SPECIAL ROOTS FOR F_4 LIE ALGREBRA :
As in the paper math-ph/0302061, our notation here is
R[A]={j1, j2, j3, j4}
replaces
R[A]={gam[1,j1], gam[2,j2], gam[3,j3], gam[4,j4]} .
for A=1,2, ... ,1152.
where
a[i]'s are simple roots of F_4.
**********************************************************************
gam[1, 1]:=0
gam[1, 2]:=a[1]
gam[1, 3]:=a[1] + a[2]
gam[1, 4]:=a[1] + a[2] + 2 a[3]
gam[1, 5]:=a[1] + a[2] + 2 a[3] + 2 a[4]
gam[1, 6]:=a[1] + 2 a[2] + 2 a[3]
gam[1, 7]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]
gam[1, 8]:=a[1] + 2 a[2] + 4 a[3] + 2 a[4]
gam[1, 9]:=a[1] + 3 a[2] + 4 a[3] + 2 a[4]
gam[1,10]:=2 a[1] + 2 a[2] + 2 a[3]
gam[1,11]:=2 a[1] + 2 a[2] + 2 a[3] + 2 a[4]
gam[1,12]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]
gam[1,13]:=2 a[1] + 4 a[2] + 4 a[3] + 2 a[4]
gam[1,14]:=2 a[1] + 4 a[2] + 6 a[3] + 2 a[4]
gam[1,15]:=2 a[1] + 4 a[2] + 6 a[3] + 4 a[4]
gam[1,16]:=3 a[1] + 3 a[2] + 4 a[3] + 2 a[4]
gam[1,17]:=3 a[1] + 4 a[2] + 4 a[3] + 2 a[4]
gam[1,18]:=3 a[1] + 4 a[2] + 6 a[3] + 2 a[4]
gam[1,19]:=3 a[1] + 4 a[2] + 6 a[3] + 4 a[4]
gam[1,20]:=3 a[1] + 5 a[2] + 6 a[3] + 2 a[4]
gam[1,21]:=3 a[1] + 5 a[2] + 6 a[3] + 4 a[4]
gam[1,22]:=3 a[1] + 5 a[2] + 8 a[3] + 4 a[4]
gam[1,23]:=3 a[1] + 6 a[2] + 8 a[3] + 4 a[4]
gam[1,24]:=4 a[1] + 6 a[2] + 8 a[3] + 4 a[4]
gam[2, 1]:=0
gam[2, 2]:=a[2]
gam[2, 3]:=a[2] + 2 a[3]
gam[2, 4]:=a[2] + 2 a[3] + 2 a[4]
gam[2, 5]:=2 a[2] + 2 a[3]
gam[2, 6]:=2 a[2] + 2 a[3] + 2 a[4]
gam[2, 7]:=2 a[2] + 4 a[3] + 2 a[4]
gam[2, 8]:=3 a[2] + 4 a[3] + 2 a[4]
gam[2, 9]:=a[1] + a[2]
gam[2,10]:=a[1] + a[2] + 2 a[3]
gam[2,11]:=a[1] + a[2] + 2 a[3] + 2 a[4]
gam[2,12]:=a[1] + 3 a[2] + 2 a[3]
gam[2,13]:=a[1] + 3 a[2] + 2 a[3] + 2 a[4]
gam[2,14]:=a[1] + 3 a[2] + 4 a[3]
gam[2,15]:=a[1] + 3 a[2] + 4 a[3] + 4 a[4]
gam[2,16]:=a[1] + 3 a[2] + 6 a[3] + 2 a[4]
gam[2,17]:=a[1] + 3 a[2] + 6 a[3] + 4 a[4]
gam[2,18]:=a[1] + 5 a[2] + 6 a[3] + 2 a[4]
gam[2,19]:=a[1] + 5 a[2] + 6 a[3] + 4 a[4]
gam[2,20]:=a[1] + 5 a[2] + 8 a[3] + 4 a[4]
gam[2,21]:=2 a[1] + 2 a[2] + 2 a[3]
gam[2,22]:=2 a[1] + 2 a[2] + 2 a[3] + 2 a[4]
gam[2,23]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]
gam[2,24]:=2 a[1] + 3 a[2] + 2 a[3]
gam[2,25]:=2 a[1] + 3 a[2] + 2 a[3] + 2 a[4]
gam[2,26]:=2 a[1] + 3 a[2] + 4 a[3]
gam[2,27]:=2 a[1] + 3 a[2] + 4 a[3] + 4 a[4]
gam[2,28]:=2 a[1] + 3 a[2] + 6 a[3] + 2 a[4]
gam[2,29]:=2 a[1] + 3 a[2] + 6 a[3] + 4 a[4]
gam[2,30]:=2 a[1] + 4 a[2] + 4 a[3]
gam[2,31]:=2 a[1] + 4 a[2] + 4 a[3] + 4 a[4]
gam[2,32]:=2 a[1] + 4 a[2] + 8 a[3] + 4 a[4]
gam[2,33]:=2 a[1] + 5 a[2] + 4 a[3] + 2 a[4]
gam[2,34]:=2 a[1] + 5 a[2] + 8 a[3] + 2 a[4]
gam[2,35]:=2 a[1] + 5 a[2] + 8 a[3] + 6 a[4]
gam[2,36]:=2 a[1] + 6 a[2] + 6 a[3] + 2 a[4]
gam[2,37]:=2 a[1] + 6 a[2] + 6 a[3] + 4 a[4]
gam[2,38]:=2 a[1] + 6 a[2] + 8 a[3] + 2 a[4]
gam[2,39]:=2 a[1] + 6 a[2] + 8 a[3] + 6 a[4]
gam[2,40]:=2 a[1] + 6 a[2] + 10 a[3] + 4 a[4]
gam[2,41]:=2 a[1] + 6 a[2] + 10 a[3] + 6 a[4]
gam[2,42]:=2 a[1] + 7 a[2] + 8 a[3] + 4 a[4]
gam[2,43]:=2 a[1] + 7 a[2] + 10 a[3] + 4 a[4]
gam[2,44]:=2 a[1] + 7 a[2] + 10 a[3] + 6 a[4]
gam[2,45]:=3 a[1] + 3 a[2] + 4 a[3] + 2 a[4]
gam[2,46]:=3 a[1] + 5 a[2] + 4 a[3] + 2 a[4]
gam[2,47]:=3 a[1] + 5 a[2] + 8 a[3] + 2 a[4]
gam[2,48]:=3 a[1] + 5 a[2] + 8 a[3] + 6 a[4]
gam[2,49]:=3 a[1] + 7 a[2] + 8 a[3] + 2 a[4]
gam[2,50]:=3 a[1] + 7 a[2] + 8 a[3] + 6 a[4]
gam[2,51]:=3 a[1] + 7 a[2] + 12 a[3] + 6 a[4]
gam[2,52]:=3 a[1] + 9 a[2] + 12 a[3] + 6 a[4]
gam[2,53]:=4 a[1] + 5 a[2] + 6 a[3] + 2 a[4]
gam[2,54]:=4 a[1] + 5 a[2] + 6 a[3] + 4 a[4]
gam[2,55]:=4 a[1] + 5 a[2] + 8 a[3] + 4 a[4]
gam[2,56]:=4 a[1] + 6 a[2] + 6 a[3] + 2 a[4]
gam[2,57]:=4 a[1] + 6 a[2] + 6 a[3] + 4 a[4]
gam[2,58]:=4 a[1] + 6 a[2] + 8 a[3] + 2 a[4]
gam[2,59]:=4 a[1] + 6 a[2] + 8 a[3] + 6 a[4]
gam[2,60]:=4 a[1] + 6 a[2] + 10 a[3] + 4 a[4]
gam[2,61]:=4 a[1] + 6 a[2] + 10 a[3] + 6 a[4]
gam[2,62]:=4 a[1] + 7 a[2] + 8 a[3] + 2 a[4]
gam[2,63]:=4 a[1] + 7 a[2] + 8 a[3] + 6 a[4]
gam[2,64]:=4 a[1] + 7 a[2] + 12 a[3] + 6 a[4]
gam[2,65]:=4 a[1] + 8 a[2] + 8 a[3] + 4 a[4]
gam[2,66]:=4 a[1] + 8 a[2] + 12 a[3] + 4 a[4]
gam[2,67]:=4 a[1] + 8 a[2] + 12 a[3] + 8 a[4]
gam[2,68]:=4 a[1] + 9 a[2] + 10 a[3] + 4 a[4]
gam[2,69]:=4 a[1] + 9 a[2] + 10 a[3] + 6 a[4]
gam[2,70]:=4 a[1] + 9 a[2] + 12 a[3] + 4 a[4]
