The Coxeter Classification 2/2: Who cares about Representation Theory?
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- Опубликовано: 29 июн 2024
- A more elegant approach to Coxeter systems.
Video 1: • The Coxeter Classifica...
00:00 Opening
01:49 Part 1: The geometry of reflections
05:38 Part 2: The general geometric representation
14:44 Part 3: When is a "reflection" actually a reflection?
26:17 Part 4: Completing the classification (again)
36:04 Conclusion, and connections to other things
The textbook mentioned in the video is Reflection Groups and Coxeter Groups by J. E. Humphreys. The book covers everything in a lot more detail, but the language used and the order in which things are proven is quite different.
Music is made by myself, except for the bit at 19:12 which is by Kevin MacLeod.
I AM NOT A LIE THEORIST BUT I AM STILL GOING TO SHOUT AT YOU!
I prefer truth practicioners more, tbh
Groups, like men, will be judged by their actions, not their words. That is why we should study their representations, not their presentations.
Incredible video.
"A couple of applications in physics." Oh boy, understatement of the century hehe
39:41 Mathematicians are just apparently not allowed to have boring names. Two more of my favorites are the Tits group, and Doob's theorem (relevant in probability theory).
Also my first formal foray into abstract algebra was a course taught by Victor Kac I took a couple years ago - he's a nice guy!
It's so crazy how so many things in math are so interrelated. I actually came at this topic from a completely different side. I came across a lot of this when studying quiver representations in relation to persistent homology. Which is where ... Dynkin diagrams show up! The whole video, I was like this is strangely familiar, and then you brought it together in the end for me.
I did some TDA, but never came across Dynkin diagrams. I'd be very happy if you could elaborate !
@@mathephilia Haha, yeah it's a bit much for a youtube comment, but you can treat persistence intervals algebraically and study their quiver representations. vrs.amsi.org.au/wp-content/uploads/sites/84/2016/03/Ainsley_Pullen_UQ.pdf This is a 12-pager where right on page one you can see them linking persistence diagrams to dynkin diagrams. Scrolling through, it looks like a decent overview. There's a whole textbook on the theory too by Oudot.
It seems like if you end up studying anything that ends up touching group theory even tangentially, you'll invariably find examples of most any manner of group you care to name...
Cool, more stuff about Coxeter groups and--WHY DID I NEVER LEARN THAT MNEMONIC FOR MATRIX MULTIPLICATION IN MATH CLASS?! THAT'S AMAZING!!
OH, P.S.: I would be thrilled to see your explanation if you ever get into the errata here, like Lorentzian/hyperbolic root lattices (and the infinitely many misbehaving "Coxeter systems" left) or more detail about root systems and Lie groups/algebras (IT IS SO HARD TO FIND GOOD SOURCES ON THAT OTHER THAN WIKIPEDIA and tbh wiki is only helpful if you already know the topic in absurd detail). I would love to be an algebraic geometer, but I need either a miracle to pay for me six figures for college again or more explanations on the internet as crystal clear as this one! Again, I really really appreciate what you're doing, and whatever your next video is about KEEP UP THE GREAT WORK!
35:27 The truncated octahedron also tiles 3D space.
The linear algebra toolbox really is wonderful. Heres one of my favorites. A basis is linearly independent and span the space if 1) there are the right number of basis vectors and 2)a linear combination of the vectors add to 0 iff the components of every vector are 0. This is super nice in showing that for example partial fraction decomposition is unique because you can just show that any given partial fraction decomposition has a single point where it is nonzero.
Having only very little higher math this and the preceding video was really interesting. It basically did cover all the mathematical background of the symmetry in crystallography, which is my bread and butter with material sciences. I did space out a bit during the middle part with all the higher math terminology (one should have a dictionary at hand), but still recognised the linear algebra stuff, the deteminants and how they are calculated, even matrix decomposition from my numerical math courses. Outstanding work, catering both to pure math people and interested laymen.
Worth noting that non-affine infinite Coxeter groups do also have objects that have representative symmetries, it's just that those objects are in hyperbolic space.
I noticed that the determinants of the C matrices seem to relate to the curvature of the shapes or tilings of they represent, does this mean that the C matrices with negative determinants represent something hyperbolic?
Yes! And this actually gives a very nice way of visualising Coxeter systems with 3 generators: take the 3 generating reflections to be the edges of a triangle and build out a triangular lattice with 2m triangles to a vertex, where m is the braid length between the reflections through that vertex. If the system is affine then you get flat geometry like in affine A2, if it’s finite then it wraps round into a sphere, and if it’s infinite and non-affine you get hyperbolic space
THIS WAS SO GOOD!!! THANK YOU!!! I've been struggling with this topic for a long time, and this was the most concise and clear video on the topic that I have ever found!! I am so grateful that you added this to the digital library (this would have helped SO MUCH when i was first starting the study years ago, I have no doubt it'll make a huge difference for somebody on their way)
26:43 a minuscule correction - Minors are determinants formed using certain matrix entries. So we don't need a word "determinant" there.
what an absolutely delightful pair of videos, thank you for making them!
Hey! Just wondering, what editing/animation software do you use to create these wonderful presentations? They're always such a pleasure to watch by the way, and super informative!!
