What's a representation? An intro to modern math's magical machinery |
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- čas přidán 1. 05. 2024
- This video is an introduction to the representation theory of finite groups. It is pretty dense, but I did my best not to include much heavy mathematics. I'll leave some links below for some more background on several of the topics in the video. None of these are necessary to understand the video, but they're definitely helpful and a good place to branch out.
3B1B's intro to group theory: • Group theory, abstract...
3B1B's Intro to linear algebra: • Vectors | Chapter 1, E...
Borcherds' representation theory (advanced): • Representation theory:...
Quanta vid on Langlands: • The Biggest Project in...
This video was made in large part for 3Blue1Brown's Summer of Math Exposition 2 #some2. Thanks to Grant Sanderson, James Schloss, and everyone else who helped organize it. See the main video for more information here: • Summer of Math Exposit...
Timestamps:
0:00 - Introduction
2:27 - Groups
4:54 - GL(V)
5:54 - bird
6:01 - Linear algebra crash course
8:48 - Cyclic group representation
12:20 - Symmetric group cycles
14:15 - A5 representation
18:25 - Summary
Credits:
Thumbnail has pics from nature and thewalrus (illustrator Jonathan Dyck)
Images of cubes and dodecahedron come from robertlovespi, who has an amazing site about tons of polyhedra
bird: • Crow searching for a n...
end credits bird gif from user arashigul on gifer - Věda a technologie
"we even teach matrices and linear algebra to cs students, so you know its not that bad" 🤣🤣🤣
Seriously though wonderful video, really gave me a better notion of what representations are. Thanks for this.
mf got us 🤣🤣🤣
Great video!
The reason the golden ratio φ appears in the A5 representation is that the geometrical construction of a regular pentagon involves the construction of a line segment of length φ.
thanks!
3:10 196,883 (the dimension of the smallest nontrivial representation of the monster)
Isn't that the number of elements of the Monster group?
18:45 "We even teach matrices and linear algebra to CS students so you know it's not that bad"
BASED
😳
Good video. However, I think at 11:19 you 'give up' on matrices a little too quickly. In fact, your choice of representation starting at 9:29 is already more complicated than it needs to be. A really useful and important fact -- especially for wrapping one's head around groups and/or representations -- is that: All finite groups are subsets of some Permutation (aka Symmetric) group. In other words: All groups can be represented as simple permutation matrices which *_only_* contain 0s and 1s! (Even more, each column contains exactly one 1, and each row also contains exactly one 1.)
This is really useful, since permutation matrices are very simple to compute with and use as examples. For example, for C4 (at 11:19), you could simply use these 4 matrices:
rho(0) = the identity matrix,
[1000
0100
0010
0001]
rho(1) =
[0100
0010
0001
1000]
rho(2) = rho(1)^2 =
[0010
0001
1000
0100]
rho(3) = rho(1)^3 =
[0001
1000
0100
0010]
These four matrices are a subset (subgroup) of S4 (the permutations of 4 objects). Yes, they require 4 dimensions, but they are incredibly simple. To reduce the number of dimensions, you have to do clever things like using negatives, nth roots, complex numbers, etc. That all takes cleverness to figure out. But if you just want to jump into groups/representations without needing a whole lot of background, using subsets of Sn (for appropriately sized n) to represent your group elements is the easiest way to do it.
It allows you to get a representation that *works correctly* right away, without much fuss. And if you have a more 'reduced' representation that is hard to understand or that you screwed up some how, you can fall back to subsets of Sn as a kind of 'debugging' aid to help you understand how the group should really work.
The only real drawback of using permutation matrices is that their dimensions are usually bigger than technically necessary. They can get out of hand for high-degree finite groups, but by that point you can start figuring out how to reduce your representations, and that's where all the complicated representation stuff starts happening. But you don't need that stuff right away!
To prove my point, for A5 at 14:45, you could instead just use the subset of S5 directly as 5x5 permutation matrices. Example, kappa((345)) would just be:
[10000
01000
00010
00001
00100]
Now, isn't that much nicer to introduce to somebody just learning about representations than the monstrosity with all those negatives and 1/2's and phi's in the 3D representation? Just sayin'!
Thanks for watching and I appreciate your comment!
I do think you're right. Permutation representations are certainly easier to understand, especially in practice. And Cayley's theorem certainly has a lot to do with that. When we learn to construct representations, we usually can't just come up with the lower dimensional irreducible ones. However, on first thought I didn't want to present a representation that had a higher dimension than the geometric object I was trying to connect it with.
