Symplectic group lattices
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- by Rudolf Scharlau and Pham Huu Tiep PDF
- Trans. Amer. Math. Soc. 351 (1999), 2101-2139 Request permission
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.References
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Additional Information
- Rudolf Scharlau
- Affiliation: Department of Mathematics, University of Dortmund, 44221 Dortmund, Germany
- Email: rudolf.scharlau@mathematik.uni-dortmund.de
- 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
- MR Author ID: 230310
- Email: tiep@math.ufl.edu
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 2101-2139
- MSC (1991): Primary 20C10, 20C15, 20C20, 11E12, 11H31, 94B05
- DOI: https://doi.org/10.1090/S0002-9947-99-02469-1
- MathSciNet review: 1653379