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arXiv · 2212.03156

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

Rafael Stekolshchik
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Abstract

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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Rafael Stekolshchik. 2022-12-06. Extending Snow's algorithm for computations in the finite Weyl groups. https://doi.org/10.1186/s13663-023-00755-w

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