I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $H_n$ to $\mathfrak S_X$, the set of bijections from a finite set $X$ to itself. I am not really interested in general representations.
More generally, I am also interested in actions of the wreath product of a symmetric group $\mathfrak S_n$ with a cyclic group $\mathbb Z_r$, also known as the complex reflection group $G(r, 1, n)$.
One example of what I have in mind are the papers by Bill Chen and Richard Stanley Derangements on the n-cube and Bill Chen, 'Induced cycle structures of the hyperoctahedral group'. I am also aware of
Anthony Henderson's 'Representations of wreath products on cohomology of De Concini-Procesi compactifications'. There is also a natural action on signed graphs, which I found in a preprint of Brian Davis 'Unlabeled Signed Graph Coloring'.
Are there any other examples? If not so, is there a good reason that permutation representations of the symmetric group are ubiquitous in the literature, but permutation representations of the hyperoctahedral group are rare?
Best Answer
Tymoczko has defined something called the "dot action" on the equivariant cohomology of regular semisimple Hessenberg varieties. I think she explains all the details only in Type A but her definition generalizes straightforwardly to other types.
For example, in Type C, you start with an indifference graph $H$ on $2n$ vertices that is palindromic (i.e., the associated unit interval order $P$ is isomorphic to the poset you get by reversing all the inequalities of $P$). Label the vertices $-n, -n+1, \ldots, -1,1,\ldots, n-1,n$ from left to right. Edges in $H$ may be identified with elements of the hyperoctahedral group: An edge connecting $i$ with $j$ represents the element that swaps $i$ with $j$ and $-i$ with $-j$.
Now define the associated moment graph $G$, whose vertices are the elements of the hyperoctahedral group. Two vertices of $G$ are adjacent if some edge of $H$ (thought of as an element of the hyperoctahedral group as explained above) takes one to the other.
For example, if $n=2$, we could take $H$ to be a path on four vertices: $$\bar{2} - \bar{1} - 1 - 2$$ The edges of $H$ define two elements of the hyperoctahedral group—the element that swaps $1$ with $-1$, and the element that swaps $1$ with $2$ and simultaneously swaps $-1$ with $-2$. Thinking of the group as acting on places (rather than letters), this means that the edges in $G$ are given by $$12 - 21 - \bar{2}1 - 1\bar{2} - \bar{1}\bar{2} - \bar{2}\bar{1} - 2\bar{1} - \bar{1}2 - 12$$ where of course we wrap around the end (i.e., the two appearances of $12$ above are really the same vertex, so $G$ is abstractly isomorphic to a cycle on eight vertices). Each vertex of $G$ is adjacent to two others, one of which is obtained by swapping the two (signed) letters and one of which is obtained by negating the letter in the first position.
The edges of $G$ are labeled with elements of the polynomial ring $R := \mathbb{C}[x_1, \ldots, x_n]$; specifically, if the letters that are swapped are $i$ and $j$ then we label the edge in $G$ with $x_i - x_j$, where we interpret $x_{-i}$ to mean $-x_i$. (This only defines the edge label of $G$ up to sign because I have not said which letter is $i$ and which letter is $j$, but for the purposes of defining the dot action, this will not matter.) In our running example, the corresponding edge labels (up to sign) are $$x_1 - x_2, 2x_2, -x_2 - x_1, 2x_1, -x_1 + x_2, -2x_2, x_2 + x_1, -2x_1.$$
An element of the equivariant cohomology ring $\mathscr{H}$ is given by specifying an element of $R$ for each vertex of $G$, subject to the condition that if $p(u)$ is the polynomial at vertex $u$ and $p(v)$ is the polynomial at vertex $v$, and $u$ and $v$ are adjacent, then $p(u) - p(v)$ must be divisible by the edge label of the edge $uv$.
Finally, the dot action on $\mathscr{H}$ is defined as follows. Let $w$ be an element of the hyperoctahedral group and let $p\in\mathscr{H}$, with $p(v)$ denoting the element of $R$ at the vertex $v$. To specify $wp$, we need to specify the value of $wp(v)$ for every vertex $v$. This is done via the formula $$(wp)(v) = w p(w^{-1}v),$$ meaning that we first apply $w^{-1}$ to the letters of $v$ to obtain a new vertex $w^{-1} v$ of $G$, and then finally letting $w$ act naturally on the variables $x_i$ (again, interpreting $x_{-i}$ as $-x_i$).