Extending Snow's algorithm for computations in the finite Weyl groups
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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 is given. The elements of W(D_4) are specified in two ways: as reduced expressions and as matrices of the faithful representation. We present 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): using Carter diagrams, using reduced expressions and using signed cycle-types. In the appendix, we provide an implementation of the algorithm in Python.
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