Thanks! The animated text is made with Manim, while the 3D animations are made in Blender.
@@josephnewton Thank you for letting me know!! :D
Out of interest, what did you use to edit the whole thing together?
@@LunizIsGlacey I just use iMovie, the program that comes with Mac computers. It's very straightforward but also very limited, so for compositing videos I sometimes had to use Blender's video-editing feature, or just carefully position things in frame before green-screening. This video was a real hodgepodge with all of the different sources!
@@josephnewton Thank you again! Good luck for all your future videos!
Keep going & give us more! (: I'm trying to do some combinatorial calculations using sympy's group theory modules atm and first got lost and then stuck. This video was really super inspirational and gave me some new ideas and avenues to pursue.
This channel really has been one of the better finds from some3 for me! I love that you are willing to make videos aimed at people who have taken some university level maths :)
Btw, as a physicist I'm really intrigued by this connection to Lie Algebras, though the appendix of this videos slightly flew above my head since I only ever learned Lie algebra from a quantum field theory textbook mostly concerned with U(1) and SU(2) & SU(3). I bet your animation style would fit a video aimed more at Lie algebras very well and there is somehow both a drought of accessible maths explainers regarding them a huge number of people interested :D
I was waiting for this to come out.
This was a fantastic pair of videos. Thank you endlessly :)
Love your quality as always
Wonderful! I've been wondering what those diagrams mean since I saw them on Wikipedia. Thanks for the explanation!
Correct me if I am wrong: at 30:35 you say that it is easy to see that there cannot be any cycles, degree-4 vertices etc from the combinatorial description, but not so easy from the positive semidefinite description. I think it is very easy. All minors of a positive semidefinite matrix are positive semidefinite. The cos-matrix of, say, a cycle is not positive semidefinite (because it is not a valid diagram). Hence a valid diagram cannot contain a cycle because this would correspond to a minor that is not PSF. Same for all other forbidden substructures. What I am saying is: it is easy to see that valid diagrams are closed under taking subgraphs.
That’s true, it’s not too hard to translate that part of the proof to use the determinant property instead. But the fact that finiteness is closed under taking subgraphs shouldn’t even need proving; the generators in a subgraph generate a subset of the whole Coxeter group, so if the whole group is finite then so is the subset automatically.
The more significant part of the proof is verifying that the matrices corresponding to cycles, pairs of edge labels, etc. do indeed give non-positive determinants. While that’s relatively straightforward, I think it’s more direct to just give a basic method to generate infinitely many elements in those cases. IMO you shouldn’t excess machinery where you don’t need it, although that’s subjective.
@@josephnewtonif you're using the basic method of generating infinitely many elements you also have to prove the word property though.
These affine Coexeter groups of affine Lie algebra define different lattices and could be useful in lattice field theories. Using lattices solutions can worked out for statistical mechanical processes like those of quantum mechanics.
So excited🎉
Great video!
Oh my gosh, this video is an odyssey. Yeah that is probably the fitting word for it.
This is extremely helpful, thank you! I hope you will cover Lie type finite groups in future as well.
Also, I wonder if it's useful to look at representations in Clifford algebras, since they give a very natural double cover of reflection groups.
Every Lie group can be written as a spin group, which is a certain subset of a Clifford algebra. Check out the paper "Lie groups as spin groups" by Doran, Hestenes, Sommen and Van Acker.
I've looked up Wilhelm Killing in Wikipedia just to be sure that he isn't a distant relative of Claude Émile Jean-Baptiste Litre ...
Ah yes, every mathematician (and engineer, computer scientist, ...) should have a Linear Algebra Toolbox™ in the house.
this made simultaneously all of the sense and none at all, but that's just maths for you
25:48 hey! That's me!
The visuals are beautiful! Are they plain LaTeX/Beamer?
Oh, just saw it, it's Blender/Manim. Nice!
Around 17:00 why doesn't
The reflection plane also move to be perpendicular to the vector?
I thought the releflection planes were defined using the normal vectors?
The concept of 'perpendicular' doesn't always exist in abstract vector spaces as explained around 17:27, and the functions aren't necessarily perpendicular reflections anymore at this point. So in the 3D animation I'm taking the reflection plane for u to be the set of vectors that are left unchanged by u, where u is the reflection function corresponding to that root vector. This was maybe something I could have explained a bit better in the video...
Hello. What programs do you use for your videos? I am a new science based channel.
I use Manim for the animated text, and Blender for the 3D parts
@@josephnewton thank you
hang on is this visualization of infinite coxeter systems related to petrie-coxeter polyhedra? I recognize the tilings from the jan misali video on all 48 regular polyhedra
Not that I know of; the symmetry groups of those ‘polyhedra’ are definitely affine Coxeter groups, but it’s not a one-to-one correspondence. For example, the infinite tilings of triangles and hexagons have the same symmetry group (it’s affine G2, since affine A2 only includes reflections along the edges of triangles), so the Coxeter group doesn’t distinguish between them
@josephnewton ah ok, interesting stuff
@@josephnewton I'm pretty sure the symmetry groups of the infinite polyhedra aren't necessarily Coxeter groups.
But why does James Joseph Sylvester have three surnames.
only 6k views, I'm baffled
25:31 not gonna lie
notch maxecemi quand il mange une golden apple :