Extending Snow’s algorithm for computations in the finite Weyl groups

In 1990, D. Snow proposed an effective algorithm for computing the orbits of finite Weyl groups. Snow’s algorithm is designed for computation of weights, W-orbits, and elements of the Weyl group. An extension of Snow’s algorithm is proposed, which allows to find pairs of mutually inverse elements together with the calculation of W-orbits in the same runtime cycle. This simplifies the calculation of conjugacy classes in the Weyl group. As an example, the complete list of elements of the Weyl group $W(D_{4})$ obtained using the extended Snow’s algorithm. The elements of $W(D_{4})$ are specified in two ways: as reduced expressions and as matrices of the faithful representation. Then we give a partition of this group into conjugacy classes with elements specified as reduced expressions. Various forms are given for representatives of the conjugacy classes of $W(D_{4})$ : with Carter diagrams, with reduced expressions, and with signed cycle-types. In the Appendix, we provide an implementation of the algorithm in Python.