I'm hoping I can turn some of this feedback into more content on representations, because it deserves a proper treatment. There's so much more I want to cover and it's really challenging (and fun) to balance that with the goal of keeping it at an introductory level. I love seeing others' takes on it as well. Cheers!
For what it is worth, I think it is important to include a "non-trvial" example to make representations more interesting
I'm a CS student and I agree with the passing remark... they never push us hard enough.
I really like advanced topics made more accessible like this, but not any simpler than it should be. I only have a minor in math from uni and this was the perfect level for me, so thank you 😊
I loved watching this as a math enthusiast and programmer, please continue :)
Amazing, it's great how you emphasize the importance of maps to more than just functions
Groups must also have associativity. [(a x b) x c = a x (b x c)] (2:51)
Besides all the other mistakes that were already pointed out (missing associativity in the definition of groups, confusing fields and vector spaces in the definition of GL_n(F) / GL(V), subset symbols instead of ∈ at 5:48...), the derivation of the symmetry group of the dodecahedron having 60 elements is also completely false - you overcount by counting rotations by 0° around different faces as different elements, as well as rotations around opposite faces, while completely neglecting rotations around vertices and edges (both of which leave no single face in its place, and aren't part of the rotations you already counted).
Don't get me wrong, I appreciate that you've taken the time to make this video to help more people get into representations, and you definitely did do a lot of things right - but next time maybe have someone else take a second look over the script before making the video, otherwise mistakes like those will inevitably happen and distract from the knowledge you're actually trying to share.
Honestly, I'm not mad he glossed over that many things, I knew about group theory, fields, galois theory beforehand and I didn't feel it was necessary to go that much into detail, when the whole point of the video was proving the power of turning abstract algebra into easier matrix manipulation. The thing with the symmetry group of the dodecahedron was a screwup, I'll give you that
I might be missing something, but pretty sure the symmetry group of the dodecahedron has 60 elements, 120 elements including reflections. Richard E. Borcherds uses the same reasoning to come to symmetry group of 60 elements for the dodecahedron. It's in his group theory playlist!
I find it interesting that the faces have order five symmetry & there's twelve faces so 5*12 = 60. Rotating about a vertex has symmetry of order three and there's twenty vertices so 3*20 = 60. Rotating about edges has symmetry of order two and there's 30 edges so 2*30 = 60.
If you don't like that reasoning then here's some different logic that I found with a quick google search!
The elements are:
4 rotations (by multiples of 2π/5) about centres of 6 pairs of opposite faces = 24
1 rotation (by π) about centres of 15 pairs of opposite edges = 15
2 rotations (by ±2π/3) about 10 pairs of opposite vertices = 20
Together with the identity this accounts for all 60 elements.
amazing video, that break with the birb was timed perfectly
This video was what made me start studying abstract algebra! Thanks for making it!
The five cubes animation was great and seeing the dodecahedron was crazy.
This is a cool video. You explain things in a way that I can actually understand. Thanks
great content. i like how you are able to simplify such complex subject into something easy to digest.
Isnt the group operation necessarily supposed to be associative as well?
Correct
and closed
Not always. You need a broader object like a ring
Yes.
Yes.
Amazing first video, i am already introduced in the topic, but i can still feel how good of an introduction this video is, thank you for this educational piece.
I like the format: dry, informative, good clear illustrations.
The way this creator has begun to define his channel is hinting me towards an isomorphism to greatness.
This is great, I've had a hard time coming to understand group theory, and your video is one of the best I've seen.
Love the Video, I currently am learning some Modern Differential Geometry where Lie-Group Representations are everywhere and I found it hard to appriciate those. Learned to know better today, definitely have a deeper appriciation for those now!
Absolutely amazing video! Subscribed.
Great video! Looking forward to many more
appreciated this video, I too once tried to do an "intro to representation theory" talk as part of the final project for one of my classes and failed. The specific thing I was struggling to understand and still don't fully get is that most proofs that graphs have certain expansion properties (and sometimes how markov chains mix) in theoretical CS involves using representation theory to analyze the eigenvalues of a matrix that is the adjacency matrix of a graph but also somehow related to a group
This was just right for me. Thanks for the good intro to this subject.
Amazing video, wish I had this when first learning groups for motivation
Exceptionally good content , make more please
wow, that construction with 5 cubes is neat!
I dropped out from the group theory course just before they introduced representations so it was really enlightening to finally understand what's that all about. xD It's so interesting, this idea of mapping difficult stuff to easier stuff is even quite philosophical... 🤔
Good explanation, but I really wish you had mentioned that groups are required to be associative. It’s perhaps their most important property. Associativity is the only reason you’re allowed to think of the operation as a transformation so that representation theory makes sense. What you described is technically called a loop.
