Computing in unitriangular matrices over finite fields
The authors describe algorithms for computations of conjugacy classes, the conjugacy vector of \(G_{n,p}\) (the sylow \(p\)-sugroup of SL\((n,p)\) formed by the upper unitriangular matrices), the character (rational or real) of the elements of \(G_{n,p}\), the cardinality of the centeralizer of each matrix of \(G_{n,p}\), the conjugacy vector of normal subset \(N_{\pi}\) corresponding to a pivot disposition \(\pi\), and the character (inert or ramification) of each entry of any matrix of \(G_{n,p}\). For \(p=2\), by using these algorithms, it is proved that the Kirillov conjecture, every matrix in \(G_{n,2}\) is conjugate to its inverse, holds for \(12\geq n\), but for \(n=13\) there exists a unique pair of inverse of conjugacy classes not conjugate. For \(n=14\), complete list of canonical matrices of the 22 counterexamples to Kirillov's conjecture is given. For \(14\geq n\), it is proved that \(A\) and \(A^5\) are conjugate, and for \(n=25\) a matrix \(A\in G_{25,2}\) is found such that \(A\) and \(A^5\) are conjugate. In addition, for \(n=32\) a matrix \(A\in G_{32,2}\) is found such that \(A\) and inverse of \(A\) are conjugate but \(A\) and \(A^5\) are not conjugate, showing that Isaac's conjecture, every real matrix in \(G_{n,2^m}\) is actually rational not true. Finally, for every odd prime \(p\), a matrix \(A\in G_{n,p}\), with \(n=2p+1\) is obtained such that \(A\) and \(A^{1+p}\) are not conjugate. Consequently, there exist irreducible characters of \(G_{6p+1,p}\) there are not \(Q(\varepsilon_p)\)-valued, where \(\varepsilon_p\) is a primitive \(p\)th root of unity. Besides, for \(p=3\) and \(13\geq n\) the conjugacy vector of \(G_{n,3}\) is computed and verified that every matrix \(A\in G_{n,3}\) is conjugate to \(A^4\). Thus, for all \(13\geq n\), the character values of \(G_{n,3}\) always lie in \(Q(\varepsilon_3)\).
scientific article; zbMATH DE number 2081245