## Richard Borcherds' Complete List of 25-Dimensional Unimodular Lattices

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and Neil J. A. Sloane, AT&T Labs-Research, (njas@research.att.com).

## The Complete List of 25-Dimensional Unimodular Lattices

Richard Borcherds
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge, England

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

• This is a table of the 25-dimensional unimodular lattices.

### Table -4.  The norm -4 vectors of II25,1.

There is a natural 1:1 correspondence between the elements of the following sets:

(1)
Orbits of norm -4 vectors u in II25,1 under Aut(II25,1).
(2)
Orbits of norm -4 vectors in the fundamental domain D of II25,1 under Aut(D).
(3)
Orbits of norm -1 vectors v of I25,1 under Aut(I25,1).
(4)
25 dimensional unimodular positive definite lattices L.
(5)
Unimodular lattices L1 of dimension at most 25 with no vectors of norm 1.
(6)
25 dimensional even lattices L2 of determinant 4.
L1 is the orthogonal complement of the norm 1 vectors of L, L2 is the lattice of elements of L of even norm, L2 is isomorphic to u^, and L is isomorphic to v^. Table -4 lists the 665 elements of any of these sets.

The height t is the height of the norm -4 vector u of D, in other words -(u,w) where w is the Weyl vector of D. The things in table -4 are listed in increasing order of their height.

Dim is the dimension of the lattice L1. A capital E after the dimension means that L1 is even.

The column ``roots'' gives the Dynkin diagram of the norm 2 vectors of L2 arranged into orbits under Aut(L2).

``Group'' gives the order of the subgroup of Aut(D) fixing u. The group Aut(L) @ Aut(L2) is of the form 2×R.G where R is the group generated by the reflections of norm 2 vectors of L, G is the group described in the column ``group'', and 2 is the group of order 2 generated by -1. If dim(L1) £ 24 then Aut(L1) is of the form R.G where R is the reflection group of L1 and G is as above.

For any root r of u^ the vector u+r is a norm -2 vector of II25,1. This vector u can be found as follows. Let X be the component of the Dynkin diagram of u^ to which u belongs and let h be the Coxeter number of X. Then r+u is conjugate to a norm -2 vector of II25,1 in D of height t-h+1 (or t-h if the entry under ``Dim'' is 24E) whose letter is the letter corresponding to X in the column headed ``norm -2's''. For example let u be the vector of height 6 and root system a22a110. Then the norm -2 vectors corresponding to roots from the components a2 or a1 have heights 6-3+1 and 6-2+1 and letters a and b, so they are the vectors 4a and 5b of table -2.

If dim(L1) £ 24 then the column ``neighbors'' gives the two even neighbors of L1+I24-dim(L1). If dim(L1) £ 23 then both neighbors are isomorphic so only one is listed, and if L1 is a Niemeier lattice then the neighbor is preceded by 2 (to indicate that the corresponding norm 0 vector is twice a primitive vector). If the two neighbors are isomorphic then there is an automorphism of L exchanging them.