Part of the Catalogue of Lattices, which is a joint project of Gabriele Nebe, RWTH Aachen university (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, AT&T Labs-Research (njasloane@gmail.com).
Richard Borcherds
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge, England
Email address
R.E.Borcherds@pmms.cam.ac.uk
Taken from R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, University of Cambridge, 1984.
The following sets are in natural 1:1 correspondence:
The height t is the height of the norm -2 vector u of D, in other words -(u,w) where w is the Weyl vector of D. The letter after the height is just a name to distinguish vectors of the same height, and is the letter referred to in the column headed ``Norm -2's'' of table -4. An asterisk after the letter means that the vector u is of type 1, in other words the lattice L is the sum of a Niemeier lattice and a1.
The column ``Roots'' gives the Dynkin diagram of the norm 2 vectors of L arranged into orbits under Aut(L). ``Group'' is the order of the subgroup of Aut(D) fixing u. The group Aut(L) is a split extension R.G where R is the Weyl group of the Dynkin diagram and G is isomorphic to the subgroup of Aut(D) fixing u.
``S'' is the maximal number of pairwise orthogonal roots of L.
The column headed ``Norm 0 vectors'' desribes the norm 0 vectors z corresponding to each orbit of roots of u^ where u is in D, as in 3.5.2. A capital letter indicates that the corresponding norm 0 vector is twice a primitive vector, otherwise the norm 0 vector is primitive. x stands for a norm 0 vector of type the Leech lattice. Otherwise the letter a, d, or e is the first letter of the Dynkin diagram of the norm 0 vector, and its height is given by height(u)-h+1 where h is the Coxeter number of the component of the Dynkin diagram of u.
For example, the norm -2 vector of type 23a has 3 components in its root system, of Coxeter numbers 12, 12, and 6, and the letters are e, a, and d, so the corresponding norm 0 vectors have Coxeter numbers 12, 12, and 18 and hence are norm 0 vectors with Dynkin diagrams E64, A11D7E6, and D10E72.
See 4.3 for more information.
Height | Roots | Group | S | Norm 0 | vectors | |
1 | a* | a1 | 8315553613086720000 | 1 | X | |
2 | a | a2 | 991533312000 | 1 | x | |
3 | a | a19 | 92897280 | 9 | a | |
4 | a | a2a112 | 190080 | 13 | aa | |
5 | a* | a124a1 | 244823040 | 25 | aA | |
5 | b | a24a19 | 3456 | 13 | aa | |
5 | c | a3a115 | 40320 | 17 | aa | |
6 | a | a29 | 3024 | 9 | a | |
6 | b | a3a25a16 | 240 | 13 | aaa | |
7 | a* | a212a1 | 190080 | 13 | aA | |
7 | b | a33a24a13 | 48 | 13 | aaa | |
7 | c | a34a18a1 | 384 | 17 | aad | |
7 | d | a4a26a15 | 240 | 13 | aaa | |
7 | e | d4a121 | 120960 | 25 | ad | |
8 | a | a36a2 | 240 | 13 | ad | |
8 | b | a4a33a23a12 | 12 | 13 | aaaa | |
8 | c | d4a29 | 864 | 13 | aa | |
9 | a* | a38a1 | 2688 | 17 | aA | |
9 | b | a42a34a1 | 16 | 13 | aaa | |
9 | c | a43a3a22a13 | 12 | 13 | aaaa | |
9 | d | d4a34a3a13 | 48 | 17 | aada | |
9 | e | a5a33a24 | 24 | 13 | aaa | |
9 | f | a5a34a16 | 48 | 17 | aaa | |
10 | a | d4a43a23 | 12 | 13 | aaa | |
10 | b | a5a42a32a2a1 | 4 | 13 | aaaaa | |
11 | a* | a46a1 | 240 | 13 | aA | |
11 | b | d44a19 | 432 | 25 | dd | |
11 | c | a5d42a33 | 24 | 17 | daa | |
11 | d | a5a5a42a3a1 | 4 | 13 | aaaaa | |
11 | e | a52d4a32a12a1 | 8 | 17 | aaaad | |
