2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58494We 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.23 pages, 6 figures. See also http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal of AlgebraGroup TheoryRepresentation Theory20Ctext