### Abstract

Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (p^{n} + 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 p^{n} = 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 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*351*(5), 2101-2139.

**Symplectic group lattices.** / Scharlau, Rudolf; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 351, no. 5, pp. 2101-2139.

}

TY - JOUR

T1 - Symplectic group lattices

AU - Scharlau, Rudolf

AU - Tiep, Pham Huu

PY - 1999

Y1 - 1999

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

UR - http://www.scopus.com/inward/record.url?scp=22644451817&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22644451817&partnerID=8YFLogxK

M3 - Article

VL - 351

SP - 2101

EP - 2139

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -