Hello,


I am investigating the Leech lattice. Lately I have discovered following. Some lattices decompose into distinct set of orthonormal frames. For example E8 lattice which contains 240 unitary vectors in dimension 8 decompose into 15 sets of 16 vectors in each set. Each set contain 16 vectors of +- orthonormal basis of R^8.


The numbers are:


* 24 = 3 \* 8, lattice in four dimension call it d4 lattice with vectors e1..e4, 1/2\* \*Sum(+-ei), i=1..4; it is root system of Lie algebra D4.
* 240 = 15\*16, E8 lattice
* 196560 = 4095\*48; Leech lattice.


My question is whether anybody knows similar decomposition of Leech lattice. I am trying to obtain one but no luck so far. Maybe it is already known fact. 


Obviously each element of Conway group Co0 transform one orthonormal frame of Leech into another. So if I know the matrix representation of Co0 then I know many examples of such frames. Each element of Co0 would define permutation on 4095 points i.e. sets of orthonormal frames, so we would have homomorphism from Co0 to S4095.


Regards,


Marek Mitros


mim\_ (at) op.pl



Leech lattice decomposition
Yes, such a decomposition exists. Here's a construction I learned from Elkies some time ago (it's mentioned in one of his papers, probably *Mordell-Weil lattices in characteristic 2, II*), using an action of the Gaussian integers Z[i] on the Leech lattice:


Let L be the Leech lattice, and consider the quotient L/(1+i)L, which has order 2^12 (since 1+i has norm 2). No minimal vector in L can be in (1+i)L (the minimal vectors in L have norm 4, so those in (1+i)L have norm 8). Thus, the minimal vectors fall into at most 2^12 - 1 classes mod (1+i)L. Suppose v and w are minimal vectors that are congruent mod (1+i)L. Then |v-w|^2 >= 8, so the inner product is at most 0. There can be at most 48 such vectors on a sphere in R^24, so each residue class mod (1+i)L contains at most 48 minimal vectors. However, 196560 = (2^12 - 1)\*48, so there must be exactly 48 in each. This gives a decomposition into 4095 orthogonal frames.


