# What is the easiest way to describe the Leech lattice explicitly?

I am aware that the Leech lattice is the unique even unimodular lattice in $\mathbb{R}^{24}$ with no norm $2$ vectors.

However I am after a way to describe this lattice explicitly without reference to Golay codes. Also I want to be able to write it in such a way so that the norm $2$ property above is simple to see. This is for a talk rather than a paper, so it has to be easy enough to write down or motivate without having to go into too much detail.

Is there such a way?

Well, http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Leech.html#GRAM gives the gram matrix. That is pretty explicit. A number of descriptions are in papers of Borcherds, available in SPLAG and http://math.berkeley.edu/~reb/papers/index.html#thesis

Meanwhile, the most active person with whom I have had contact is Daniel Allcock, see http://www.ma.utexas.edu/users/allcock/

Not widely known: the Leech lattice has a left-handed and a right-handed version. Any lattice with roots possesses an automorph with negative determinant, just take the reflection in that root. Or reflexion. Now, in dimension 2, people distinguish "opposite" classes of quadratic forms because of the importance as inverses in the class group. In odd dimension it does not matter. For even dimension four and higher, pretty much everyone uses the lax definition of equivalence class, although i know some people who occasionally fiddle with the strict version in dimension 4. So, the strict Niemeier class number is 25.

Let's see. Among other surprising features, the Leech lattice has covering radius $\sqrt 2,$ which is very small. Any even lattice (ignore unimodular) with covering radius strictly below $\sqrt 2$ has (lax) class number one, and so is of dimension no larger than ten. As it happens, all such also have strict class number one, which mostly comes down to either odd dimension or possessing roots.

Well, enough for the moment. Lots more where that came form.

EEDDITTTT: perhaps what you want is pages 129-130 in Lattices and Codes, second edition, by Wolgang Ebeling: take the hyperbolic lattice $II_{25,1}$ and vector $$w = (0,1,2,3,\ldots,23,24|70)$$ This $w$ is isotropic, as the sum of the squares of the first bunch of numbers is 4900. Then the Leech lattice is isomorphic to $$w^\perp/ \langle w \rangle.$$

A third edition of Ebeling's book is coming out. He did not have time to put in my stuff about covering radius and class number.

BOOKS:

THOMPSON

EBELING

• Do you know if there is any simple pattern to the generators? I don't fancy having to write this matrix on a blackboard :p – fretty Feb 11 '13 at 22:55
• @fretty, edited in a little. Hard to imagine there is any truly simple description. There could be (I have no idea) a set of 24 vectors in the standard integer lattice $\mathbb Z^{25}$ or $\mathbb Z^{26}$ or so that exhibit the desired inner products. However, maybe not. There are Lie algebra root lattices for which that trick does not work. – Will Jagy Feb 11 '13 at 23:13
• I have actually seen this characterisation of the lattice but cannot quite see the norm 2 property from it. I have a simpler construction from Conway/Sloane that uses $I_{25,1}$ and takes the perp of $w$ where $w = (3,5,7,...,47,51|145)$ but I can't see how to explain from this the properties of the Leech lattice. – fretty Feb 11 '13 at 23:30