### Abstract

We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups script G sign that act irreducibly on all fundamental representations of script G sign.

Original language | English (US) |
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Pages (from-to) | 379-427 |

Number of pages | 49 |

Journal | Pacific Journal of Mathematics |

Volume | 202 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2002 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Pacific Journal of Mathematics*,

*202*(2), 379-427. https://doi.org/10.2140/pjm.2002.202.379