This is a BRILLIANT video. It's well motivated, with the classification of finite simple groups lurking in the background, and with the tangible examples. It's complete and cohesive, making the general classification argument visual and crystal clear while sweeping the finicky details under the rug. And it motivates you to understand the higher math of representation theory to have a better insight into the whole problem. As a math PhD student who has taken a graduate course in representation theory, this is the most fun I've had with groups in a while, maybe ever!!
This was a wonderful dive into a specific part of group theory that'd always gone over my head -- I am THRILLED for the representation theory!! I'm impressed at how well you described this with such accessible mathematics/simple tools!!
One of the most interesting math videos I've seen lately! Also your presentation of these is basically perfect, both the sterile LaTeX graphics and the cardboard cutouts. (Also have to applaud your comedy, that last "try to figure it out tune" at 30:27 got me laughing) Looking forward to the second part!
I've seen a lot of group theory videos over the years while never actually trying to study it. This is by far the most digestible explanation I've seen. Well done.
What a journey this video was. I had to pause multiple times and ponder about what's being said because it was going a bit fast but in the end I did end up with a nice overviewing understanding of the topic. I always was fascinated by the classification of finite simple groups but never had the time and energy to study it properly so this was a nice taste of what that's like. The introduction reminded me strongly of when I studied formal grammars and the Chomsky hierarchy at the university. I guess Coxeter systems can be thought of as a type of a context-free grammar.
9:10 the name "Word Property" totally makes sense as the construction of representatives reminded me immediately of minimal automata and the Myhill-Nerode theorem, and I guess the minimal automaton is nothing else but the Cayley graph
Absolutely fantastic video! I loved it! 🤩 I am coming at Coxeter Groups from the geometric side, being interested in higher dimensional geometry. i *almost* understand most of these groups and how Coxeter-Dynkin diagrams work, but the branching groups have always blown my mind. It is really interesting and very informative to hear an explanation from a group theory perspective. The reason that the Demicube Group is Group D is because Groups B and C are dual to one another. Group B (The Cubic Group or Measure Polytope Group) is dual to Group C (The Octahedral Group or the Orthoplex Group). They are the same with the edge labels in reverse order. So a Cube is (*)4()-() or ()-()4(*) while an Octahedron is (*)-()4() or ()4()-(*). Since they are just the reverse of each other, Group C is a bit redundant. The F4 Group is my favorite for just the reason you mentioned - that there are no analogous shapes in higher or in lower dimensions! There's just somethin' special about 4D, I suppose. Gosset Polytopes, the E6-8 Group, just blows my mind. I cannot really conceptualize what these beasties must look like. I really like how you can use these Coxeter-Dynkin Diagrams to figure out the properties of polyhedra and higher dimensional polytopes! If you think of the nodes as "activating" the mirrors that each one represents, then reflections of a point across those mirrors results in (for example): the cube (*)4()-(), truncated cube (*)4(*)-(), cuboctahedron ()4(*)-(), small rhombicuboctahedron (*)4()-(*), great rhombicubactahedron (*)4(*)-(*), truncated ocatehedron ()4(*)-(*), and finally the octahedron ()4()-(*)! Anyway, thanks so much for your video - it helped to clarify a lot of things for me! 😁
I actually wanted to see the combinatorial side of these coxeter stuff and this video was great. Can't wait for the representation theory side. (Also that music 0_0)
The president of the university I attend (Free University of Berlin) recently held a talk on the 24-cell and one of its interesting properties is that it has the same number of 0-dimensional faces (vertices) as 3-dimensional faces and the same number of 1-dimensional faces as 2-dimensional faces.
And when you compute # 0-dim faces - # 1-dim faces + # 2-dim faces - # 3-dim faces you get 0 - as is the case for all 4-dim polygons and moreover for all even-dimensional ones. And the odd dimensional polygons give you 2 as alternating sum. These facts can be proven using the notion of the so-called "Euler characteristic".
This is greatly done. I wouldn't have thought that you can rule out almost all inifinite diagrams by just looking at ways to construct infinite sequences. I'm also relieved that this method was not feasible for E8. Otherwise, it would have been possible to distinguish with a human brain by the help of combinatorics between roughly 600 million words and infinitely many.
I've been working with Coxeter groups for months as part of writing the higher-dimensional twisty puzzle software Hyperspeedcube but never really understood the full picture. This video finally made it all click! Thank you so much!
28:45 I have some understanding of the 24 cell’s amazingness. You can construct it by attaching hyperpyramids to the cells of a tesseract; if you do the analogous thing in 3d, you get a rhombic dodecahedron, which is not completely regular but a very cool shape. It turns out to be more regular than you expect in 4D, ultimately because the vector (1, 1, 1, 1) has length 2.
as someone who has only been using these diagrams as a system to classify all those high dimensional symmetries and shapes it's nice to see some of the connections it has with less horrific things like Words
Another way to get simplifies from the cube (coordinates [0,1]^3) is slice it up into coordinate inequalities x ≤ y ≤ z, y ≤ x ≤ z etc. These form funny simplices of which there are 6 inside the cube. It's like cutting through the cube along the planes x=y, y=z, ... I think. Intersections / unions of the inequalities take faces / unions of the simplices. I believe this is directly connected to the general cube-simplex thing. The funny simplices you get this way are also related to associahedra: take one of them and label its 3 length-one edges a,b,c. Then the diagonal edges are ab, bc, abc, and the faces correspond to a*b=ab, a*bc=abc, a*b*c=abc etc.
