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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

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5 Answers 5

up vote 18 down vote accepted

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.

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This is great! Do you by any chance know a familiar name for the subgroup of Co0 that preserves the Z[i]-module structure? –  S. Carnahan Mar 18 '10 at 17:46
    
I'm not sure. It'll be the centralizer of multiplication by i, so when we mod out by -1, it will be the centralizer of an involution in Co_1. There are three conjugacy classes of involutions in Co_1, but I haven't checked which one this is. The centralizers are 2^(1+8).O_8^+(2) for 2A, (2^2 x G_2(4)):2 for 2B, and 2^11:M_12:2 for 2C. –  Henry Cohn Mar 18 '10 at 19:26
    
Thank you for this answer ! I need some time to analyze it to create explicite decomposition. Do you know where I can find representation of Conway group Co0 as subgroup of SO(24) i.e. real 24x24 matrices ? In Atlas of finite groups there are representations over finite fields. Regards, Marek –  Marek Mitros Mar 19 '10 at 8:11
    
Magma can compute it: AutomorphismGroup(Lattice("Lambda",24)) It will give the group to you as a subgroup of GL_24(Z), but conjugating by a basis matrix for the lattice will put it in SO(24). –  Henry Cohn Mar 22 '10 at 18:13
    
I see you are also posting on the GAP forum. I would be interested in working with this too, since I also work with the GAP software. Could you possibly bring me up to speed with respect to the 4095 crosses in the Conway group construction, is this like the 54 crosses of Curtis's Kitten for M12 (or the related MOG construction for M24?) Thanks PGH –  Paul Hjelmstad Dec 31 '11 at 22:24
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I don't have an answer to your full question, but I can say something about the permutation representation.

Assuming a decomposition into frames existed, it cannot be preserved by the action of Co0, because it would yield a subgroup of Co0 of index at most 4095 as a stabilizer of a frame, or equivalently, a subgroup of Co1 of index at most 4095 as a stabilizer of a distinguished 24-tuple of the 98280 pairs of antipodal norm 4 points in the Leech lattice. However, according to the ATLAS, the largest maximal subgroup of Co1 is the group Co2, of index 98280.

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Yes, I've made mistake with this homomorphism. –  Marek Mitros Mar 19 '10 at 8:17
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Whilst I think this is probably not quite the construction you´re after, one configuration of vectors in the Leech lattice that sounds very much like what you desribe are the crosses (or ´frames´ or ´frames of reference´). Representing the (real) Leech lattice in the usual way (see Conway and Sloane, Chapter 10) then we can define the standard cross to be each of the 48 vectors of the form (\pm8,0^23) which is simply an orthonormal basis for R^24 rescaled and with the negatives thrown in for good measure. The stabilizer of such a configuration is the subgroup of monomial matrices (again, assuming we´re in the usual representation) which is isomorphic to 2^12:M_24, the Mathieu group M_24 acting on the binary Golay code in the usual way. The other 8292374 crosses are naturally viewed as the images of the above (standard) cross under the action of words in Conway´s zeta elements (or eta, depending on which account you´re reading - again see Conway and Sloane) of length at most 4.

The most detailed acount of the crosses is probably RT Curtis 1972 Cambridge PhD thesis ´The Mathieu group M_24 and related topics´ or his paper in the journal of symbolic computation last year.

Hope this helps,

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I will look into Conway, Sloane when I stop in the library. I am also trying to decompose Leech lattice in following way. Take any 4-frame from Leech lattice. The question is when generated D4 lattice (24 vectors) is subset of Leech. Similar question: when E8 lattice generated by given 8-frame in Leech is subset of Leech ? If I remember correctly the paper "Pieces of eight" create Leech lattice from 3 perpendicular E8 lattices. Then if we know all E8 sublattices of Leech we could - hopefully - combine them into decomposition into 4095 "crosses" = 24-frames. Regards, Marek –  Marek Mitros Mar 19 '10 at 8:24
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I dont know about a decomposition into 48, but there is extensive work in coding theory, just search leech lattice decoder, lots of papers by conway&sloane, Forney, and Be'ery&Vardy. The methods i am familiar are the original in terms of the golay code, a checkerboard decomposition with hexacode, and with e8. The decompositions lead to faster decoding schemes for encoded words that are sent over noisy channels, leech lattice is definitely a favourite of communication theorists.

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Hi again,

Now I think I could perform decomposition using Wilson definition of Leech lattice using octonions (2008). I this definition Leech lattice is easily seen as union of 819 E8 sublattices; 819 = 3*(1+16+16*16). Having this we can decompose each E8 lattice into crosses. Last step (little vague) would be to find decomposition of 819 E8 lattices into 273 triples.

But now I am struggling with following problem. Consider 2A class in Co1 having 819*759*75 elements. Each element a from 2A have two representatives in Co0. Element a corresponds to E8 sublattice in Leech defined as {v: av=-v} where I call by a also proper preimage in Co0. Now the opposite having E8 sublattice L in Leech I want to find element a(E8) in 2A class. My straightforward function build c*d*c^-1 where d is diagonal matrix changing sign in octad [1..8]. But I have not obtained Co0 element.

My goal is to find relation between Order(ab) for a,b in 2A and corresponding geometry of two E8 sublattices. The Order(ab) can be 2,3,4,5,6.

Take any other sporadic group g and certain conjugacy class cg of order 2 elements. Is it known possible values of Order(ab) for a,b in cg ?

Regards, Marek

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Why don't you post this as a new question? –  Someone Feb 8 '12 at 14:02
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