Find sphere points in a lattice

edit

Here is a snippet of code that should do:

sage: Q = RootSystem('E8').weight_lattice()
sage: B = Q.basis()
sage: the_set = set()
sage: finished = False
sage: i = 1
sage: value = 4
sage: while not finished:
....:     smaller_or_eq_values = False
....:     int_vectors = IntegerVectors(n=i,k=8)
....:     for vect in int_vectors:
....:         weight = sum(vect[j]*B[j+1] for j in range(8)) # Create the weight
....:         wns = weight.norm_squared()
....:         if wns <= value:
....:             smaller_or_eq_values = True
....:             if wns == value:
....:                 print(weight)
....:                 the_set.add(weight)
....:     if not smaller_or_eq_values:
....:         finished = True
....:     i += 1

You will get a set the_set which above contains all the (positive) weights that have the norm squared to be equal to value=4.

The norm squared seem to deliver the same as the symmetric form:

sage: B[1]+B[2]
Lambda[1] + Lambda[2]
sage: a_weight = B[1]+B[2]
sage: a_weight.symmetric_form(a_weight)
22
sage: a_weight.norm_squared()
22