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Adam Cross

Editor - Newspaper, Swing Vote Chronicle

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What is Root System, the Dynkin Diagram, and Combinatorics Mathematics?

I am not a mathematician I am a writer but this equation, from the bits I understand, might be relevant to a project I am currently invested in.

Can someone please explain this to me as though you are speaking to a 6-year-old (or an English major)?

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    Jun 1 2012: Combinatorics in mathematics consists of strategies for counting things. A question like how many different committees of three people can be formed from a club of twenty people is a question in combinatorics. Hoiw many different card hands of five cards drawn from a normal card deck contain at least one face card and exactly two hearts is a question in combinatorics.
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    May 31 2012: I am interested in this because I am working on an idea. I need support from a mathematical equation because I am finished with the other aspect and I have hit a wall. I believe that root systems and the Dynkin diagrams are just what I have been looking for.

    Thanks for taking the time to help me Toke.
  • May 31 2012: (Continuing my previous comment:)

    The dimension n of the space is called the "rank" of the root system. You can see drawings of all rank 2 root systems in the Wikipedia article "root system". You can also sort of draw rank 3 root systems (since we live in a 3-dimensional world, humans are quite good at understanding pictures of 3-dimensional things). But higher rank root systems are harder to draw. You sometimes see pictures of higher rank root systems, for example this picture of E6 (E6 has rank 6): http://en.wikipedia.org/wiki/File:Up_1_22_t0_E6.svg. This is not really a picture of the root system (but rather of a "projection" of the root system): While these kind of pictures are pretty, they can be a bit misleading when trying to understand what a root system really is.

    So, Dynkin diagrams, which are relatively simple, are interesting because they *classify* root systems, which are relatively more complex. Now, root systems are interesting because they classify even more complicated mathematical structures: Lie algebras, Lie groups and other stuff. To be precise root systems classify the so called semisimple Lie algebras. That is, given a semisimple Lie algebra you can construct its corresponding root system. And given a root system you can recover the semisimple Lie algebra it corresponds to.

    Lie groups, which are closely related to Lie algebras, describe the symmetries of geometric objects. Lie groups are important in physics (as well as mathematics).
  • May 31 2012: It's a bit hard to explain these concepts without being able to draw examples. I would suggest you look at for example Wikipedia for drawings of Dynkin diagrams and root systems. Also, it might be easier to explain if you can say something about why you are interested in these concepts.

    A Dynkin diagram is basically a bunch of circles/dots with lines between them. Between each pair of dots there can be 0, 1, 2 or 3 lines. If there are 2 or 3 lines there will also be an arrow pointing towards one of the dots. A Dynkin diagram satisfies some specific rules about how you can draw the lines between the dots. Following these rules it is relatively easy to make a list of all possible Dynkin diagrams. You can see such a list on Wikipedia. The list of connected Dynkin diagrams consists of four infinite families A-D where for each positive whole number n you get a Dynkin diagram with n dots, plus 5 extra diagrams called E6, E7, E8, F4 and G2.

    The point of introducing Dynkin diagrams is that they *classify* root systems. That is, given a root system you can construct its corresponding Dynkin diagram, and given two different root systems their corresponding Dynkin diagrams are also different. You can also go backwards and construct a root system given its Dynkin diagram. You can think of a Dynkin diagram as a recipe for how to construct a root system. This means that having a list of all possible Dynkin diagrams is basically the same as having a list of all possible root systems.

    A root system is basically a bunch of arrows in n-dimensional space all starting at the same point. Mathematicians call such arrows "vectors". These vectors have to follow some rules saying something about the symmetry between the vectors. The dimension n of the space equals the number of dots in the corresponding Dynkin diagram. So for example if there are 2 dots in the Dynkin diagram the space is 2-dimensional, so it is possible to draw the root system on a piece of paper.