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

TL;DR

This paper extends Snow's algorithm to efficiently compute pairs of inverse elements and W-orbits in finite Weyl groups, simplifying conjugacy class calculations with an implementation in Python.

Contribution

The paper introduces an extension to Snow's algorithm that enables simultaneous computation of inverse pairs and W-orbits, improving the process of classifying conjugacy classes in Weyl groups.

Findings

Successfully computed all elements of W(D_4) using the extended algorithm.

Provided multiple representations of conjugacy class elements, including Carter diagrams and reduced expressions.

Implemented the algorithm in Python for practical use.

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.