## Root Systems

Okay, now to lay out the actual objects of our current interest. These are basically collections of vectors in some inner product space , but with each vector comes a reflection and we want these reflections to play nicely with the vectors themselves. In a way, each point acts as both “program” — an operation to be performed — and “data” — an object to which operations can be applied — and the interplay between these two roles leads to some very interesting structure.

First off, only nonzero vectors give rise to reflections, so we don’t really want zero to be in our collection . We also may as well assume that spans , because it certainly spans *some* subspace of and anything that happens off of this subspace is pretty uninteresting as far as goes. These are the things that would just be silly not to ask for.

Now, the core requirement is that if , then the reflection should leave invariant. That is, if is any vector in , then

is also a vector in . In particular, this means that we have to have . But we don’t want any other scalar multiples of to be in , because they’d just give the same reflection again and that would be redundant.

Of course, we could just throw in more and more vectors as we need to make invariant under all of its reflections, and each new vector introduces not only new images under the existing reflections, but whole new reflections we have to handle. We want this process to stop after a while, so we’ll insist that is a *finite* collection of vectors. This is probably the biggest constraint on our collections.

We have one last condition to add: we want to ask that for every pair of vectors and in , we have . In other words, the length of the projection of onto must be a half-integral multiple of the length of . This makes it so that the displacement from to is some integral multiple of . This provides a certain rigidity to our discussion.

So, let’s recap:

- is a finite, spanning set of vectors in which does not contain .
- If then the only scalar multiples of in are .
- If then the reflection leaves invariant.
- If and are in , then is an integer.

A collection of vectors satisfying all of these conditions is called a “root system”, and the vectors in are called “roots” for ABSOLUTELY ARBITRARY REASONS THAT HAVE ABSOLUTELY NOTHING TO DO WITH ANYTHING. As far as we’re concerned for now.

So yeah: “root system”. Just ’cause…

Our lofty goal, for the immediate future, is to classify all the possible root systems.

Oh boy. This should be interesting.

Are you going to do the motivation of, say, classifying complex simple Lie algebras?

Comment by Akhil Mathew | January 20, 2010 |

What are these “Lie algebras”, of which you speak? I just see a bunch of vectors being reflected around in regular old -dimensional spaces. These “Lie algebras” sound big and scary.

đ

Comment by John Armstrong | January 20, 2010 |

[…] Weyl Group of a Root System Let’s take a root system in the inner product space . Each vector in gives rise to a reflection in , the group of […]

Pingback by The Weyl Group of a Root System « The Unapologetic Mathematician | January 21, 2010 |

[…] Category of Root Systems As with so many of the objects we study, root systems form a category. If is a root system in the inner product space , and is a root system in the […]

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Pingback by Bases for Root Systems « The Unapologetic Mathematician | February 1, 2010 |

[…] Existence of Bases for Root Systems We’ve defined what a base for a root system is, but we haven’t provided any evidence yet that they even exist. Today we’ll not only […]

Pingback by The Existence of Bases for Root Systems « The Unapologetic Mathematician | February 2, 2010 |

[…] A very useful concept in our study of root systems will be that of a Weyl chamber. As we showed at the beginning of last time, the hyperplanes for […]

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[…] Lemmas on Simple Roots If is some fixed base of a root system , we call the roots “simple”. Simple roots have a number of nice properties, some of […]

Pingback by Some Lemmas on Simple Roots « The Unapologetic Mathematician | February 4, 2010 |

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[…] of (Possible) Root Systems At long last, we can state the classification of irreducible root systems up to isomorphism. We’ve shown that for each such root system we can construct a connected […]

Pingback by The Classification of (Possible) Root Systems « The Unapologetic Mathematician | February 19, 2010 |

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Pingback by Proving the Classification Theorem I « The Unapologetic Mathematician | February 22, 2010 |

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Pingback by Construction of B- and C-Series Root Systems « The Unapologetic Mathematician | March 4, 2010 |

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Pingback by Construction of the G2 Root System « The Unapologetic Mathematician | March 8, 2010 |

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Pingback by Construction of E-Series Root Systems « The Unapologetic Mathematician | March 10, 2010 |

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Pingback by Root Systems Recap « The Unapologetic Mathematician | March 12, 2010 |

[…] Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a […]

Pingback by Coxeter groups « Annoying Precision | June 27, 2010 |

I like to say “both verb and noun” — keep it closer to elementary school level when possibleđ

Comment by isomorphismes | March 11, 2015 |

“program and data”, on the other hand, keeps it closer to the topics in

GĂ¶del Escher Bach, which a lot of math people are also fans of.Comment by John Armstrong | March 11, 2015 |

Reblogged this on Human Mathematics.

Comment by isomorphismes | May 15, 2015 |