11 | f | a53a24 | 48 | 13 | aa | |
11 | g | d5a36a1 | 48 | 17 | aad | |
11 | h | a6a42a32a2a1 | 4 | 13 | aaaaa | |
12 | a | a54a2 | 24 | 13 | ad | |
12 | b | d5a44a2 | 8 | 13 | aaa | |
12 | c | a6d4a43 | 6 | 13 | aaa | |
12 | d | a6a52a3a22 | 4 | 13 | aaaa | |
13 | a* | a54d4a1 | 48 | 17 | aaA | |
13 | b | d5a52d4a3a1 | 4 | 17 | aaada | |
13 | c | d5a53a13a1 | 12 | 17 | aaae | |
13 | d* | d46a1 | 2160 | 25 | dD | |
13 | e | a62a5a4a12 | 4 | 13 | aaaa | |
13 | f | a7a5a42a3 | 4 | 13 | aaaa | |
13 | g | a7a5d4a32a12 | 4 | 17 | aaaaa | |
14 | a | a6a6d5a4a2 | 2 | 13 | aaaaa | |
14 | b | a63d4 | 12 | 13 | aa | |
14 | c | a7a6a5a4a1 | 2 | 13 | aaaaa | |
15 | a* | a64a1 | 24 | 13 | aA | |
15 | b | d53a5a3 | 12 | 17 | ade | |
15 | c | d6d44a13 | 24 | 25 | ddd | |
15 | d | d6a52a5a3 | 4 | 17 | aada | |
15 | e | a7d52a32a1 | 4 | 17 | aaad | |
15 | f | a72d42a1 | 8 | 17 | aad | |
15 | g | a8a53 | 6 | 13 | aa | |
15 | h | a8a6a5a3a2 | 2 | 13 | aaaaa | |
16 | a | a73a2 | 12 | 13 | ad | |
16 | b | a8a6d5a4 | 2 | 13 | aaaa | |
17 | a* | a72d52a1 | 8 | 17 | aaA | |
17 | b | e6a53d4 | 12 | 17 | aae | |
17 | c | a7d6d5a5 | 2 | 17 | daaa | |
17 | d | a72d6a3a1 | 4 | 17 | aada | |
17 | e | a8a72a1 | 4 | 13 | aaa | |
17 | f | a9d5a5d4a1 | 2 | 17 | aaaaa | |
17 | g | a9a7a42 | 4 | 13 | aaa | |
18 | a | e6a63 | 6 | 13 | aa | |
18 | b | a9a8a5a2 | 2 | 13 | aaaa | |
19 | a* | a83a1 | 12 | 13 | aA | |
19 | b | d63d4a13 | 6 | 25 | ddd | |
19 | c | a7e6d52a1 | 4 | 17 | eaad | |
19 | d | d7a7d5a5 | 2 | 17 | aaad | |
19 | e | d7a72a3a1 | 4 | 17 | aaad | |
19 | f | a9a7d6a1a1 | 2 | 17 | aaaad | |
19 | g | a10a7a6a1 | 2 | 13 | aaaa | |
20 | a | a82e6a2 | 4 | 13 | aaa | |
20 | b | a10a8d5 | 2 | 13 | aaa | |
21 | a* | a92d6a1 | 4 | 17 | aaA | |
21 | b | a11d6a5a3 | 2 | 17 | aaaa | |
21 | c | a11a8a5 | 2 | 13 | aaa | |
21 | d* | d64a1 | 24 | 25 | dD | |
21 | e | a9e6d6a3 | 2 | 17 | aaad | |
23 | a | d7e62a5 | 4 | 17 | ead | |
23 | b | d8d62d4a1 | 2 | 25 | dddd | |
23 | c | a9d72 | 4 | 17 | da | |
23 | d | a9d8a7 | 2 | 17 | daa | |
23 | e | a11d7d5a1 | 2 | 17 | aaad | |
24 | a | a112a2 | 4 | 13 | ad | |
24 | b | a12e6a6 | 2 | 13 | aaa | |
25 | a* | a11d7e6a1 | 2 | 17 | aaaA | |
25 | b | a13d6d5 | 2 | 17 | aaa | |
25 | e* | e64a1 | 48 | 17 | eE | |
26 | a | a13a10a1 | 2 | 13 | aaa | |
27 | a* | a122a1 | 4 | 13 | aA | |
27 | b | e7d63 | 3 | 25 | dd | |
27 | c | a9a9e7 | 2 | 17 | ada | |
27 | d | d9a9e6 | 2 | 17 | ada | |
27 | e | a11d9a5 | 2 | 17 | aad | |
27 | f | a14a9a2 | 2 | 13 | aaa | |
29 | a | a11e7e6 | 2 | 17 | daa | |
29 | d* | d83a1 | 6 | 25 | dD | |
31 | a | d82e7a1a1 | 2 | 25 | ddde | |
31 | b | d10d8d6a1 | 1 | 25 | dddd | |
31 | c | a15d8a1 | 2 | 17 | aad | |
33 | a* | a15d9a1 | 2 | 17 | aaA | |
33 | b | a15e7a3 | 2 | 17 | aad | |
33 | c | a17a8 | 2 | 13 | aa | |
35 | a | e73d4 | 6 | 25 | de | |
35 | b | a13d11 | 2 | 17 | da | |
36 | a | a18e6 | 2 | 13 | aa | |
37 | a* | a17e7a1 | 2 | 17 | aaA | |
37 | d* | d10e72a1 | 2 | 25 | ddD | |
39 | a | d12e7d6 | 1 | 25 | ddd | |
45 | d* | d122a1 | 2 | 25 | dD | |
47 | a | d10e8e7 | 1 | 25 | edd | |
47 | b | d14d10a1 | 1 | 25 | ddd | |
47 | c | a17e8 | 2 | 17 | da | |
48 | a | a23a2 | 2 | 13 | ad | |
51 | a* | a24a1 | 2 | 13 | aA | |
61 | d* | d16e8a1 | 1 | 25 | ddD | |
61 | e* | e83a1 | 6 | 25 | eE | |
63 | a | d18e7 | 1 | 25 | dd | |
93 | d* | d24a1 | 1 | 25 | dD |