Representations of Finite Groups | Definitions and simple examples.
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- Опубликовано: 8 сен 2024
- We define the notion of a representation of a group on a finite dimensional complex vector space. We also explore one and two dimensional representations of the cyclic group Zn.
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Hey, just wanted to say your content is excellent. I was a math undergrad and seriously considered doing a math doctorate. Instead, five years later I run Data Science R&D for a financial services firm. Your content brings me back to the good old days 5 years back, and its a really nice way to relax after a hard day. Keep up the good work, and if you set up a Patreon I will absolutely be the first to signup.
How much math do you use as a data scientist? Do u use abstract algebra, functional analysis, differential geometry, topology ?
@@lisaking3996 Quite an extensive amount. Probability theory, Linear algebra, and functional analysis is admitted our bread and butter, I’m regularly setting up Eigenspaces for a PCA or designing adjoint operators. Dynamics is great for certain. Agent based simulatoons. Topological theory is surprisingly good for making certain regulatory filings about how “new” or “similar” an algorithm is - I’ll use homeomorphisms to show our implementations of certain models, while secret and proprietary, are similar enough to open source implementations to not require extensive testing or rework. And abstract algebra is incredibly useful for writing patents.
I would even argue that most data scientists are algebraists in our ultimate method if not practice.
@@Ttarler really interesting. I really love topology and algebra. But I have never heard of using topology and algebra in filing patent and showing similarity between algorithms. Can you provide links to some references from which I can learn more details? Thx
@@Ttarlerthat sounds awesome. Im really interested in this. Pls send info
I was reading something on Dirichlet’s theorem after one semester of representation theory, the application of character theory in number theory is really amazing!
There is any website aur youtube channel dealing with detail on representation theory
This is basically the newest topic in which I've become interested, and lo and behold, you post a video about it. Thanks!
I like you used linear algebra knowledge especially at the end of the video! Nice work man!
thanks for your videos very useful I am very thankful to you for explaining the concert so simply . It is probably the simplest way I found the explanation of representation in group theory. Thank you so much sir I am obliged to receive your guidance🙂🌷🙏🏻🙏🏻.
Thanking you,
Yours sincerely,
Shreyansh
I don’t know why, but I love representation theory, even this simple introductory stuff
I love this area of mathematics. I look forward to the other videos.
This is exactly the topic I've been trying to learn the past 6 months! Thank you thank you!
Any book resources to expand on this? I have a degree in physics so math isn't a problem.
ICTP has a RUclips playlist of lecture videos on Representation Theory. One good book is Groups and Characters by Victor Hill. Also Artin's book has a good chapter.
A book I love is Sagan - The symmetric group : representations, combinatorial algorithms, and symmetric functions
I am keen to study the representation theory. This video helped me to under the basics. Nice job!!
Representation Theory is such a fine topic. Severely enhanced by the fact its name is so fitting.
Very nice. Thanks.
Very good !
Rotation matrices are also good ways of embedding the dihedral group.
"Underscore" is a curious way of saying "subscript"
More of this please
Great video! What is this series goal?
Could this be GL_2(R) -- also 2-dimensional but different vector space? Thanks
Yes. I was wondering why he made it GL_2(C).
there is no difference; it's a matter of choice but it's better to work in ℂ. The dimension in ℝ is just doubling that of ℂ.
Since 𝑈(𝑛) and 𝑆𝑈(𝑛) are defined using complex matrices (as a group of unitary 𝑛×𝑛 matrices), it's IMHO the natural thing to consider it as a subgroup of 𝐺𝐿(𝑛, ℂ)
@@Zantorc ℂ is more general and later you will know that every finite group has unitary representations
Good lectur!
You rock!
I have a question about the matrix of Phi_1 when you do the 2 dimensional representation of Zn. Don't you need to have defined the notion of angle in C^2 to work out the rotation matrix?
At 2:35, in the definition of the trivial representation, the identity is not the identity element of V, it's the identity operator on V, right? Usually, I guess people write L(V) instead of V. Anyway, its a choice of notation if the context is clear. :)
Excellent channel! Keep up good work! You need to setup a donations page - I would gladly donate!
exactly !!
You say we "will explore in the upcoming videos" but perhaps next time you could tell us what the phrase/word for this connection is? You are leaving me curious and I don't know how to google this.... thank you
Look up reducible vs irreducible representations, representation isomorphisms, and direct sums of representations
@@noahtaul very useful thank you
@@noahtaul yea I was wondering if Euler's formula was closely tied to the rotation matrix which seemed really interesting to consider
at 9:16 wouldn’t it be phi(m)?
Plz upload more videos on representation theory bruh
You say "underscore," when some mathematicians say "sub." Any differences?
I am only intermediate in mathematics and I am not advanced enough to maths on board. I love learning notations.
I still don't get it. I thought representation was the set where some group act on.
Why does he say 1 plus 1 equals zero at aroubd 3:15 in the video?
Because he is working in Z/2Z under which the addition must be understand (mod 2). Basically 1+1=0 (mod 2)
Please solve .
It is given that
M = 1+1/2 +1/3 +1/4 +....+1/23= n/23!
where n is a natural number.
What is the remainder when n is divided by 13?
please make video...
Multiply by 23! Then the only term without a 13 on the top is 23!/13, so 23.22.21....14.12 ......3.2.1. Note that 14 to 23 equal 1 to 10 (mod 13). So answer is 12.11.(10! Squared). And note that 2.7=3.9=4.10=5.8=1 mod 13. So total is (12.11).(6.6)=(2).(-3)=-6=7 (mod 13)
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