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BRX TH-374
Octonions: Invariant Representation of the Leech
Lattice
Geo rey Dixon
Department of Mathematics or Physics
Brandeis University
Waltham, MA 02254
email: [email protected]
Department of Mathematics
University of Massachusetts
Boston, MA 02125
email: [email protected]
April 10, 1995
Abstract
The Leech lattice, , is represented on the space of o ctonionic 3-
24
vectors. It is built from two o ctonionic representations of E , and is
8
reached via . Itisinvariant under the o ctonion index cycling and
16 doubling maps. processed by the SLAC/DESY Libraries on 10 Apr 1995. 〉
PostScript
supp orted in part by alien invaders. 0 HEP-TH-9504040
1. Intro duction.
My interest in the Leech lattice arose from [1], in whichitwas made clear
that is in some sense the most select of all lattices. In [2] I made it clear
24
that I felt that the real division algebras (reals, complexes, quaternions, and
o ctonions) are the most select of all algebraic ob jects, and it was this selectness
that inspired myinterest in these algebras. The same may b e said of .
24
However, like many p eople, I generally nd other p eople's understandings of
mathematical ob jects p erplexing, and I only gain any degree of understanding
myself by re-representing the ob jects in a form with which I am comfortable.
That is what I have done here.
The form itself is based on the o ctonion multiplication I employed in [2],
and in a recent series of pap ers [3-6]. The multiplication is one of four for
which index cycling and doubling are automorphisms. These invariances are
translated to my representation of .
24
I'm going to ask the interested reader to lo ok at [3-6] for background mate-
rial on the relationship of the o ctonions to E and . The notation I employ
8 16
here, and in those previous pap ers, is the same I employed in [2]. The o ctonion
algebra is denoted O.
2. Three E 's in .
8 24
In [3] I employed the following sets to pro duce renumb erings of the o ctonion
pro duct:
= fe g; 16
0 a
p
= f(e e )= 2: a; b distinctg; 112
1 a b
= f(e e e e )=2:a; b; c; d distinct;
2 a b c d
(1)
e (e (e e )) = 1g; 224
a b c d
p
P
7
= f( e )= 8: oddnumb er of +'sg;
3 a
a=0
a; b; c; d 2f0; :::; 7g: 128
The numb ers down the right side are the orders of these sets of o ctonion units
7
(ie., each of unit norm, so elements of S , the o ctonion 7-sphere). It should b e
noted that these sets dep end on the o ctonion pro duct chosen; mychoice is that
used in [2-6]. 1
From these sets we de ne
ev en
E = [ ;
0 2
8
(2)
odd
E = [ :
1 3
8
Each of these contains 240 elements and is the inner shell (normalized to unity)
ev en 3
of an E lattice [4].From E we de ne in O :
8
8
1 ev en
= f
24 8
1 3
That is, consists of all the elements of O zero in exactly two comp onents,
24
ev en
the third comp onent an elementofE . This set is the rst rung in the ladder
8
to a full Leech lattice, (in particular, we will end up with an inner shell
24
for consisting of 196560 elements of unit norm in the 24-dimensional space
24
3 1
O ). The set accounts for 3 240 = 720 elements.
24
3. Three 's in .
16 24
De ne
1
2 odd
p
= f
24 8
2
(4)
1
y
e ;a2f0; :::; 7gg: AB =
a
2
This constitutes that subset of the inner shell of the full each elementof
24
which has exactly two nonzero comp onents. For each pair of comp onents chosen
to b e nonzero, there are 16 240 di erent combinations satisfying the pro duct
condition in (4). Since there are three ways all together to ll exactly two
comp onents, this subset accounts for 3 16 240 = 11520 elements.
1 2
In addition, the subset of [ consisting of elements zero in one of the
24 24
three comp onents (so the order of such a subset would b e 240 + 240 + 3840 =
4320) is a representation of the inner shell of the lattice (see [5-6]).
16 2
4. Inner Shell.
24
3
All that remains is to nd the set of elements of the inner shell of that
24
24
are nonzero in all three comp onents. The order of this set must b e
2
196560 720 11520 = 184320 = 3 16 240:
A rotation of a representation of the Leech lattice develop ed in [1] gavemea
3
guess as to how to construct a representation of consistent with the de ni-
24
1 2
tions of and . A desire for it to b e as symmetric as p ossible led to the
24 24
representation b elow.
3
Before presenting the representation of , one last word: in general, if
24
< A;B;C >2 as represented here, then so are < A; B; C>with all
24
p ossible sign combinations, and all six p ermutations of each of these elements.
In addition, the representation b eing constructed will b e invariant under b oth
index cycling and index doubling, so given a particular element many other el-
ements may b e easily constructed via these op erations (see [2]).
3
All of can b e constructed from the following two elements in linear
24
1 2
combination with and :
24 24
1 1 1 1
(1 + e + e + e ); (1 + e + e + e ); ( 1+ e + e +e )+ e >: U = <
3 5 6 3 5 6 3 5 6 7
4 4 4 2
even even odd
with 1 with 1 with 1
(5)
1 1 1 1
V = < ( 1+e +e +e ); ( 1+e +e +e ); (e +e +e e )+ >:
3 5 6 3 5 6 1 2 4 7
4 4 4 2
odd odd odd
with 1 with 1 w/o 1
(6)
There are several p oints to make here. First, it will b e observed that
kU k = kV k =1:
Also, since
e e = e ;
3 5 6
these three o ctonions form a quaternionic triple. Hence
1 1 1
(1+e +e +e ); (e +e +e e ) 2 : (7)
3 5 6 1 2 4 7 2
4 4 2 3
The rst two comp onents of U haveaneven numb er of "+" signs and contain
the identity ("with 1"; the p oint of all this is to develop a pattern). The third
1
comp onent is the sum of an elementof , which has an o dd numberof"+"
2
2
1
signs and contains the identity, and an elementof with a distinct index.
