Here’s a fun challenge for people confined due to coronavirus.
The E8 lattice is a thing of beauty, taking full advantage of the magic properties of the number 8. The octahedron has 8 sides. Wouldn’t it be cool if you could build the E8 lattice from the humble octahedron?
David Harden thinks he’s found a way! But I haven’t carefully checked that it works: it takes some calculations. Can you check it?
He wrote about it here:
• David L. Harden, What other lattices are obtainable from this noncommutative ring?, MathOverflow, 14 March 2020.
Let me dive in and explain it.
You start with the octahedron. You take the double cover of its group of rotational symmetries and think of this as a group of unit quaternions. You get 48 very special quaternions this way. You then take integer linear combination of these quaternions: call the set of these It turns out that every element of is of the form
where are rational.
Not every quaternion of this form is in But that’s okay: what we have is enough to let us think of as sitting inside which is an 8-dimensional vector space over the rational numbers.
To see this lattice as E8, we use a special inner product on If
then their usual quaternion inner product is
for some rational numbers and We then define the inner product on by
So, we have a lattice in an 8-dimensional rational vector space with an inner product… and David claims that it’s a copy of the E8 lattice!
To prove this, it’s enough to show that
1) the inner product of any two vectors in is an integer,
and that either
2a) the inner product of any vector in with itself is even, and if is any list of generators for then the determinant of the 8 × 8 matrix is 1
2b) there are no vectors in of length 1, and 240 vectors in whose inner product with themselves is 2.
Either of these characterizes E8. David has actually checked 1) and both 2a) and 2b), but some of his calculations take work, so I hope some of you can check them again.
Let me help you out a bit.
If we start with the octahedron and take the double cover of its group of rotational symmetries, you get something called the binary octahedral group, which has 48 elements. I described it here:
• John Baez, The binary octahedral group, Azimuth, 29 August 2019.
I even described how to think of its elements as unit quaternions. We get 8 like this:
and 16 like this:
These 24 are the vertices of a wonderful shape called the 24-cell, drawn here by Greg Egan:
The remaining 24 elements of the binary octahedral group form the vertices of a second 24-cell! Here they are:
Starting from the 48 elements of the binary octahedral group, we can easily get ahold of 8 generators of the lattice David Harden chose these:
I think it’s pretty easy to see that these generate the lattice I also think it’s pretty easy to check 1). For this we need to check that the inner product of any two of these vectors is an integer. But we have to use the correct inner product: the one described above!
So, for example, the ordinary quaternion inner product of and is
After doing a bunch of these, I’m convinced that all the inner products are integers. So the hard part is checking 2a) or 2b).
Can you do it? The matrix of inner products needed for checking 2a) is called the Gram matrix, and David Harden has computed it, but you could compute it yourself and check that its determinant is 1.
By the way, all this is similar to the construction of the E8 lattice from the icosahedron, explained in Conway and Sloane’s book and later here:
• John Baez, From the icosahedron to E8.
There you start with the icosahedron. You take the double cover of its group of rotational symmetries, which is called the binary icosahedral group, and think of this as a group of unit quaternions. You then take integer linear combination of these quaternions: call the set of these It turns out that every element of is of the form
where are rational.
Not every quaternion of this form is in But that’s okay: this is enough to let us think of as sitting inside which is an 8-dimensional vector space over the rational numbers. And since the rationals sit in the reals, you can think of as a lattice in an 8-dimensional real vector space.
To see this lattice as , we put this norm on
And we get a copy of the E8 lattice this way, since 1) and 2a) hold, and for that matter also 2b).
The lattice is actually a subring of the quaternions, which Conway and Sloane call the icosians. Harden’s lattice is also a subring of the quaternions, and he has dubbed it the octians.
It’ll be cute to see an octahedron giving E8, since they are both connected to the number eight! But even if this construction really works, I have no idea what it ‘really means’.
Is this construction new?