### Abstract

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem.” Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GLn(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in q with integral coefficients

Original language | English |
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Pages (from-to) | 135-155 |

Number of pages | 22 |

Journal | Journal of Algebra |

Volume | 233 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2000 |

Externally published | Yes |

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## Cite this

Murray, S. (2000). Conjugacy classes in maximal parabolic subgroups of the general linear group.

*Journal of Algebra*,*233*(1), 135-155. https://doi.org/10.1006/jabr.2000.8431