0
2
The rst two comp onents of V haveanoddnumb er of "+" signs and contain
1
the identity . The third comp onent is the sum of an elementof , which has
2
2
1
an o dd numb er of "+" signs and no identity ("w/o 1"), and an elementof
0
2
with a distinct index.
Note also that the elements U and V are invariant under index doubling,
but index cycling leads to six other elements for each. For example, cycling the
indices of U by 3 leads to
1 1 1 1
< (1 + e + e + e ); (1 + e + e + e ); ( 1+ e + e +e )+ e >:
6 1 2 6 1 2 6 1 2 3
4 4 4 2
Index doubling on this leads to
1 1 1 1
(1 + e + e + e ); (1 + e + e + e ); ( 1+ e + e +e )+ e >: <
5 2 4 5 2 4 5 2 4 6
4 4 4 2
Many other unit norm elements may b e constructed from U and V by adding
1 2
and subtracting elements of and . Requiring that the result have unit
24 24
norm, and that the rst two comp onents of the result b e the same as those of
U leads to the following 16 elements (note that this means we are mo difying U
1
with elements of nonzero in the third comp onent):
24
1 1 1 1
< (1 + e + e + e ); (1 + e + e + e ); ( 1+e +e +e ) e >;
3 5 6 3 5 6 3 5 6 7
4 4 4 2
1 1
(1 e + e + e ) e >;
3 5 6 2
4 2
1 1
(1 + e + e e ) e >;
3 5 6 4
4 2
1 1
(1 + e e + e ) e >;
3 5 6 1
4 2
(8)
where each of the pairs are indep endent. (Note that this set of 16 elements
is invariant under index doubling.) In general we will nd that if the rst two
1
comp onents are elements of , and the overall element has unit norm, then
2
2
there will b e 16 p ossible forms for the third comp onent similar to those shown in
(8) ab ove. Moreover, the rst two such comp onents must di er bymultiplication
from the left by some e ;a2f0; :::; 7g, just as was the case for the nonzero
a
1
2
comp onents of is 224, and there are three p ositions . Since the order of
2
24
2
1 1
+ , this accounts for 3 16 16 224 for the p eculiar comp onent from
0 2
2 2
3
new unit elements in .
24 4
Let
1 1 1 1
0
V = < ( 1+ e + e +e ); ( 1+ e + e +e ); ( e e e +e ) >;
3 5 6 3 5 6 1 2 4 7
4 4 4 2
which is the same as V with the sign of the third comp onentchanged. Take the
di erence,
1 1 1
0
W = U V = < ; ; (1 + e + e + e + e + e + e + e ) >:
1 2 3 4 5 6 7
2 2 4
(9)
As was true of U and V , the element W is index doubling invariant. And it has
unit norm. Maintaining the unit norm prop erty, and mo difying only the third
1
comp onent with elements of , leads to the following 16 variations:
24
1 1 1
0
; ; (1 + e + e + e + e + e + e + e ) >; W = U V = <
1 2 3 4 5 6 7
2 2 4
1
(1 e e e + e e + e + e ) >;
1 2 3 4 5 6 7
4
1
(1 + e e e e + e e + e ) >;
1 2 3 4 5 6 7
4
1
(1 + e + e e e e + e e ) >;
1 2 3 4 5 6 7
4
1
(1 e + e + e e e e + e ) >;
1 2 3 4 5 6 7
4
1
(1 + e e + e + e e e e ) >;
1 2 3 4 5 6 7
4
1
(1 e + e e + e + e e e ) >;
1 2 3 4 5 6 7
4
1
(1 e e + e e + e + e e ) >
1 2 3 4 5 6 7
4
(10)
(this is another set closed under index doubling, and in this case, also index
cycling). In general, for any element whose rst two comp onents are drawn
1
from (and all such combinations are o ccur), there will b e 16 p ossible third
0
2
y
1
comp onents (again, assuming overall unit norm), each an elementof .So
3
2
1
, and for each of these 16 second that means 16 rst comp onents from
0
2
1
comp onents from , and for each combination 16 third comp onents from
0
2
y y
1 1
.Taking into account p ermutations of the element to the rst and
3 3
2 2
second comp onents, wehave accounted for 3 16 16 16 new unit elements
3 3
of . And that's all there are. So the order of is, as exp ected
24 24
3 16 16 (16 + 224) = 3 16 16 240: 5
The combined elements of
1 2 3
[ [
24 24 24
constitute an inner shell of a representation of the Leech lattice, .
24
The interested reader should nowhave enough information to b e able to get
a feeling for how this representation works. I am nding it considerably easier
to play with than the other representations I've found. Further developments
will b e revealed as they b ecome available (as, presumably, will b e my reason for
pursuing this line of questioning).
I'd liketoacknowledge several electronic conversations with Tony Smith, who
maintains a fascinating Web site at Georgia Tech: www.gatech.edu/tsmith/home.html
References
[1] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups,
(Springer-Verlag, 2nd Ed., 1991).
[2] G.M. Dixon, Division Algebras: Octonions, Quaternions, Complex Num-
bers, and the Algebraic Design of Physics, (Kluwer, 1994).
[3] G.M. Dixon, Octonion X-Pro duct Orbits, hep-th 9410202.
[4] G.M. Dixon, Octonion X-Pro duct and Octonion E Lattices, hep-th
8
9411063.
[5] G.M. Dixon, Octonions: E Lattice to , hep-th 9501007.
8 16
[6] G.M. Dixon, Octonion XY-Pro duct, hep-th 9503053. 6