gam[2,71]:=4 a[1] + 9 a[2] + 12 a[3] + 8 a[4]
gam[2,72]:=4 a[1] + 9 a[2] + 14 a[3] + 6 a[4]
gam[2,73]:=4 a[1] + 9 a[2] + 14 a[3] + 8 a[4]
gam[2,74]:=4 a[1] + 10 a[2] + 12 a[3] + 6 a[4]
gam[2,75]:=4 a[1] + 10 a[2] + 14 a[3] + 6 a[4]
gam[2,76]:=4 a[1] + 10 a[2] + 14 a[3] + 8 a[4]
gam[2,77]:=5 a[1] + 7 a[2] + 8 a[3] + 4 a[4]
gam[2,78]:=5 a[1] + 7 a[2] + 10 a[3] + 4 a[4]
gam[2,79]:=5 a[1] + 7 a[2] + 10 a[3] + 6 a[4]
gam[2,80]:=5 a[1] + 9 a[2] + 10 a[3] + 4 a[4]
gam[2,81]:=5 a[1] + 9 a[2] + 10 a[3] + 6 a[4]
gam[2,82]:=5 a[1] + 9 a[2] + 12 a[3] + 4 a[4]
gam[2,83]:=5 a[1] + 9 a[2] + 12 a[3] + 8 a[4]
gam[2,84]:=5 a[1] + 9 a[2] + 14 a[3] + 6 a[4]
gam[2,85]:=5 a[1] + 9 a[2] + 14 a[3] + 8 a[4]
gam[2,86]:=5 a[1] + 11 a[2] + 14 a[3] + 6 a[4]
gam[2,87]:=5 a[1] + 11 a[2] + 14 a[3] + 8 a[4]
gam[2,88]:=5 a[1] + 11 a[2] + 16 a[3] + 8 a[4]
gam[2,89]:=6 a[1] + 9 a[2] + 12 a[3] + 6 a[4]
gam[2,90]:=6 a[1] + 10 a[2] + 12 a[3] + 6 a[4]
gam[2,91]:=6 a[1] + 10 a[2] + 14 a[3] + 6 a[4]
gam[2,92]:=6 a[1] + 10 a[2] + 14 a[3] + 8 a[4]
gam[2,93]:=6 a[1] + 11 a[2] + 14 a[3] + 6 a[4]
gam[2,94]:=6 a[1] + 11 a[2] + 14 a[3] + 8 a[4]
gam[2,95]:=6 a[1] + 11 a[2] + 16 a[3] + 8 a[4]
gam[2,96]:=6 a[1] + 12 a[2] + 16 a[3] + 8 a[4]
gam[3, 1]:=0
gam[3, 2]:=a[3]
gam[3, 3]:=a[3] + a[4]
gam[3, 4]:=a[2] + a[3]
gam[3, 5]:=a[2] + a[3] + a[4]
gam[3, 6]:=a[2] + 2 a[3]
gam[3, 7]:=a[2] + 2 a[3] + 2 a[4]
gam[3, 8]:=a[2] + 3 a[3] + a[4]
gam[3, 9]:=a[2] + 3 a[3] + 2 a[4]
gam[3,10]:=2 a[2] + 3 a[3] + a[4]
gam[3,11]:=2 a[2] + 3 a[3] + 2 a[4]
gam[3,12]:=2 a[2] + 4 a[3] + 2 a[4]
gam[3,13]:=a[1] + a[2] + a[3]
gam[3,14]:=a[1] + a[2] + a[3] + a[4]
gam[3,15]:=a[1] + a[2] + 2 a[3]
gam[3,16]:=a[1] + a[2] + 2 a[3] + 2 a[4]
gam[3,17]:=a[1] + a[2] + 3 a[3] + a[4]
gam[3,18]:=a[1] + a[2] + 3 a[3] + 2 a[4]
gam[3,19]:=a[1] + 2 a[2] + 2 a[3]
gam[3,20]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]
gam[3,21]:=a[1] + 2 a[2] + 3 a[3]
gam[3,22]:=a[1] + 2 a[2] + 3 a[3] + 3 a[4]
gam[3,23]:=a[1] + 2 a[2] + 5 a[3] + 2 a[4]
gam[3,24]:=a[1] + 2 a[2] + 5 a[3] + 3 a[4]
gam[3,25]:=a[1] + 3 a[2] + 3 a[3] + a[4]
gam[3,26]:=a[1] + 3 a[2] + 3 a[3] + 2 a[4]
