Symplectic group lattices
| Autor: | Pham Huu Tiep, Rudolf Scharlau |
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| Rok vydání: | 1999 |
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| Zdroj: | Transactions of the American Mathematical Society. 351:2101-2139 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/s0002-9947-99-02469-1 |
| Popis: | Let p p be an odd prime. It is known that the symplectic group S p 2 n ( p ) Sp_{2n}(p) has two (algebraically conjugate) irreducible representations of degree ( p n + 1 ) / 2 (p^{n}+1)/2 realized over Q ( ϵ p ) \mathbb {Q}(\sqrt {\epsilon p}) , where ϵ = ( − 1 ) ( p − 1 ) / 2 \epsilon = (-1)^{(p-1)/2} . We study the integral lattices related to these representations for the case p n ≡ 1 mod 4 p^{n} \equiv 1 \bmod 4 . (The case p n ≡ 3 mod 4 p^{n} \equiv 3 \bmod 4 has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or p p -modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index. |
| Databáze: | OpenAIRE |
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