Orthogonal Complements of Root Lattices in E_8 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:50:24Z http://mathoverflow.net/feeds/question/68356 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68356/orthogonal-complements-of-root-lattices-in-e-8 Orthogonal Complements of Root Lattices in E_8 unknown (google) 2011-06-21T09:42:43Z 2011-06-21T12:19:08Z <p>I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $E_8$ lattice. Is the orthogonal complement of $L$ in $E_8$ unique up to isomorphism, or for different primitive embeddings could I get non-isomorphic complements? </p> http://mathoverflow.net/questions/68356/orthogonal-complements-of-root-lattices-in-e-8/68368#68368 Answer by Tony Pantev for Orthogonal Complements of Root Lattices in E_8 Tony Pantev 2011-06-21T12:11:22Z 2011-06-21T12:11:22Z <p>You can get different orthogonal complements for different embeddings. There are two different embeddings of $A_{7}$ in $E_{8}$ so that for the first embedding the orthogonal complement is the lattice $A_{1}$, and for the second embedding the orthogonal complement is the lattice $\langle 8 \rangle$. </p>