Very nice intro to representation theory. Can we talk about representations of groups in Aut(any object) or Aut( monoids) or even SL_2(Z); where Z is the set of integers? (I mean does it have to be always linear representation). I think the answer is yes but I do not know why we do not treat them in research as much as linear case!! Thank you!
Life of Brian yes, the rotation group SO(2) with real entries is isomorphic to exp(i*theta)=z for real theta, that is the unit complex numbers with the usual complex multiplication, i.e. { z complex ; |z|=1}.
Either by thinking of them as linear transformations that rotate and reflect a square onto itself (D4) or merely rotate it onto itself (Z4), or another approach is to use the left regular representation to get 8x8 permutation matrices for the elements of D4 and 4x4 permutation matrices for Z4 as in here: ruclips.net/video/lNozOtU02bs/видео.html
That's an excellent question. For finite groups of order N the answer is yes: the permutation representation always gives a degree-N representation just by identity matrices whose columns have been permuted according to the group's Cayley table. For infinite groups the question is more subtle!
These videos need to be watched and re-watched until every bit of detail is disgested. Thank you Matt!
Thanks for providing examples for understanding the representation of a group, it's the best intro I've watched
wow man this is mind blowing for me because i v never though about groups behaving this is so visual!! so beautiful great content man !!
Thank you so much for these videos!!!
WOW! this was terrific. Do you have any recommendations for introduction into representation theory, I'm considering studying this in graduate school
Very nice intro to representation theory.
Can we talk about representations of groups in Aut(any object) or Aut( monoids) or even SL_2(Z); where Z is the set of integers? (I mean does it have to be always linear representation). I think the answer is yes but I do not know why we do not treat them in research as much as linear case!!
Thank you!
its really helpful......plz make a vedio on FG module also
Thank you so much. Video was comprehensive and helped me a lot
This rotation in 90º might have something to do with multiplying by i?
Life of Brian yes, the rotation group SO(2) with real entries is isomorphic to exp(i*theta)=z for real theta, that is the unit complex numbers with the usual complex multiplication, i.e. { z complex ; |z|=1}.
Wow that’s cool
Can we call 2nd matrix at 6:43 as phi(2)?
Ibrahim Hamim Indeed!
How do you know the matrices representing D4 and Z4?
Either by thinking of them as linear transformations that rotate and reflect a square onto itself (D4) or merely rotate it onto itself (Z4), or another approach is to use the left regular representation to get 8x8 permutation matrices for the elements of D4 and 4x4 permutation matrices for Z4 as in here: ruclips.net/video/lNozOtU02bs/видео.html
do all groups have a representation? and what the representation of z5={0,1,2,3,4}
That's an excellent question. For finite groups of order N the answer is yes: the permutation representation always gives a degree-N representation just by identity matrices whose columns have been permuted according to the group's Cayley table. For infinite groups the question is more subtle!
شكرا
It is difficult to follow as you speak too fast
You can reduce speed of video in settings