhil Mathew Thu, 09 Jul 2009 20:23:53 +0000 http://hilbertthm90.wordpress.com/?p=480#comment-418 I was just actually looking at an example of this phenomenon. Say $latex H \subset G$ is a subgroup of a finite group $latex G$. If $latex M$ is a representation of $latex H$, then $latex aM$ is a representation of $latex aHa^{-1}$, for $latex a \in G$. This representation can be written in a form resembling the discussion above. The representation $latex aM$ is isomorphic to the representation $latex M^a$ given by mapping $latex aha^{-1}, m$ to $latex hm$. I was just actually looking at an example of this phenomenon. Say H \subset G is a subgroup of a finite group G. If M is a representation of H, then aM is a representation of aHa^{-1}, for a \in G.

This representation can be written in a form resembling the discussion above. The representation aM is isomorphic to the representation M^a given by mapping aha^{-1}, m to hm.

]]> By: hilbertthm90 http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/#comment-417 hilbertthm90 Thu, 09 Jul 2009 04:04:32 +0000 http://hilbertthm90.wordpress.com/?p=480#comment-417 Maybe I should switch the "the" to an "an"? I did not expect such a fuss over this. This was the point of Zygmund, "...the embedding G ->S_n is noncanonical". I just wanted another representation, I didn't need it to be nice or canonical or unique. Maybe I should have just posted something more standard... :( Maybe I should switch the “the” to an “an”? I did not expect such a fuss over this. This was the point of Zygmund, “…the embedding G ->S_n is noncanonical”. I just wanted another representation, I didn’t need it to be nice or canonical or unique. Maybe I should have just posted something more standard… :(

]]> By: Qiaochu Yuan http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/#comment-415 Qiaochu Yuan Wed, 08 Jul 2009 22:47:54 +0000 http://hilbertthm90.wordpress.com/?p=480#comment-415 Is it even obvious that the "almost trivial representation" is unique? One could imagine a group with two embeddings into S_n (for n minimal) related by an outer automorphism in which different group elements get called even or odd. Is it even obvious that the “almost trivial representation” is unique? One could imagine a group with two embeddings into S_n (for n minimal) related by an outer automorphism in which different group elements get called even or odd.

]]> By: Zygmund http://hilbertthm90.wordpress.com/2009/07/05/representation-theory-i/#comment-412 Zygmund Mon, 06 Jul 2009 11:53:24 +0000 http://hilbertthm90.wordpress.com/?p=480#comment-412 "We have an arsenal of examples already, but probably don’t even realize it. The trivial representation is just sending every group element to the identity transformation. What I like to call the “almost trivial representation” (term my own, so don’t use this somewhere and expect people to know what you are talking about), is to embed G in S_n for some large n, which we know is possible. Then under this embedding, a group element is either even or odd. If it is even, send it to the identity transformation. If it is odd, send it to the negative identity transformation. Probably a better way to say this is: a representation of S_n is, \phi(x)=sgn(x)1_V. " But strictly speaking, we don't know that this is different from the trivial representation, and even if it is, it's not functorial because the embedding G->S_n is noncanonical. One could generalize this: given a homomorphism G->H, there is a functor Rep(H)->Rep(G). “We have an arsenal of examples already, but probably don’t even realize it. The trivial representation is just sending every group element to the identity transformation. What I like to call the “almost trivial representation” (term my own, so don’t use this somewhere and expect people to know what you are talking about), is to embed G in S_n for some large n, which we know is possible. Then under this embedding, a group element is either even or odd. If it is even, send it to the identity transformation. If it is odd, send it to the negative identity transformation. Probably a better way to say this is: a representation of S_n is, \phi(x)=sgn(x)1_V. ”

But strictly speaking, we don’t know that this is different from the trivial representation, and even if it is, it’s not functorial because the embedding G->S_n is noncanonical. One could generalize this: given a homomorphism G->H, there is a functor Rep(H)->Rep(G).

]]>