Weyl Group of Bn as Automorphisms of n-Cube: Isomorphism and Conjugacy

The Weyl groups are important for Lie algebras. Lie algebras arise in the study of Lie groups, coming from symmetries of differential equations, and of differentiable manifolds. The Weyl groups have been used to classify Lie algebras up to isomorphism. The Weyl group associated to a Lie algebra of type Bn and the group of graph automorphisms of the n-cube Aut(Qn) are known to be isomorphic to Z2 ⋊ Sn. We provide a direct isomorphism between them via correspondence of generators. Geck and Pfeiffer have provided a parametrization of conjugacy classes and an algorithm to compute standard representatives. We believe we have a more transparent account of conjugacy in the Weyl group by looking at Aut(Qn). We give a complete description of conjugacy in the automorphism group. We also give an algorithm to recover a canonical minimal length (in the Weyl group sense) representative from each conjugacy class, and an algorithm to recover that same representative from any other in the same conjugacy class. Under the correspondence with the Weyl group, this representative coincides precisely with the minimal length representative given by Geck and Pfeiffer, leading to an easier derivation of their result.