Homework 5. (Due March 20.)

Let V be a finite dimensional vector space over Q, the rational numbers,
with a fixed symmetric bilinear form (,) with values in Q such that
(v,v)>0 for all v in V. A finite subset R of V-\{0\} (or the triple (V,R,(,)))
is said to be a root system if
R spans V
if a\in R then -a\in R
if a\in R, c\in Q,ca\in R then c=1 or c=-1
if a,b\in R then 2(b,a)/(a,a)\in Z (=integers) and b-2(b,a)/(a,a)a\in R.

Now 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\in{2,3,...}. 

(1) Let E' ={c_1e_1+...c_ne_n in E with c_i\in Q, c_1+...+c_n=0}.
Let R be the set of all e=c_1e_1+...+c_ne_n in E' with c_i\in Z, (e,e)=2.
Show that (E',R,(,)) is a root system.

(2) Let R be the set of all e=c_1e_1+...c_ne_n in E with c_i\in Z, 
(e,e)=2. Show that (E,R,(,)) is a root system.

(3) Let R be as in (2). Let R'=R union with {e_1,...,e_n,-e_1,...,-e_n}.
Show that (E,R',(,)) is a root system.

(4) Let R be as in (2). Let R'=R union with {2e_1,...,2e_n,-2e_1,...,-2e_n}.
Show that (E,R',(,)) is a root system.

(5) Assume that n=3. Let E' be as in (1). 
Let R be the set of all e=c_1e_1+c_2e_2+c_3e_3 in E' with c_i\in Z, (e,e)\in{2,6}.
Show that (E',R,(,)) is a root system.