Which lattices have more than one minimal periodic coloring? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:15:06Z http://mathoverflow.net/feeds/question/11453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11453/which-lattices-have-more-than-one-minimal-periodic-coloring Which lattices have more than one minimal periodic coloring? Steve Huntsman 2010-01-11T18:20:00Z 2010-04-27T13:27:58Z <p>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 <a href="http://en.wikipedia.org/wiki/Lattice%5Freduction" rel="nofollow">reduced basis</a> 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.)</p> <p>The root lattice $A_n$ has many minimal periodic colorings if $n+1$ is not prime (I have sketched this <a href="http://blog.eqnets.com/2009/08/17/a-minimal-periodic-coloring-theorem-part-1/" rel="nofollow">here</a>, 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):</p> <p><img src="http://eqnets.files.wordpress.com/2009/08/blog0071.png?w=300" alt="alt text" /> The generic ("cyclic") coloring.</p> <p><img src="http://eqnets.files.wordpress.com/2009/08/blog008.png?w=300" alt="alt text" /> A nontrivial example.</p> <p>The lattices $D_n$ are also trivially 2-colored. </p> <p>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.</p> <p>(A related question: does every minimal periodic coloring of $A_n$ arise from a group of order $n+1$?)</p> http://mathoverflow.net/questions/11453/which-lattices-have-more-than-one-minimal-periodic-coloring/22721#22721 Answer by Robby McKilliam for Which lattices have more than one minimal periodic coloring? Robby McKilliam 2010-04-27T13:27:58Z 2010-04-27T13:27:58Z <p>When you say "reduced basis", I assume you mean that two lattice points are connected in your graph if the distance between them is the <em>minimum distance</em> of the lattice (i.e. the shortest distance between any two lattice points).</p> <p>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 $A_8$, </p> <p>$E_8 = A_8 + (A_8 + g) + (A_8 + 2g)$</p> <p>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 <a href="http://books.google.com.au/books?id=gd9CcFclBRIC&amp;lpg=PP1&amp;ots=Kf3Ur1gWc1&amp;dq=martinet%252C%2520perfect%2520lattices&amp;pg=PA153#v=onepage&amp;q&amp;f=false" rel="nofollow">Martinet, Perfect Lattices in Euclidean Spaces</a>. 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 $E_8$.</p>