# Phan systems study group III

Sukru lead this week’s discussion and worked through the root systems of type . Before covering the maths I should first mention that Sukru feels that we should come up with a better title for the study group as he doesnt think that it is very reflective of what we are actually doing. We are yet to agree on a new one so I have stuck with the old one.

Recall from the first study group post that a *root system* is a finite spanning subset of equipped with a positive definite symmetric bilinear form such that

- is invariant under for each . ( is the reflection through the hyperplane orthogonal to .)
- given , the only scalar multiples of in are and .
- is an integer for all (crystallographic condition).

We begin with a general construction. Let be a basis for . Let *I* be the -lattice of *B* and let be the set of all vectors in *I* which have certain length or lengths (two lengths). This set is finite, spans *E* and does not contain (as we don’t allow vectors of length .) The certain lengths are chosen so that the only scalar multiples of in are and the squared lengths divide 2. Note that for all we have . Since each preserves I (by the crystallographic condition) and preserves lengths it follows that preserves . Hence is a root system.

**Type :** Choose to be the subspace orthogonal to . Let and define . Then

and in fact and we can take to be a set of simple roots. (check the first study group post for definition of simple roots). Let and set . Then

.

Also, . Thus we can define an isomorphism , .

The Coxeter graph is

**Type :** Let and

.

Then a set of simple roots is . In particular, we can write . Clearly the first are long roots and is a short root.

Let . Then and corresponds to a signed permutation which changes the sign of i and fixes all the other j. In fact . Moreover, . Thus .

The Coxeter graph is

**Type , : **Let and

.

This is the dual of so we get the same Weyl group. A set of simple roots now has short roots and 1 long root.

**Type , : **Here and

.

A set of simple roots is .

As seen in type , for , we have and . Thus . Now

and

Hence induces the signed permutation of which interchanges and and multiplies both by . Thus includes all the changes of signs which change an even number of signs. Hence .

The Coxeter graph is

In your general construction, ‘the squared lengths divide 2’ is too strong a condition, and is not satified by . Instead it should just be ‘such that ‘.