The Coxeter Classification 1/2: Combinatorics is hard

A very hands-on approach to Coxeter systems.
#SoME3

00:00 Opening
02:23 Part 1: Introducing Coxeter systems
10:14 Part 2: Proving things are infinite
15:23 Part 3: Symmetry groups of polyhedra
25:00 Part 4: The exceptional systems
32:18 Conclusion

Some casual overviews of finite groups, if you want a primer for this video:

If you're looking for more detailed proofs of results like the Word Property, I recommend the textbook Combinatorics of Coxeter Groups by Björner and Brenti which covers these in the first 3 chapters.

Small corrections thanks to commentors:
*The word 'prism' at 19:16 should be 'pyramid', or equivalently an asymmetric prism so that reflecting through the length is not a symmetry.

[MrSamwise]

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!!

[StylishHobo]

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.

[lexinwonderland]

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!!

[silentobserver]

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!

[Macieks]

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.

[sichelsam]

Very good video. Never thought group theory could be visually explained so appealing!

[tanchienhao]

The best video I’ve seen on coxeter graphs! Short and to the point. Can’t wait for the rep theory one

[JediJerry]

It’s amazing how just the right use of music is able to get me to burst into laughter in a video about combinatorics. Excellent video.

[maxqutekerman]

This is the most coherent introduction to group classification I've seen so far.

[toxicore]

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

[WildStar]

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! 😁

[HactarCE]

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!

[Nathan-czuk]

really nice video, i like the setup of doing it the 'inelegant' way first, then revisiting it from a cleaner perspective. looking forward to part 2

[Number_Cruncher]

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.

[username-]

I love that, great work and effort. I love your thought process and I wouldn't mind a video on All 5 platonic solids! Thanks so much

[nice]

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)

[smiley_]

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.

[LunizIsGlacey]

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.

[matematicke_morce]

Great video! I know these diagrams from Wikipedia articles about polytopes and have always wondered what exactly they mean

[poke]

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