### 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) |
---|---|

Pages (from-to) | 379-427 |

Number of pages | 49 |

Journal | Pacific Journal of Mathematics |

Volume | 202 |

Issue number | 2 |

State | Published - 2002 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*202*(2), 379-427.

**Irreducibility of tensor squares, symmetric squares and alternating squares.** / Magaard, Kay; Malle, Gunter; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 202, no. 2, pp. 379-427.

}

TY - JOUR

T1 - Irreducibility of tensor squares, symmetric squares and alternating squares

AU - Magaard, Kay

AU - Malle, Gunter

AU - Tiep, Pham Huu

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0036408236&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036408236&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036408236

VL - 202

SP - 379

EP - 427

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -