## Solutions to the Exercises

Exercise 1: Does w=[1,2,4,3,2,1,3,4] represent an involution in W(F4)?

Yes. Here's one solution:

w:=[1,2,4,3,2,1,3,4];
reduce([op(w),op(w)],F4);

[ ]

Exercise 2: Find a reduced expression for the reflection corresponding to the highest root of E6.

Solution:

v0:=interior_pt(E6);
r:=highest_root(E6);
vec2fc( reflect(r,v0), E6, 'w'); w;

[2, 4, 3, 1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2]

Exercise 3: Determine the number of double cosets of the form W(H3)\W(H4)/W(A3).

Solution: The number of H\G/K double cosets is the inner product of the permutation characters for G/K and G/H.

chi1:=perm_char([1,2,3],H4);
chi2:=perm_char([2,3,4],H4);
cprod(chi1, chi2, H4);

15

Exercise 4: For which exceptional root_systems R is it true that rho(R) belongs to the lattice generated by the roots?

Solution: All except E7.

for R in [G2,F4,E6,E7,E8] do root_coords(rho(R), R) od;

[5, 3]
[11, 21, 15, 8]
[8, 11, 15, 21, 15, 8]
[17, 49/2, 33, 48, 75/2, 26, 27/2]
[46, 68, 91, 135, 110, 84, 57, 29]

Exercise 5: Determine the decomposition of the tensor product of the spin representation and the adjoint representation of SO9.

Solution:

w:=weights(B4);
r0:=highest_root(B4);
qtensor(w[1], r0, B4);

X[1, 0, 1, 0] + X[1, 0, 0, 1] + X[1, 0, 0, 0]

Back to the Coxeter/Weyl tutorial.

Copyright 1998 by John R. Stembridge, University of Michigan, Ann Arbor MI 48109.