Induced cycle structures of the hyperoctahedral group
W. Y. C. Chen · University of North Texas Digital Library (University of North Texas) · 1992
The hyperoctahedral group B{sub n} is treated as the automorphism group of the n-dimensional hypercube, denoted Q{sub n}, which is nowadays understood to be a graph on 2{sup n} vertices. It is well-known that B{sub n} can be represented by the group of signed permutations. In other words, any signed permutation induces a permutation on the vertices of Q{sub n} which preserves adjacencies. Moreover, signed permutations also a permutation group on the edge of Q{sub n}, denoted H{sub n}. We study the cycle structures of both B{sub n} and H{sub n}. The technique proposed here is to determine the induced cycle structure of a signed permutation by the number of fixed vertices or fixed edges of a signed permutation in the cyclic group generated by a signed permutation of given type. Here we directly define the type of a signed permutation by a double partition based on its signed cycle decomposition. In this way, we obtain explicit formulas for the number of induced cycles on vertices as well as edges of Q{sub n} of a signed permutation in terms of its type. By further exploring the connection between cycle indices and the structure of fixed points, we obtain the cycle indices of both B{sub n} and H{sub n}. Our formula for the cycle index of B{sub n}is much more natural and considerably simpler than that of Harrison and High. Meanwhile, the cycle structure of H{sub n} seems to have been untouched before, although it is well motivated by nonisomorphic edge colorings of Q{sub n} as well as by the recent interest in symmetries of computer networks.