gam[3,27]:=a[1] + 3 a[2] + 5 a[3] + a[4]
gam[3,28]:=a[1] + 3 a[2] + 5 a[3] + 4 a[4]
gam[3,29]:=a[1] + 3 a[2] + 6 a[3] + 2 a[4]
gam[3,30]:=a[1] + 3 a[2] + 6 a[3] + 4 a[4]
gam[3,31]:=a[1] + 4 a[2] + 5 a[3] + 2 a[4]
gam[3,32]:=a[1] + 4 a[2] + 5 a[3] + 3 a[4]
gam[3,33]:=a[1] + 4 a[2] + 6 a[3] + 2 a[4]
gam[3,34]:=a[1] + 4 a[2] + 6 a[3] + 4 a[4]
gam[3,35]:=a[1] + 4 a[2] + 7 a[3] + 3 a[4]
gam[3,36]:=a[1] + 4 a[2] + 7 a[3] + 4 a[4]
gam[3,37]:=2 a[1] + 2 a[2] + 3 a[3] + a[4]
gam[3,38]:=2 a[1] + 2 a[2] + 3 a[3] + 2 a[4]
gam[3,39]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]
gam[3,40]:=2 a[1] + 3 a[2] + 3 a[3] + a[4]
gam[3,41]:=2 a[1] + 3 a[2] + 3 a[3] + 2 a[4]
gam[3,42]:=2 a[1] + 3 a[2] + 5 a[3] + a[4]
gam[3,43]:=2 a[1] + 3 a[2] + 5 a[3] + 4 a[4]
gam[3,44]:=2 a[1] + 3 a[2] + 6 a[3] + 2 a[4]
gam[3,45]:=2 a[1] + 3 a[2] + 6 a[3] + 4 a[4]
gam[3,46]:=2 a[1] + 4 a[2] + 4 a[3] + 2 a[4]
gam[3,47]:=2 a[1] + 4 a[2] + 5 a[3] + a[4]
gam[3,48]:=2 a[1] + 4 a[2] + 5 a[3] + 4 a[4]
gam[3,49]:=2 a[1] + 4 a[2] + 7 a[3] + 2 a[4]
gam[3,50]:=2 a[1] + 4 a[2] + 7 a[3] + 5 a[4]
gam[3,51]:=2 a[1] + 4 a[2] + 8 a[3] + 4 a[4]
gam[3,52]:=2 a[1] + 5 a[2] + 6 a[3] + 2 a[4]
gam[3,53]:=2 a[1] + 5 a[2] + 6 a[3] + 4 a[4]
gam[3,54]:=2 a[1] + 5 a[2] + 7 a[3] + 2 a[4]
gam[3,55]:=2 a[1] + 5 a[2] + 7 a[3] + 5 a[4]
gam[3,56]:=2 a[1] + 5 a[2] + 9 a[3] + 4 a[4]
gam[3,57]:=2 a[1] + 5 a[2] + 9 a[3] + 5 a[4]
gam[3,58]:=2 a[1] + 6 a[2] + 8 a[3] + 4 a[4]
gam[3,59]:=2 a[1] + 6 a[2] + 9 a[3] + 4 a[4]
gam[3,60]:=2 a[1] + 6 a[2] + 9 a[3] + 5 a[4]
gam[3,61]:=3 a[1] + 4 a[2] + 5 a[3] + 2 a[4]
gam[3,62]:=3 a[1] + 4 a[2] + 5 a[3] + 3 a[4]
gam[3,63]:=3 a[1] + 4 a[2] + 6 a[3] + 2 a[4]
gam[3,64]:=3 a[1] + 4 a[2] + 6 a[3] + 4 a[4]
gam[3,65]:=3 a[1] + 4 a[2] + 7 a[3] + 3 a[4]
gam[3,66]:=3 a[1] + 4 a[2] + 7 a[3] + 4 a[4]
gam[3,67]:=3 a[1] + 5 a[2] + 6 a[3] + 2 a[4]
gam[3,68]:=3 a[1] + 5 a[2] + 6 a[3] + 4 a[4]
gam[3,69]:=3 a[1] + 5 a[2] + 7 a[3] + 2 a[4]
gam[3,70]:=3 a[1] + 5 a[2] + 7 a[3] + 5 a[4]
gam[3,71]:=3 a[1] + 5 a[2] + 9 a[3] + 4 a[4]
gam[3,72]:=3 a[1] + 5 a[2] + 9 a[3] + 5 a[4]
gam[3,73]:=3 a[1] + 6 a[2] + 7 a[3] + 3 a[4]
