T\dim S | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|
r4_d7_1 | 0 | 0 | 0 | 1 | |||
r4_d7_2 | 0 | 0 | 0 | 1 | |||
r4_d7_3 | 0 | 0 | 0 | 1 | |||
r4_d7_4 | 0 | 0 | 0 | 1 | |||
r4_d7_5 | 0 | 0 | 0 | 1 | |||
r4_d7_6 | 0 | 0 | 0 | 1 | |||
r4_d7_7 | 0 | 0 | 0 | 0 | |||
r4_d8_1 | 0 | 0 | 1 | 1 | 1 | ||
r4_d8_2 | 0 | 0 | 8 | 4 | 1 | ||
r4_d8_3 | 0 | 0 | 3 | 3 | 1 | ||
r4_d8_5 | 0 | 3 | 14 | 3 | 1 | ||
r4_d8_6 | 0 | 2 | 14 | 4 | 1 | ||
r4_d8_7 | 0 | 0 | 0 | 0 | 0 | ||
r4_d8_8 | 0 | 0 | 0 | 0 | 0 | ||
r4_d8_9 | 0 | 1 | 0 | 0 | 0 | ||
r4_d8_12 | 0 | 12 | 34 | 9 | 1 | ||
r4_d8_17 | 0 | 0 | 0 | 0 | 0 | ||
r4_d9_1 | 3 | 2 | 3 | 1 | 1 | ||
r4_d9_2 | 2 | 19 | 17 | 3 | 1 | ||
r4_d9_5 | 19 | 4 | 0 | 0 | 0 | ||
r4_d9_6 | 0 | 0 | 0 | 0 | 0 | ||
r4_d9_7 | 1 | 2 | 1 | 0 | 0 | ||
r4_d10_1 | 2 | 3 | 2 | 1 | 1 | ||
r4_d10_2 | 20 | 24 | 7 | 2 | 1 |
The table of non-isomorphic DHO of rank 5 consists of the above simply connected DHO and their quotients.
There are 55 DHO in dimension 10, 647 DHO in dimension 11, 3228 DHO in dimension 12,
154 DHO in dimension 13, 16 DHO in dimension 14 and the well known 4 DHO in dimension 15.
Finaly there is a table identifying the Extensions of bilinear DHO of rank 4 in the table of non-isomorphic DHO of rank 5.
Say you are interested to see which universal covers arrise from prolongations but not from extensions.
You look in the table with simply connected DHO of rank 5. All DHO which have no entry in the column "remark" are such universal covers.
Example 7.3
Below the essential data for the DHO of Rank 6 in Example 7.3. The DHO has a hyperplane induced subDHO, isomorphic to the
DHO of rank5 in dimension 10 with ID 55.
ID
GAP_id
|Aut|
dim(P)
bilinear
1
r6_d11_1
384
10
1
Non-splitting DHO
Among the quotients of DHO of rank 5 there is one non-splitting DHO in dimension 12 (a quotient of the Tanigushi DHO) and 125 non-splitting DHOs in dimension 11 (quotients of the Veronesean and Tanigushi DHO).
References