Using representation theory to understand averaging processes: an example I have been reading about some elementary representation theory (of finite groups), and have been puzzled by the following question, which of all of the questions I have worked on thus far is the most natural to me, and is perhaps the only one which seems like it'd be of significant interest "external to the subject".  So I'd like to remedy my complete confusion about how to bring representation theory to bear on this.
The question:

Let G be the group of rotations of a cube, and  $_{\mathbb{C}[G]}V = _{\mathbb{C}[G]}\mathbb{C}X$, where $X$ is the set of vertices of the cube, be the permutation module arising from the action of G on the vertices of the cube. Let $T:V \rightarrow V$ be the map replacing the number at a vertex by the average of the numbers at the three adjacent vertices. Let $w= (w_{1}, \cdots, w_{8}) \in V$. I want to understand how to compute $lim_{n \rightarrow \infty} T^{n}w$, using respresentation theory.

What I've considered:
For full disclosure, this is part of a homework assignment, but not my homework assignment, and I am just aiming to gain some understanding, not turn anything in. So far, I have:


*

*Come to understand that $G \cong S_{4}$, via the action of $G$ on the main diagonals.

*Written down the character table for $S_{4}$, using the irreducible representations already known to me ( the trivial, and sign), then forming the rest with the aide of tensor product, complements in the regular representation, and orthonormality of the irreducible characters. I should note that I have avoided thinking explicitly about these other irreducible reps at this point.

*Decomposed $_{\mathbb{C[S_{4}]}} V$ into irreducible submodules, by computing the character of $V$ directly with pictures (using the fact that $\chi (g) = \#$ (fixed points of g in X) for a permutation representation $_{\mathbb{C}[G]}\mathbb{C}X$), and then using the fact that the irreducible characters form an orthonormal basis for class functions.


Edit: I have seen this question but, like Ron, I fail to understand how the first answer involves my considerations about the representation (or rather, analogous considerations about the relevant permutation module).
 A: Think first about two special cases: if you start with a vector $w$ whose coordinates are all equal to one another, then $T^n w=w$ for all $n$. On the other hand, if you start with a vector $w$ whose coordinates have the property that if $i$ and $j$ are adjacent vertices then $w_i=-w_j$, then $T^n w=(-1)^n w$ for all $n$.
As it turns out, the limiting behavior is a superposition of these two cases, and one can use representation theory to see why that is. The philosophy is as follows: it's easy to compute powers of a diagonal matrix. For a general operator $T$, the best you can hope for is to be able to diagonalize it explicitly; we will do this here with the advantage that the extra structure allows us to organize the calculation much more efficiently.
The observation that makes this go is that the operator $T$ may be expressed as the following average of the $4$-cycles in $S_4$
$$T=\frac{1}{6} ((1234)+(1243)+(1324)+(1342)+(1423)+(1432)).$$
Since this is a class sum, it is central in the  group algebra of $S_4$ and hence acts as a scalar on each irreducible representation. Now that you have computed the character table you know which scalar: $T$ acts on an irrep with character $\chi$ by the scalar $\chi(1234)/\chi(1)$. 
Explicitly, the $T$ action is by $1$ on the trivial representation, $-1$ on the sign representation, $-1/2$ on the reflection representation, $1/2$ on the tensor product of the sign representation by the reflection representation, and $0$ on the two-dimensional irrep.  
You can therefore write $w$ as a sum of eigenvectors for $T$ by using the projecting idempotents for the various $S_4$ irreps; since you have decomposed the permutation representation you are studying you know it contains the trivial representation, the sign representation, and the two three-dimensional irreps all with multiplicity one. In the large $n$ limit, only the trivial and sign representations contribute, since $(1/2)^n$ becomes very small. It follows that
$$T^n w=\pi(w)+(-1)^n \sigma(w)+\epsilon(w),$$ where $\pi(w)$ is the vector whose coordinates are all equal to the average value of the coordinates of $w$, $$\sigma(w)=\frac{1}{24} \sum_{g \in S_4} \mathrm{det}(g) g w$$ is the projection of $w$ onto the alternating space of vectors in $V$, and $\epsilon(w)$ is a very small vector. 
In particular, the limit $\lim_{n \to \infty} T^n w$ exists (and is equal to $\pi(w)$) if and only if $\sigma(w)=0$. 
