### Abstract

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L_{2}(q) and ^{2}B_{2}(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (p^{n} - 1)/2 of the symplectic group Sp_{2n} (p) (p = 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(p^{n} - 1). A summary of currently known globally irreducible representations is given.

Original language | English (US) |
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Pages (from-to) | 85-123 |

Number of pages | 39 |

Journal | Geometriae Dedicata |

Volume | 64 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1997 |

### Keywords

- Euclidean integral lattice
- Even unimodular lattice
- Finite simple group
- Globally irreducible representation

### ASJC Scopus subject areas

- Geometry and Topology