Symplectic group lattices

Rudolf Scharlau; Pham Huu Tiep

doi:10.1090/s0002-9947-99-02469-1

Symplectic group lattices

Rudolf Scharlau, Pham Huu Tiep

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pⁿ + l)/2 realized over Q(p), where e = (-1)^(p-1)/2. We study the integral lattices related to these representations for the case pn = 1 mod 4. (The case pⁿ = 3 mod 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.

Original language	English (US)
Pages (from-to)	2101-2139
Number of pages	39
Journal	Transactions of the American Mathematical Society
Volume	351
Issue number	5
DOIs	https://doi.org/10.1090/s0002-9947-99-02469-1
State	Published - 1999
Externally published	Yes

Keywords

Finite symplectic group
Integral lattice
Linear code
Maslov index
P-modular lattice
Self-dual code
Unimodular lattice
Weil representation

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/s0002-9947-99-02469-1

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title = "Symplectic group lattices",

abstract = "Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pn + l)/2 realized over Q(p), where e = (-1)(p-1)/2. We study the integral lattices related to these representations for the case pn = 1 mod 4. (The case pn = 3 mod 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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AU - Scharlau, Rudolf

AU - Tiep, Pham Huu

PY - 1999

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N2 - Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pn + l)/2 realized over Q(p), where e = (-1)(p-1)/2. We study the integral lattices related to these representations for the case pn = 1 mod 4. (The case pn = 3 mod 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.

AB - Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pn + l)/2 realized over Q(p), where e = (-1)(p-1)/2. We study the integral lattices related to these representations for the case pn = 1 mod 4. (The case pn = 3 mod 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.

KW - Finite symplectic group

KW - Integral lattice

KW - Linear code

KW - Maslov index

KW - P-modular lattice

KW - Self-dual code

KW - Unimodular lattice

KW - Weil representation

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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Symplectic group lattices

Abstract

Keywords

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