Thanks for the video man. I saw some comments pointomg out at your mistakes, I just want you to know that it's not that big of a deal for the uneducated public. I personally lack a formal education on this topic (only lineal algebra) and now I feel like I can come to understand it better with self study. This video values clarity over rigor and I'm thankful for that, it's not supposed to be a science article after all.
Although maybe not the most rigorous treatment you maintained my attention and attracted me to a subject that I thought would be a lot more complex than it is, at least the gist of it.
Excellent video! Thank you.
Excellent job!!!
Actually, this video is wonderful, and I thank you very much for this effort, but I expected more, and I am still waiting for more of your videos, I know that it is very difficult, so thank you
Truly excellent.
Mind blowing 😮
Splendid! I understood almost everything but the mapping part though
Great stuff
That is very nice ..very important..very clear.. Thank you
Thanks a lot! It works for me!
This was a brilliant video - super engaging! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)
Great video! One small nitpick, at 8:30 I think you made a typo with the matrix multiplication shown at the bottom of the screen. At a_21 I think you meant for it to be 1 not -1 b/c the resulting matrix from what you have yeilds (-2,-5) not the desired (-2,-1).
excellent intro, my compliments. pity that you did not continue further into the topic.
hi, wonderful video!! what is the font you're using? I love it
Wonderful!
Now I want a Megaminx-shaped(Dodecahedron) rubics cube where you can turn only along the internal cubes!
But really good video!
Great video.
Fantastic!
Amazing video
I’d recommend watching Another Roof’s series on Set Theory before this, as I noticed I was constantly thinking about those videos to make sense of the beginning of this one
Maybe a technical detail you could mention is, that a vector space is more abstract and can be fairly easy defined by a few axioms, or even from the group axioms. And R and C are not the vector space V, they can be the fields over which the scalar multiplication is defined.
I hope you make more videos.
post more videos please!!!
Great introduction to representations! Indeed, transformations from one mathematical field to another are exremely important. Subscribed!
Thank you for the video, I found it very helpful at my level of math self study. Ignore the negative energy from the nit pickers
For your next video you might wanna:
1) think about how much each part of the video adds or takes away from the whole, and if it adds, but doesn't add a lot, maybe it's not worth it if it's not integral to the concept
2) use keyframes and the gain setting, or the cutting tool, in a video editor to take out any noises
3) show, don't tell, when you're trying to get a point across
glhf
Please do more on the fermat thing. 😃
In addition the other errors you're probably tired of reading: get a pop filter for your mic. You were thumping the mic whenever you made a T sound. And then make another video, this was GREAT.
What’s the link between group homomorphisms and topological homeomorphisms? I mean they sound similar, and one professor on yt described homeomorphisms in the same way you described homomorphisms, which is that they allow you to deform a difficult problem into a simpler one, and solve the simple case instead.
16:35
edit:
ohh i just read the comment of Peabrainiac, ok, to exclude overcounting of 0° rotations, and include rotations 180° around edges,
1*{0°}+2*V/2+1*E/2+4*F/2=1+2*20/2+1*30/2+4*12/2=1+20+15+24=60
ok now it's back to being good, point was there was miscalculation
unedited:
rotations of dodecahedron sequence of 5 rotations around 12 faces, but opposite faces are parallel which means for every 1 rotation there is double counted rotation form parallel face, so 5*12/2=30
but there are unaccounted 30 rotations around vertices, sequence of 3 around 20 vertices with double counting the the opposite, so exactly missing 3*20/2=30
awesome vid so far :D
Opposite faces on a dodecahedron don't line up, so there are 10 rotations that matter, not 5. So you can look at it as 12x5 or as 6x10. If you start looking at vertices then you have an isocahedron, which also has 60 (20x3 or 10x6) as it's the dual of the dodecahedron.
@@SkorjOlafsen (edited) rotations from opposite faces do not need faces to align to be the same rotation. important part is axis of rotation and angle of rotation. the dual doesn't change the matter that dodecahedron has symmetries by rotating around faces, vertices and edges. It still has 5 (0°,72°,144°,216°,288°) rotations around each face double counted, 3 (0°,120°,240°) around each vertices double counted and 2 (0°,180°)rotations around middle of each edge (axis goes perpendicular to the edge to center and to middle of opposite edge) also double counted .
if we inscribe icosahedron then we have 3 (0°,120°,240°) rotational symmetries around each face double counted, then 2 (0°,180°) around edges and 5 rotational symmetries (0°,72°,144°,216°,288°) around each vertices double counted which gives us 1*{0°}+(3-1)*F/2+(2-1)*E/2+(5-1)*V/2=1*1+2*20/2+1*30/2+4*12/2=1+20+15+24=60
the same 60.
