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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?

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  • $\begingroup$ As a finite subgroup of $GL(n,\mathbb{Z})$, the hyperoctahedral group acts on many lattices, i.e., it permutes a finite set of basis vectors. Are such actions interesting to you as well? $\endgroup$ – eins6180 Jan 29 '16 at 14:11
  • $\begingroup$ Your choice of tags does not look very good to me. From waht you say, representation-theory is not what you are looking for! Why not group theory, or finite groups, or Coxeter groups? $\endgroup$ – Derek Holt Jan 29 '16 at 14:15
  • $\begingroup$ @DerekHolt: quite right! $\endgroup$ – Martin Rubey Jan 29 '16 at 14:36
  • $\begingroup$ @eins6180: I guess so! I admit I did not think of this yet! $\endgroup$ – Martin Rubey Jan 29 '16 at 14:37
  • $\begingroup$ Well, then for a start you could have a look at Section 3.4 of Conway & Sloane: Sphere Packings, Lattices and Groups, 3rd ed., 2010. $\endgroup$ – eins6180 Jan 29 '16 at 14:53
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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 $-x_i$).

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These references come to mind:

  • Actions of the hyperoctahedral group are studied with respect to space groups (aka crystallographic groups). A (Euclidean) space group is a discrete cocompact group of isometries. Bieberbach proved that the set of all translations in these groups form a lattice of full rank. If you mod out this lattice, you're left with the finite group point group that acts on the lattice. For some space groups the point group is the hyperoctahedral group. In the dimension $n=3$, for example, there are $219$ space groups (up to isomorphism) but they split into a total of $230$ if you take orientation of isometries into acount. The numbers 221, 222, 223, 224, 225, 226, 227, 228, 229, and 230 have the hyperoctahedral group as their point group. See the International Tables for Crystallography for the actual groups and the complete classification in dimension $n=3$.

  • The hyperoctahedral group is a Weyl group (as D. Holt mentioned in his comment), and as such it acts on the chambers of its root system. This is described in many places, see, for example, Humphreys: Reflection Groups and Coxeter Groups.

  • Stanley studied the action of the hyperoctahedral group on the face lattice of the cross polytope.

  • Geissinger & Kinch studied representations of the hyperoctahedral group in detail, see here. In particular, they compared its representation theory to the one by the symmetric group.

There are probably many more references, but I don't know them from the top of my head.

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The group $H_{n}$ acts faithfully as a group of permutations of the non-zero vectors on an $n$-dimensional vector space over $\mathbb{Z}/p\mathbb{Z}$ for any odd prime $p$. The action on $1$-dimensional subspaces has kernel of order two. An analogous statement is true for $(\mathbb{Z}/r\mathbb{Z}) \wr S_{n}$ if we restrict to primes $p \equiv 1$ (mod $r$)- the action on $1$-dimensional subspaces having kernel of order $r$.

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  • $\begingroup$ This looks very interesting! Can you say where this comes up? $\endgroup$ – Martin Rubey Jan 29 '16 at 18:53
  • $\begingroup$ You just take an $n$-dimensional (linear) representation of $H_{n}$ as the group of all monomial matrices with non-zero entries $\pm 1$. This may be viewed as a linear representation over any field of odd characteristic. So for $p$ an odd prime, this gives a permutation action of $H_{n}$ on the $p^{n}-1$ non-zero vectors in an $n$-dimensional vector space over the field of $p$ elements. Only the identity of $H_{n}$ fixes all vectors . $\endgroup$ – Geoff Robinson Jan 29 '16 at 19:00
  • $\begingroup$ Sorry, possibly I was not clear enough: what I'd be (particularly) interested in is whether this action comes up in the literature - apart from being a very enjoyable and natural example! $\endgroup$ – Martin Rubey Jan 29 '16 at 19:36
  • $\begingroup$ I'm sure it must, since it is natural, but I know no reference. $\endgroup$ – Geoff Robinson Jan 29 '16 at 19:50
  • $\begingroup$ In case you run across a reference, please let me know! $\endgroup$ – Martin Rubey Jan 30 '16 at 14:03
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Given $w \leq n$, consider the graph $G$ whose vertex set is $F_2^n \times \binom{[n]}{w}$, and two vertices $(v,S),(v',S')$ are connected if $v|_S = v'|_S$ or $v|_{S'} = v'|_{S'}$, where $v_S \in F_2^w$ is the restriction of $v$ to the coordinates in $S$. An independent set in this graph is known as a $w$-witness code. The hyperoctahedral group acts on $G$, and this action can be used to facilitate computing the Lovász theta function of $G$, which upper bounds the size of an independent set in $G$ (and so the size of a $w$-witness code). See Makriyannis and Meyer, Some constructions of maximal witness codes.

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