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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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 $\langle 8 \rangle$.
answered Jun 21, 2011 at 12:11
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$\begingroup$ Sorry, I should have said that PRIMITIVE embeddings. I think then that the discriminant groups of $L$ and $ L^\perp $ will be isomorphic, so at least I still have some hope. I've edited the question to reflect this. $\endgroup$– user4192Commented Jun 21, 2011 at 12:21
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$\begingroup$ According to Dynkin's classification of root sublattices in $E_{8}$, there are only five root subblattices in $E_{8}$ (modulo the Weyl group action) that admit more than one embedding in $E_{8}$. Moreover each of the five admits exactly two non-equivalent embeddings. The five exceptions (and their possible complements) are: $\endgroup$ Commented Jun 21, 2011 at 13:00
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1$\begingroup$ $A_{1}^{\oplus 4}$, ($D_{4}$, $A_{1}^{\oplus 4}$); \linebreak $A_{3}\oplus A_{1}^{\oplus 2}$, ($A_{3}$, $A_{1}^{\oplus 2}\oplus \langle 4\rangle$); \linebreak $A_{3}^{\oplus 2}$, ($A_{1}^{\oplus 2}$, $\langle 4 \rangle^{\oplus 2}$), \linebreak $A_{5}\oplus A_{1}$, ($A_{2}$, $A_{1}\oplus \langle 6 \rangle$); \linebreak $A_{7}$, ($A_{1}$, $\langle 8\rangle$). \linebreak So for the primitive embeddings you get a uniquely determined orthogonal complement. $\endgroup$ Commented Jun 21, 2011 at 13:09
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1$\begingroup$ The original paper is: Dynkin, E. B. Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), 349–462. There is also an English translation: E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras. AMS Translations, volume 6, (1957), 111-244. The part you need is in Table 11 in Chapter II. $\endgroup$ Commented Jun 21, 2011 at 14:06
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2$\begingroup$ The first time I encountered all this was in the excellent paper: Oguiso, Keiji; Shioda, Tetsuji The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40 (1991), no. 1, 83–99. Among other things they show that both embeddings of all five exceptions do occur as narrow Mordell-Weil lattices of rational elliptic surfaces. $\endgroup$ Commented Jun 21, 2011 at 14:06