Cyclotomic fields and lattices

Adam P. Goucher discusses lattices constructed from cyclotomic fields. In particular, one possible lattice that can be constructed is the Leech lattice. As the Leech lattice has a notorious lack of simple constructions, it’s worth considering this in more detail.

If $\zeta_n$ is a primitive $n$th root of unity, the $n$th cyclotomic ring $\mathbb Z[\zeta_n]$ is the set of numbers of the form $a+b\zeta_n+c\zeta_n^2+\cdots$, where $a$, $b$, $c$, and so on are integers. Elements of $\mathbb Z[\zeta_n]$ can be added and multiplied. The cyclotomic field is similar, but allows the coefficients to be rationals.

Each number in $\mathbb Z[\zeta_n]$ has a minimal polynomial. This is a polynomial with integer coefficients, leading coefficient 1, and having the number as one of the roots. If the minimal polynomial has second coefficient $a$ and degree $m$, then the trace is defined to be $-\frac{a}{m}$. Equivalently, the trace is the average of all the roots of the minimal polynomial.

Inner products

Lattices of the same dimension are distinguished by their inner product. The inner product is essentially a way to compute lengths and angles. For lattices in Euclidean space, the inner product is the Euclidean dot product.

Over $\mathbb Z[\zeta_n]$, an inner product can be defined by $\langle u,v\rangle=\operatorname{tr}(\alpha u v^*)$ where $\alpha$ is a fixed number and $v^*$ is the conjugate. This has the property of symmetry: $\langle \zeta_n u,\zeta_n v\rangle=\langle u,v\rangle$. Thus, any lattice constructed in this way will have quite a lot of symmetry. For example, using $\mathbb Z[\zeta_{84}]$, the lattice has only 24 dimensions, yet has an order 84 symmetry.

For this to actually define an inner product, $\langle u,v\rangle=\langle v,u\rangle$. This happens if and only if $\alpha$ is real. Additionally, $\langle u,u\rangle>0$ for all nonzero numbers $u$.

One simple example is $\alpha$ being a positive integer. Are there other examples?

There’s a large class of examples we don’t need to worry about. For any $u$, $uu^*>0$, and also $\operatorname{tr}(uu^*)>0$ as well. The condition $uu^*$ ends up as a linear condition with irrational coefficients on the components of $uu^*$ unless $n=3$ or $6$, so we do not need to care about it. If we want $\operatorname{tr}(\alpha uu^*)>0$, this ends up as a linear condition on the components of $uu^*$. If it has to be always satisfied, then either there’s another linear constraint on $uu^*$, or the condition $\operatorname{tr}(\alpha uu^*)>0$ is exactly the same as $\operatorname{tr}(uu^*)>0$. (In hindsight, this was kind of tautological.) Since both are linear in $uu^*$, this implies $\operatorname{tr}(\alpha uu^*)=\operatorname{tr}(\alpha)\operatorname{tr}(uu^*)$. By polarization, this means $\operatorname{tr}(\alpha uv^*)=\operatorname{tr}(\alpha)\operatorname{tr}(uv^*)$. Therefore, we just get a scalar multiple of the inner product $\langle u,v\rangle=\operatorname{tr}(uv^*)$. Thus, we will restrict attention to this specific inner product.

Some lattices

We will restrict ourselves to the cyclotomic rings of even order, since the cyclotomic ring of order $n$ for $n$ odd is the same as the cyclotomic ring of order $2n$. ($\zeta_{2n}=-\zeta_n^{\frac{n+1}{2}}$)

For 4, we have the basis $\{1,i\}$ with realization $1=(1,0),i=(0,1)$.This is the ordinary square lattice. For 6, we get the hexagonal lattice. For 8, we have a 4-dimensional lattice. We can pick the basis $\{1,\zeta_8,i,\zeta_8^3\}$. These basis vectors are all at right angles to each other. For 10, we get another 4D lattice, with 4 basis vectors all at $\cos^{-1}\left(-\frac{1}{4}\right)$ angles to each other. Interestingly, these together with the negative of their sum form a regular simplex. 12 yields another 4D lattice. Picking the basis $\{1,\zeta_{12},\zeta_{12}^2,i\}$, we find the pairwise angles as follows: $$\begin{matrix} – & 90 ^ \circ & 60 ^ \circ & 90 ^ \circ \\ 90 ^ \circ & – & 90 ^ \circ & 120 ^ \circ \\ 60 ^ \circ & 90 ^ \circ & – & 90 ^ \circ \\ 90 ^ \circ & 120 ^ \circ & 90 ^ \circ & – \end{matrix}$$ By drawing a Dynkin diagram, we identify this lattice as $A_2^2$, i.e. two copies of the hexagonal lattice.

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