First time in my life I could understand something in maths.
14:53 I'm crying "No I will not explain"
"We even teach it to CS students" lol 🤣
Now seriously, nice video. I always found this topic fascinating and I have seen both videos you cited and none was satisfactory enough. Too shallow or to fast into the abyss. Your video had small scope but the right pacing.
I would only try to fix the audio for the next one. I'm sure there is a way to filter that high pitch tone. It was a little distracting.
Either way, it was a good one. Congrats!
That was basically my linear algebra 1 course))
How can we find the result matrix from the input group?
cool good job
so the idea of langlands is to have representation of different types of numbers into geometries?
Great video, I learned a lot, but I did find some errors.
Some errors:
2:28 - The group definition requires that the operation on the set is associative.
5:45 - A vector space V is not just R or C, in fact, these are usually what vector spaces are over (fields). Every finite-dimensional vector space does indeed have a matrix representation given a certain basis, but V can be infinite-dimensional as well. This is more a technical note, focusing only on GL(R) and GL(C) is totally fine!
Because the vector-space-over-a-field-ness is kinda baked into the linear group, I've always seen it as GL_n(F). Maybe Wiles' paper uses a notation where F and n are already collapsed into the prebuilt vector space V.
Funnily enough, as much as throwing different size square matrices together may seem nonsensical, I've seen it done! When calculating the actual matrix multiplication, you could extend the smaller square matrix to the size of the larger one by filling new cells with the elements of an identity matrix. That could lead to a dimensionless GL(F).
It's unfortunate that this notation technically overlaps with GL(V), because any field is also a vector space over itself, but it should always be clear from context what's going on.
@@ilonachan I agree that GL(F) is the more natural choice. However, the video uses GL(V) where V is a vector space.
@@rjthescholar177 oh I totally get it! Just wanted to share this thing I learned about in a seminar recently that I thought was cool, because the notation reminded me of it. GL(V) makes a lot of practical sense though, maybe I'll just write GL(R³) or sth in the future...
I did a video on the three identities of zero.
Nice
The acting in the beginning was really cute ^v^
the crow break 👌
Hi there, is there a way I can contact you personally (for example, a DM on Twitter or an email address)? Great job.
Thumbs up
I thought that all possible groups had been classified. It's one of the biggest pieces of work in mathematics that took over 30 years but it's now complete.
Yes, we have given names to them all. We can say "this group is that kind of group", after 30 years of work. Group theory is hard.
I think you have a typo at 8:44, the first column of your basis vectors should be (-1, 1) not (-1, -1)
👍
Bird does cool wooo sound. Wooo.
18:49 i feel attacked
the prequel to pascals triangle
Don't know why youtube pushed this video to me, maybe because I watched a bunch of videos on AdS/CFT Duality, which could be a good example of representation.
42
Is the bird okay?
I think you made a mistake at 5:26. V would be R to the power of the dimension of the vectorspace, i.e. the number of rows and columns of the matrix. So the correct way of saying htat would be that the vectorspace has the ground field R or C.
There is a general phi function for each prime with a novel sequential difference. The square root of the prime, plus a counting number, divided by that sequential difference. For instance phi, as the square root of 5 is the first twin, is divided by 2, returning a cycle of two mantissas. Eleven is the first square prime and its square root, plus a counting number, when divided by 4, yields a repeating cycle of four mantissas. Likewise for 29, the first sexy prime and its recurring cycle of six mantissas. Despite having an unknown finite limit, it is guaranteed that sequential differences among primes climb to at least seventy million, we find there would be so many mantissas in its cycle, too. We can see the universal matrix producing the self-similarity to manifest integer abundances out of this complex array.
You got 41 k views on your 1st video
'seems like its a legit paper... it's in the annals'.... lmao
How the youtube know that I am studing representation?
Me-: I finally found a video on CZcams that explains the FLT proof
My mind:_ It's been 20 minutes and there are only 2 minutes left.
Me-: I just have to be patient, maybe he will explain the proof in the last two minutes.
Video -: And this was what the arrow you see in the proof means
hhhhhhhhhhhhh
feel like everyone knows this shit but its impossible to put it into words
liked because of crow
The low thumps in the audio make the video really hard to concentrate on
Yantra Chakshur Vidya
Nice