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

DOI:
https://doi.org/10.1090/S0002-9947-99-02469-1

Published electronically:
January 26, 1999

MathSciNet review:
1653379

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an odd prime. It is known that the symplectic group has two (algebraically conjugate) irreducible representations of degree realized over , where . We study the integral lattices related to these representations for the case . (The case has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or -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

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

Email:
tiep@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02469-1

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