Stability of the E8-form (Ex)
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− | Show an isometry of unimodular forms | + | Show there exists an isometry of unimodular forms |
− | $\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle$ | + | $\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$ |
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+ | where $\lambda$ is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry $({\mathbb Z}^8,\lambda) \cong E_8$). | ||
Latest revision as of 06:08, 8 January 2019
Show there exists an isometry of unimodular forms
where is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry ).