# MAPLE. April 28, 2009.
#
# Higman triality scheme from U_3(5) (3 copies of Hoffman-Singleton)
d := 4;
v := [1, 30, 42, 70, 7];
verts := 150;

Q := matrix([
[1 ,  21 ,  84 ,   42 ,   2],
[1 ,   7 ,   0 ,   -7 ,  -1],
[1 ,   1 ,  -6 ,    2 ,   2],
[1 ,  -3 ,   0 ,    3 ,  -1],
[1 ,  -9 ,  24 ,  -18 ,   2]
]);

#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
#  if we are only given the cosines (normalized column 1 of Q).
L := matrix([
[0 ,   21  ,   0    ,   0    ,  0  ],
[1 ,    4  ,  16    ,   0    ,  0  ],
[0 ,    4  ,   9    ,   8    ,  0  ],
[0 ,    0  ,  16    ,   4    ,  1  ],
[0 ,    0  ,   0    ,  21    ,  0  ]
]);

         [0    30     0     0    0]
         [                        ]
         [1     8    14     7    0]
         [                        ]
 L_1 :=  [0    10     0    20    0]
         [                        ]
         [0     3    12    12    3]
         [                        ]
         [0     0     0    30    0]

         [0     0    42     0    0]
         [                        ]
         [0    14     0    28    0]
         [                        ]
 L_2 :=  [1     0    35     0    6]
         [                        ]
         [0    12     0    30    0]
         [                        ]
         [0     0    36     0    6]

         [0     0     0    70    0]
         [                        ]
         [0     7    28    28    7]
         [                        ]
 L_3 :=  [0    20     0    50    0]
         [                        ]
         [1    12    30    23    4]
         [                        ]
         [0    30     0    40    0]

           [0    0    0    0    7]
           [                     ]
           [0    0    0    7    0]
           [                     ]
 L_4 :=    [0    0    6    0    1]
           [                     ]
           [0    3    0    4    0]
           [                     ]
           [1    0    6    0    0]


       [ 1      30      42      70      7]
       [                                 ]
       [ 1      10       2     -10     -3]
       [                                 ]
 P :=  [ 1       0      -3       0      2]
       [                                 ]
       [ 1      -5       2       5     -3]
       [                                 ]
       [ 1     -15      42     -35      7]


           [1    21    84     42     2]
           [                          ]
           [1     7     0     -7    -1]
           [                          ]
      Q := [1     1    -6      2     2]
           [                          ]
           [1    -3     0      3    -1]
           [                          ]
           [1    -9    24    -18     2]

         [0    21     0     0    0]
         [                        ]
         [1     4    16     0    0]
         [                        ]
 Ls1 :=  [0     4     9     8    0]
         [                        ]
         [0     0    16     4    1]
         [                        ]
         [0     0     0    21    0]

         [0     0    84     0    0]
         [                        ]
         [0    16    36    32    0]
         [                        ]
 Ls2 :=  [1     9    54    18    2]
         [                        ]
         [0    16    36    32    0]
         [                        ]
         [0     0    84     0    0]


         [0     0     0    42    0]
         [                        ]
         [0     0    32     8    2]
         [                        ]
 Ls3 :=  [0     8    18    16    0]
         [                        ]
         [1     4    32     4    1]
         [                        ]
         [0    21     0    21    0]


           [0    0    0    0    2]
           [                     ]
           [0    0    0    2    0]
           [                     ]
   Ls4 :=  [0    0    2    0    0]
           [                     ]
           [0    1    0    1    0]
           [                     ]
           [1    0    0    0    1]