Coxeter and Weyl Packages

Yes. Here's one solution:

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

` [ ] `

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]
`

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

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

` 15 `

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]
`

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.

Last modified Mon Jun 22 00:06:34 EDT 1998