I had come into this video thinking it might be about one thing I had done some work on as an amateur based on those graphs (I don't know what it's called, but they are graphs related to endgames in dots and boxes that can be manipulated in ways I forget to determine which player has a surefire winning strategy), but it ended up being about something that was related to another area of study I had done amateur work on, namely the group of permutations of doubly-magic squares of rank 4. It turns out that I was able to generate that group by four order 2 elements, though I think there were relations other than braid relations, so it wasn't actually a Coxeter system. But it was the same kind of thing. In those relational systems where there might be relations with 3 elements, it was a lot harder to determine when a relation was reducible, and it generally meant going back to the definition of the permutation and seeing if it equaled a simpler one once you actually rearranged the elements.
There are similar properties that make Coxeter diagrams containing the E7~ and E8~ represent infinite Coxeter groups. And similar things for the BDn and CDn diagrams. It all has to do with the distinction between spherical/finite Coxeter groups on the one hand, and the affine and hyperbolic Coxeter groups on the other hand. An important generalization is the compact Coxeter groups, that are "locally" spherical/finite, which roughly means all Coxeter subdiagrams of their diagram are spherical (in reality we should have sub Coxeter groups, which are sometimes "hidden" when looking at diagrams). These are either affine or compact hyperbolic. The affine ones are all classified. So are the compact hyperbolic ones nowadays.
Very pleased to see this video. Takes me tight back to my postgrad year. It was only a few weeks ago I stumbled across another video covering Hilary code and the Leech lattice.
I very much enjoyed this video. I also had no clue what was happening, ha. Any chance you could give, like. A middle school understanding of group theory? I'm missing a lot of background already...
Coxeter groups are useful for homotopy theory such as with n-simplexes (simplicial homotopy) and n-cubes because they are Salvetti complexes (CW complexes). Weyl groups are coxeter groups which at least suggestion a relation to the classification of finite simple groups like with the Weyl group of the infinite dimensional Kac-Moody algebra of the monster group.
I'm currently doing a project classifying high-dimensional polytopes, and after watching this video I almost wanna kiss my man Joseph. How do you only have 57.9k subs?
Is this an extremely sophisticated hyperdo commercial? Now I gets why lego nerds exist. This is an absolute bangers for math nerds. Is there a real E8 model? Take my $1000. For real.
Jesus man, 22:14 smacked me in the face at 1 in the morning, that is the LAST sound I expected lmfao Maybe I should have though, considering the baba from earlier in the video
31:25 if you label the nodes in the straight part xabcdefg from right to left, and then say h is the node above the e, would the sequence xabcdefghefdcedhefgbcdefhedcbabcdefghedcfedhefgbcdefhedcba, applied repeatedly, work? To be fair, I haven't checked it yet, I got this sequence by messing around with the coordinates of the actual root system but I'm not at all confident in my technique haha
I have no idea how to check that, but I’m impressed nonetheless! If everything after the x is the longest element in E8 then I think it should work; in the description of affine systems I give in the 2nd video I’m pretty sure the extra root for which you shift the reflection plane is always the lowest root of the corresponding finite root system, meaning its plane is the longest element of the finite Coxeter group. In this case that’s x and the finite system is E8, so if you start with that root then the longest element in E8 flips it to its negative, and then x flips it again but sends it further out, and then repeating sends it out infinitely in a straight line.
@@josephnewton That's pretty much what I tried to do, although my idea was to "recreate" the root x using the other roots (I'm not aware of much of the theory behind this, it just seemed like it would work)
Yep, that graph only shows the ‘parabolic’ subgroups which means they’re generated by the generators of the containing group. The Bn->Dn subgroup doesn’t fall into that category because one of the generators is aba, which isn’t a generator of Bn. Finding all the subgroup relations between finite Coxeter groups is much more complicated
It’s in chapter 3 of Combinatorics of Coxeter Groups by Björner and Brenti; that book goes into more detail on the generator description of Coxeter groups than the Humphreys one
What's the problem with prisms? I think the number of generators of its Coxeter symmetry group still equals the dimension. For example the triangular prism has diagram o-o o, so three nodes = three dimensions.
That's right, my mistake. The real problem would be pyramids (or prisms with some asymmetry that prevents reflecting them across their length), where the reflection group is nothing more than the reflection group of the base.
Here is my favorite related open problem: try to classify the inscribed polyhedra in dimension 3 all of which have all faces centrally symmetric (that is, they are zonotopes). As far as this is knows, there are 17, and they have a classification quite parallel to the Coxeter groups because they emerge from projections of root systems. That is, you are in dimension 3, but you are witnessing the FULL ADE-classification from all dimensions! Well, the conjecture is, that these 17 are all of them, but that is open, and probably very hard.
Lovely video! Packed full of fun references and jokes too. I've been fascinated by group theory for a long time but never really could get my head around the Coxeter groups (and Dynkin diagrams which I believe are related? Hopefully this can kickstart further exploration haha!). This is a really great introductory video and has demystified a lot of it! Definitely subscribing for the next one.
Brilliant! Should it be noted that 1) vertices i and j are adjacent if and only if m_ij ≥3 2) An edge is labelled with the value of m_{ij} whenever the value is m_{ij} ≥ 4.
