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Globally Irreducible Representations of Finite Groups and Integral Lattices

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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 2B2(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 (pn-1)/2 of the symplectic group Sp2n(p) (p ≡ 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pn−1). A summary of currently known globally irreducible representations is given.

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  1. Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH, 43210, U.S.A.

    Pham Huu Tiep

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Tiep, P.H. Globally Irreducible Representations of Finite Groups and Integral Lattices. Geometriae Dedicata 64, 85–123 (1997). https://doi.org/10.1023/A:1004917606870

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