Matrices that commute with Elements from the Symmetric-Group and the Hyperoctahedral Group I am wondering whether people would have references for simple examples about:


*

*Matrices that commute with elements of the Symmetric-Group, e.g. matrix $A$ as below, and the Hyperoctahedral Group, i.e. matrix $B$ as below.

*Efficient decomposition and projection into the "symmetric space"
$$
A = 
\begin{bmatrix}
     1&2&3&7&8&9\\
     4&5&6&10&11&12\\
    12&11&10&6&5&4\\
     9&8&7&3&2&1
\end{bmatrix}\quad
B = 
\begin{bmatrix}
     1&2&3&7&8&9\\
     4&5&6&10&11&12\\
    -12&-11&-10&-6&-5&-4\\
     -9&-8&-7&-3&-2&-1
\end{bmatrix}
$$
I have already worked with matrices that are invariant to cyclic permutations (circulant matrices), for which the efficient transformation is described by the discrete Fourier transformation.
It would be very helpful if you could point me at some books/papers/posts with as many examples as possible. I am currently trying to avoid books like Linear Representations of Finite Groups by J.P Serre.
Thank you very much in advance.
EDIT:
By invariance I mean that if one forms a permutation matrix $P_n\in\mathbb{R}^{n\times n}$ with ones on the anti-diagonal, i.e.
$$
P_4 = \begin{bmatrix} 0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0 \end{bmatrix},
$$
and if one pre- and post-multiplies $A$ or $B$ by $P_4$ and $P_6$, respectively, it holds that:
$$P_4 A = AP_6, \qquad P_4 B = -BP_6.$$
As far as I know, such matrices come with transformation matrices $T$ that "sparsify" the original matrix and map a vector $x$ into the "symmetric space". Moreover, in most cases the product $T x$ can be computed very efficiently (that is my interest in this).
I probably use lots of clumsy formulations above and would also be happy if you could correct the terms I am using which would make my search for literature easier.
 A: Without correcting the various sloppy statements I made before (e.g. irreducible  basis should be irreducible representations), the solution was rather obvious. Given $\tilde{A}$ from my previous post, for which I know the decomposition, one first needs to bring $A$ into a similar form to $\tilde{A}$. This can be done by pre- and post-multiplying $A$ by the invertible $S_l$ and $S_r$, respectively, where
$$
S_l=I_2\oplus P_2,\qquad\qquad
S_r=I_3\oplus P_3,
$$
and where $\oplus$ denotes the block-diagonal concatenation. Then $S_l A S_r$ has the same form than $\tilde{A}$ and is decomposed as in this post.
Finally, the terms to look for in the literature, are:
Linear Representation Theory, Irreducible Representations and Character Tables.
A: Following doesn't answer my question, but I think that it might be related. Following matrix $\tilde{A}$,
$$
\tilde{A} = 
\begin{bmatrix}
     1&2&3&7&8&9\\
     4&5&6&10&11&12\\
     7&8&9&1&2&3\\
     10&11&12&4&5&6
\end{bmatrix},
$$
commutes with $P_2\otimes I_2$, where $\otimes$ denotes the Kronecker product. Two block-wise flips around the centered vertical and horizontal axis yield the same matrix again. A matrix with this property is decomposed into two blocks of size $2\times 3$ by $(T\otimes I_2)^T\tilde{A}(T\otimes I_3)$, where
$$
T = \begin{bmatrix} 1&1\\ -1&1\end{bmatrix}
$$
represents a projection onto - please correct me if I am wrong - the irreducible basis of the symmetric-group $S_2$, which is also referred to as the cyclic group or denoted as $\mathbb{Z}_2$. As mentioned in Joriki's comment to my initial question, this represents an average and a difference of two directions. An academic example of a dynamical system with this property is a string of e.g. 4 identical mass-spring-damper systems, for which the 1st "symmetric movement" is a movement where all masses move in one direction and the 2nd  "symmetric movement" is one where the 2 left-hand masses move in one and the two right-hand masses move in the other direction.
The difference between $A$ and $\tilde{A}$ is that one needs to flip all elements of $A$ around the centered vertical and horizontal axis to obtain $A$ again, rather than a block-wise flip.
Does somebody here think that there is also such an irreducible basis for $A$?
