Leech lattice

Leech lattice

The

Leech lattice

is the unique even (

http://planetmath.org/EvenLattice

)

unimodular lattice

of

dimension

(

http://planetmath.org/Dimension2

) 24 having no elements of norm 2. Its

automorphism group

(

http://planetmath.org/EquivalentCode

) is the largest

Conway group

C

⁢

o

0

(sometimes denoted by

⋅

0

). The

quotient

of

C

⁢

o

0

by its center is called

C

⁢

o

1

, a sporadic simple group.

The construction of the Leech lattice below depends on the existence of the

extended binary Golay code

𝒢

24

(for a construction of the latter, see

miracle octad generator

).

1

Construction of the Leech lattice

Let

Ω

=

{

1

,

2

,

…

,

24

}

and assume we have constructed the Golay

𝒢

24

on

Ω

. The Leech lattice

Λ

is the set of all points

1

8

⁢

(

x

1

,

x

2

,

…

,

x

24

)

in

ℝ

24

where each

x

i

is an

integer

, such that

•

For some integer

m

, we have

x

i

≡

x

j

≡

m

(

mod

2

)

for all

i

,

j

∈

Ω

;

•

For any integer

n

, the set of

coordinates

{

i

∈

Ω

:

x

i

≡

n

(

mod

4

)

}

is in

𝒢

24

;

•

∑

i

∈

Ω

x

i

≡

4

⁢

m

(

mod

8

)

.

2

Properties of the Leech lattice

1. The Leech lattice

Λ

is an unimodular lattice; in other words:

•

The set

Λ

spans all of

ℝ

24

as an

ℝ

-

vector space

.

•

For any

x

,

y

∈

Λ

, the

scalar product

x

⋅

y

is an integer.

•

For any

x

∈

Λ

, the norm

x

⋅

x

is an

even integer

.

•

The volume of the

fundamental parallelogram

of

Λ

is 1.

2. Let

Λ

⁢

(

n

)

=

{

x

∈

Λ

:

x

⋅

x

=

2

⁢

n

}

. Then

|

Λ

⁢

(

0

)

|

=

1

,

|

Λ

⁢

(

1

)

|

=

0

,

|

Λ

⁢

(

2

)

|

=

196560

,

|

Λ

⁢

(

3

)

|

=

16773120

,

|

Λ

⁢

(

4

)

|

=

398034000

.

3. The automorphism group (

http://planetmath.org/EquivalentCode

)

Aut

⁢

(

Λ

)

is the largest Conway group

C

⁢

o

0

with order

8 315 553 613 086 720 000

=

2

22

⋅

3

9

⋅

5

4

⋅

7

2

⋅

11

⋅

13

⋅

23

.

4. The group

C

⁢

o

0

acts transitively (

http://planetmath.org/LeftAction

) on the sets

Λ

⁢

(

2

)

,

Λ

⁢

(

3

)

,

Λ

⁢

(

4

)

. For

n

=

2

,

3

, the imprimitivity blocks of the action of

C

⁢

o

0

on

Λ

⁢

(

n

)

are the sets

{

x

,

-

x

}

where

x

∈

Λ

⁢

(

n

)

. The imprimitivity blocks of the action of

C

⁢

o

0

on

Λ

⁢

(

4

)

are sets of 48 vectors called

. Any two distinct vectors in a are either or

orthogonal

, and are

congruent

(

http://planetmath.org/QuotientGroup

) modulo

2

⁢

Λ

.

5. Any vector in

Λ

is modulo

2

⁢

Λ

to a vector in

Λ

⁢

(

n

)

for one of

n

=

0

,

2

,

3

,

4

. The imprimitivity blocks of the action of

C

⁢

o

0

on these sets account for all classes (

http://planetmath.org/EquivalenceClass

) of

Λ

/

2

⁢

Λ

:

1

+

|

Λ

⁢

(

2

)

|

/

2

+

|

Λ

⁢

(

3

)

|

/

2

+

|

Λ

⁢

(

4

)

|

/

48

=

2

24

=

|

Λ

/

2

⁢

Λ

|

.

References

1

J. H. Conway and N. J. A. Sloane.

Sphere Packings

, Lattices, and Groups. Springer-Verlag, 1999.

Title

Leech lattice

Canonical

name

LeechLattice

Date of creation

2013-03-22 18:43:23

Last modified on

2013-03-22 18:43:23

Owner

monster (22721)

Last modified by

monster (22721)

Numerical id

7

Author

monster (22721)

Entry type

Definition

Classification

msc 20D08

Classification

msc 20B25

Classification

msc 11H56

Classification

msc 11H06

Classification

msc 51E10

Related topic

BinaryGolayCode

Related topic

MiracleOctadGenerator

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