Computing in unitriangular matrices over finite fields Original Research Article

Let image be the Sylow p-subgroup of SL(n,p) formed by the upper unitriangular matrices. The aim of this paper is to describe algorithms for the computation of the number of conjugacy classes, the conjugacy vector of image, the character (rational or real) of the elements of image, the cardinality of the centralizer of each matrix of image the conjugacy vector of the normal subset Nπ corresponding to a pivot disposition π, and the character (inert or ramification) of each entry of any matrix of image. For p=2, by using these algorithms, we have proved that Kirillovʹs conjecture, every matrix of image is conjugate to its inverse, holds for nless-than-or-equals, slant12, but for n=13 there exists a unique pair of inverse conjugacy classes not conjugate. A representative pair of these conjugacy classes is given in [J. Algebra 202 (1998) 704]. For n=14, we give the complete list of the canonical matrices of the 22 counterexamples to Kirillovʹs conjecture. For nless-than-or-equals, slant14, we have proved that A and A5 are conjugate and for n=25 we have found a matrix image such that A and A5 are not conjugate. In addition, for n=32 we have found a matrix image such that A and A−1 are conjugate but A and A5 are not conjugate. So, Isaacsʹ conjecture, every real matrix in image is actually rational, is not true.