Conjugacy classes in maximal parabolic subgroups of general linear groups

2000/01/06 by Scott H. Murray, Murray, Scott H. · 2 citations
Mathematics · #20C #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20C

paper · pdf · doi:10.48550/arxiv.math/0001031

23 pages, 6 figures. See also http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal of Algebra

arxiv created 2000/01/06 · openalex publication_date 2000/01/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 GL_n(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.

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