gam[3,74]:=3 a[1] + 6 a[2] + 7 a[3] + 4 a[4]
gam[3,75]:=3 a[1] + 6 a[2] + 9 a[3] + 3 a[4]
gam[3,76]:=3 a[1] + 6 a[2] + 9 a[3] + 6 a[4]
gam[3,77]:=3 a[1] + 6 a[2] + 10 a[3] + 4 a[4]
gam[3,78]:=3 a[1] + 6 a[2] + 10 a[3] + 6 a[4]
gam[3,79]:=3 a[1] + 7 a[2] + 9 a[3] + 4 a[4]
gam[3,80]:=3 a[1] + 7 a[2] + 9 a[3] + 5 a[4]
gam[3,81]:=3 a[1] + 7 a[2] + 10 a[3] + 4 a[4]
gam[3,82]:=3 a[1] + 7 a[2] + 10 a[3] + 6 a[4]
gam[3,83]:=3 a[1] + 7 a[2] + 11 a[3] + 5 a[4]
gam[3,84]:=3 a[1] + 7 a[2] + 11 a[3] + 6 a[4]
gam[3,85]:=4 a[1] + 6 a[2] + 8 a[3] + 4 a[4]
gam[3,86]:=4 a[1] + 6 a[2] + 9 a[3] + 4 a[4]
gam[3,87]:=4 a[1] + 6 a[2] + 9 a[3] + 5 a[4]
gam[3,88]:=4 a[1] + 7 a[2] + 9 a[3] + 4 a[4]
gam[3,89]:=4 a[1] + 7 a[2] + 9 a[3] + 5 a[4]
gam[3,90]:=4 a[1] + 7 a[2] + 10 a[3] + 4 a[4]
gam[3,91]:=4 a[1] + 7 a[2] + 10 a[3] + 6 a[4]
gam[3,92]:=4 a[1] + 7 a[2] + 11 a[3] + 5 a[4]
gam[3,93]:=4 a[1] + 7 a[2] + 11 a[3] + 6 a[4]
gam[3,94]:=4 a[1] + 8 a[2] + 11 a[3] + 5 a[4]
gam[3,95]:=4 a[1] + 8 a[2] + 11 a[3] + 6 a[4]
gam[3,96]:=4 a[1] + 8 a[2] + 12 a[3] + 6 a[4]
gam[4, 1]:=0
gam[4, 2]:=a[4]
gam[4, 3]:=a[3] + a[4]
gam[4, 4]:=a[2] + a[3] + a[4]
gam[4, 5]:=a[2] + 2 a[3] + a[4]
gam[4, 6]:=a[2] + 2 a[3] + 2 a[4]
gam[4, 7]:=a[1] + a[2] + a[3] + a[4]
gam[4, 8]:=a[1] + a[2] + 2 a[3] + a[4]
gam[4, 9]:=a[1] + a[2] + 2 a[3] + 2 a[4]
gam[4,10]:=a[1] + 2 a[2] + 2 a[3] + a[4]
gam[4,11]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]
gam[4,12]:=a[1] + 2 a[2] + 3 a[3] + a[4]
gam[4,13]:=a[1] + 2 a[2] + 3 a[3] + 3 a[4]
gam[4,14]:=a[1] + 2 a[2] + 4 a[3] + 2 a[4]
gam[4,15]:=a[1] + 2 a[2] + 4 a[3] + 3 a[4]
gam[4,16]:=a[1] + 3 a[2] + 4 a[3] + 2 a[4]
gam[4,17]:=a[1] + 3 a[2] + 4 a[3] + 3 a[4]
gam[4,18]:=a[1] + 3 a[2] + 5 a[3] + 3 a[4]
gam[4,19]:=2 a[1] + 3 a[2] + 4 a[3] + 2 a[4]
gam[4,20]:=2 a[1] + 3 a[2] + 4 a[3] + 3 a[4]
gam[4,21]:=2 a[1] + 3 a[2] + 5 a[3] + 3 a[4]
gam[4,22]:=2 a[1] + 4 a[2] + 5 a[3] + 3 a[4]
gam[4,23]:=2 a[1] + 4 a[2] + 6 a[3] + 3 a[4]
gam[4,24]:=2 a[1] + 4 a[2] + 6 a[3] + 4 a[4]
*******************************************************************************
R[ 1]={1, 1, 1,1}
R[ 2]={1, 1, 1,2}
R[ 3]={1, 1, 2,1}