Classes of conjugate elements of unitriangular matrix groups (Q2759419)

Let \(\text{UT}(n,p)\) be the group of all upper unitriangular matrices of \(n\)-th order over a finite prime field. The authors propose new algorithms for finding representatives for all classes of conjugate elements of the group \(\text{UT}(n,2)\). The quantity of all classes of conjugate elements of the group \(\text{UT}(n,2)\) is given by \(t_n(2)=n!\sum\rho^{k_1}(n_1)\cdots\rho^{k_s}(n_s)/((n_1!)^{k_1}\cdots(n_s!)^{k_s}\cdot k_1!\cdots k_s!)\), where summation is taken over all positive integers \(k_1,\dots,k_s\), \(n_1,\dots,n_s\), (\(s\geq 1\)) such that \(n=k_1n_1+\cdots+k_sn_s\), \(1\leq n_1<n_2<\cdots<n_s<n\), \(\rho(n_i)\) is the quantity of classes of conjugate elements of the group \(\text{UT}(n_i,2)\), (\(i=1,\dots,s\)). The authors present numerical results concerning these problems.