Richard Borcherds' Complete List of Even 25-Dimensional Lattices of Determinant 2

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).

Taken from R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, University of Cambridge, 1984.

• This is a table of the 121 even 25-dimensional lattices of determinant 2.

Table -2.  The norm -2 vectors of II_25,1.

The following sets are in natural 1:1 correspondence:

(1)

Orbits of norm -2 vectors in II_25,1 under Aut(II_25,1).

(2)

Orbits of norm -2 vectors u of D under Aut(D).

(3)

25 dimensional even bimodular lattices L.

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

INDEX FILE | ABBREVIATIONS