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Added primitivity to hypotheses.
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user4192
user4192
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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?

I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to isomorphism, or for different embeddings could I get non-isomorphic complements?

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?

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user4192
user4192
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Orthogonal Complements of Root Lattices in E_8

I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to isomorphism, or for different embeddings could I get non-isomorphic complements?