How did you compute the trace of a group element without writing out its corresponding matrix? Is there another interpretation for trace aside from the sum of its diagonal elements?
I think for most cases he just knows the matrices in his head. First, the identity group element will always map to the identity matrix (of dimension equal to the vector space, of course); thus, the character corresponding to the identity element will always be dim(V). In the case of the one dimensional representations, the trace of the matrix is the number itself, so that's mental. I think he explains/draws out the matrix in all other cases. As to your second question: Yes, and there are a lot of them and they mean many different things. My favorite, and the one that may show the most promise in understanding representation theory, is that the trace of a matrix is the sum of its eigenvalues. In fact, this can be generalized with the determinant with the knowledge that the (negative) trace is the (n-1)th order term in the matrix's characteristic polynomial (en.wikipedia.org/wiki/Characteristic_polynomial#Properties), and the determinant is the 0th order coefficient. The determinant itself is the product of the eigenvalues; in fact, every term in the characteristic polynomial is a sum-of-products, or elementary symmetric polynomial (en.wikipedia.org/wiki/Elementary_symmetric_polynomial), of the eigenvalues; e.g, for n=3, the trace is e1 + e2 + e3, then the next term has coefficient e1e2 + e2e3 + e1e3, and finally the determinant with e1e2e3, up to a sign. (The sign can be determined with the other formula on the characteristic polynomial page, to do with exterior product; I like the geometric algebra version, but in ANY CASE this is three digressions deep.) There are other excellent interpretations of trace; there are a handful more on the wikipedia page (en.wikipedia.org/wiki/Trace_(linear_algebra) ) but I think this one is also cool th-cam.com/video/B2PJh2K-jdU/w-d-xo.htmlsi=ZNepGTALFCIjhF9Q.
There is a subject called classification which studies the question of when we can classify all models of a theory. If I remember correctly group theory is one of the theories where you cannot classify all models. A simpler answer is that the groups we know are such a complicated mess that it seems very unlikely that there is any reasonable classification.
Saving this to my library! So many concepts have suddenly clicked upon watching.
Sir, you are a saint
At 12:24 the invariant subspace should be all vectors (a, 0) and not (0, a).
Yes, or the matrix needs to act on the right. But in that case it should be written as a row vector for sanity's sake.
The man. The legend.
Indeed! 😄
Wow! Terrific channel, glad I found it
My dream is to find a video that explains in 15 minutes what's the point of Lie groups/algebras, specifically regarding physics applications. One day.
It's about rotation n shit😅
@@harshv871 yup, that's the 2 second version of the video right there
16:25 When you write v as that sum, a natural question to ask is what is gv in the first place?
So glad you created this series :)
How did you compute the trace of a group element without writing out its corresponding matrix? Is there another interpretation for trace aside from the sum of its diagonal elements?
I think for most cases he just knows the matrices in his head. First, the identity group element will always map to the identity matrix (of dimension equal to the vector space, of course); thus, the character corresponding to the identity element will always be dim(V). In the case of the one dimensional representations, the trace of the matrix is the number itself, so that's mental. I think he explains/draws out the matrix in all other cases.
As to your second question: Yes, and there are a lot of them and they mean many different things. My favorite, and the one that may show the most promise in understanding representation theory, is that the trace of a matrix is the sum of its eigenvalues. In fact, this can be generalized with the determinant with the knowledge that the (negative) trace is the (n-1)th order term in the matrix's characteristic polynomial (en.wikipedia.org/wiki/Characteristic_polynomial#Properties), and the determinant is the 0th order coefficient. The determinant itself is the product of the eigenvalues; in fact, every term in the characteristic polynomial is a sum-of-products, or elementary symmetric polynomial (en.wikipedia.org/wiki/Elementary_symmetric_polynomial), of the eigenvalues; e.g, for n=3, the trace is e1 + e2 + e3, then the next term has coefficient e1e2 + e2e3 + e1e3, and finally the determinant with e1e2e3, up to a sign. (The sign can be determined with the other formula on the characteristic polynomial page, to do with exterior product; I like the geometric algebra version, but in ANY CASE this is three digressions deep.)
There are other excellent interpretations of trace; there are a handful more on the wikipedia page (en.wikipedia.org/wiki/Trace_(linear_algebra) ) but I think this one is also cool th-cam.com/video/B2PJh2K-jdU/w-d-xo.htmlsi=ZNepGTALFCIjhF9Q.
Is there a text to accompany these videos?
Thank you for your awesome lectures! (and work)
Sir, Which book do you recommend for learning Representation Theory?
"Linear representations of finite groups" by J.-P. Serre.
@@richarde.borcherds7998 thank you sir
Why can't we classify all groups? Is there a proof of this?
There is a subject called classification which studies the question of when we can classify all models of a theory. If I remember correctly group theory is one of the theories where you cannot classify all models.
A simpler answer is that the groups we know are such a complicated mess that it seems very unlikely that there is any reasonable classification.
Groups acting on mathematical objects are interesting
Classic mann
Excuse me sir, if you don not mind how can I contact you please
ye
Go Bears