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Symplectic group lattices

Authors: Rudolf Scharlau and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 351 (1999), 2101-2139
MSC (1991): Primary 20C10, 20C15, 20C20, 11E12, 11H31, 94B05
Published electronically: January 26, 1999
MathSciNet review: 1653379
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Abstract: Let $p$ be an odd prime. It is known that the symplectic group $Sp_{2n}(p)$ has two (algebraically conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized over $\mathbb{Q}(\sqrt{{\epsilon}p})$, where ${\epsilon}= (-1)^{(p-1)/2}$. We study the integral lattices related to these representations for the case $p^{n} \equiv 1 \bmod 4 $. (The case $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 $p$-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.

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Additional Information

Rudolf Scharlau
Affiliation: Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany

Pham Huu Tiep
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, University of Florida, Gainseville, Florida 32611

Keywords: Integral lattice, unimodular lattice, $p$-modular lattice, finite symplectic group, Weil representation, Maslov index, linear code, self-dual code
Received by editor(s): December 10, 1996
Published electronically: January 26, 1999
Additional Notes: Part of this work was done during the second author’s stay at the Department of Mathematics, Israel Institute of Technology. He is grateful to Professor D. Chillag and his colleagues at the Technion for stimulating conversations and their generous hospitality. His work was also supported in part by the DFG
Article copyright: © Copyright 1999 American Mathematical Society

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