hyperoctahedralGroupActors -- standard action of the hyperoctahedral group

The hyperoctahedral group is the Weyl group of type B. It can be realized as the automorphism group of the hypercube, as the group of signed permutation matrices, or as the semidirect product $\mathbb{Z}_2^n \rtimes S_n$ of the symmetric group $S_n$ acting on $\mathbb{Z}_2^n$ by permutations. The standard action on a polynomial ring in $n$ variables is the multiplication action of the signed $n\times n$ permutation matrices on the vector of the variables.

This function returns a list of of matrices, each representing an element of the hyperoctahedral group acting on the variables of the polynomial ring in the input as a signed permutation. This simplifies the setup for hyperoctahedral group actions with the action command.

The output list contains one element for each conjugacy class of the hyperoctahedral group. The conjugacy classes are in bijection with the bipartitions of $n$, i.e., pairs of partitions of two integers adding up to $n$, where $n$ is the number of variables. The first partition gives the cycle type of a permutation of an initial subset of variables without signs changes. The second partition gives the cycle type of a permutation of the remaining variables with each cycle containing a single sign change.