# Which lattices have more than one minimal periodic coloring?

The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of the graph $G = (V,E)$ with $V = L$ and $(x,y) \in E$ if $x$ and $y$ differ by a reduced basis element. (NB. I am not quite sure that this graph is the proper one to consider in general, so comments on this would also be nice.)

The root lattice $A_n$ has many minimal periodic colorings if $n+1$ is not prime (I have sketched this here, and some motivation is in the last post in that series); if $n+1$ is prime, then it has essentially one $n+1$-coloring. Two minimal periodic colorings for $A_3$ are shown below (for convenience, compare the tops of the figures): The generic ("cyclic") coloring. A nontrivial example.

The lattices $D_n$ are also trivially 2-colored.

So: are there other lattices that admit more than one minimal periodic coloring? I'd be especially interested to know if $E_8$ or the Leech lattice do.

(A related question: does every minimal periodic coloring of $A_n$ arise from a group of order $n+1$?)

There is a simple way to generate a $24$ colouring of $E_8$ using the $8$ colouring you have for $A_8$. It so happens that $E_8$ is isomorphic to the union of 3 translations of $A_8$,
$E_8 = A_8 + (A_8 + g) + (A_8 + 2g)$
where $g = \left( \tfrac{8}{3}, -\left(\tfrac{1}{3}\right)^8 \right)$. That is, $g$ is a vector with one $\tfrac{8}{3}$ and eight $-\tfrac{1}{3}$'s. See Martinet, Perfect Lattices in Euclidean Spaces. So you just need 3 independently coloured $A_8$'s. The resultant colouring will be periodic if the colourings for $A_8$ are periodic. It's likely that this is not the best colouring possible for $E_8$.