Homework 6. (Due Apr.3)

1. Let R be a root system with a fixed set of simple roots \Pi.
Define R^+,R^-,W,l:W \to{0,1,2,...} as in the lectures.

(a) Show that there is a unique element w_0 in W such that w_0(R^+)=R^-.

(b) Show that l(w_0) equals the number of elements in R^+.
Show that if w in W  is not w_0 then l(w)<l(w_0).

(c) Show that the map from W to {1,-1} which sends w to 
1 if l(w) is even and to -1 if l(w) is odd is a group homomorphism.

2. Let E be a two dimensional Q-vector space. Let (,) be a positive definite
symmetric bilinear form on E with values in Q.
Let R be a root system in E and let {a_1,a_2} be a set of simple roots for R.

(a) Assume that (a_1,a_1)=1,(a_2,a_2)=1,(a_1,a_2)=-1/2. Write a list of all 
vectors in R as linear combinations of a_1,a_2.

(b) Assume that (a_1,a_1)=1,(a_2,a_2)=2,(a_1,a_2)=-1. Write a list of all 
vectors in R as linear combinations of a_1,a_2.

(c) Assume that (a_1,a_1)=1,(a_2,a_2)=3,(a_1,a_2)=-3/2. Write a list of all 
vectors in R as linear combinations of a_1,a_2.

(Hint: Use the fact that any root is of the form w(a_i) for i=1 or 2
where w is a product of reflections r_j with respect to a_j with j=1 or 2.)

3) Let E be a Q-vector space with basis e_1,e_2,...,e_n. Let (,) be the symmetric bilinear
form on E with values in Q given by (e_i,e_j)=1 if i =j, (e_i,e_j)=0 otherwise.
Assume that n>3. Let R be the set of all e=c_1e_1+...c_ne_n in E with c_i integer, 
(e,e)=2. (Recall that (E,R,(,)) is a root system.) Find a set of simple roots
for R.