Replacing entries of dice by average of it neighbours I am interested in Representation Theory. I came across the following answer while reading this question on Mathoverflow.

An example from Kirillov's book on representation theory: write numbers $1,2,3,4,5,6$ on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe (approximately) the numbers on the faces after many iterations.

However I do not understand how to approach this problem via representation theory. 

Let the value on each of the six sides of the dice be $x_1, x_2, ... , x_6$
You can create equations for each new value. So for example, the value $x_1$ will be replaced by the average of $x_2, x_3, x_4$ and $x_5$
We can say that $x_{1,n+1} = \frac 1 4 x_{2,n} + \frac 1 4 x_{3,n}+ \frac 1 4 x_{4,n}+ \frac 1 4 x_{5,n} $
You now need to write down the other five equations.
The equation I have given you can be represented by the first row of the matrix below:
$\begin{bmatrix} x_{1,n+1}\\ x_{2,n+1} \\ x_{3,n+1}\\x_{4,n+1}\\x_{5,n+1}\\x_{6,n+1} \end{bmatrix} = \begin{bmatrix} 0 & 0.25 &0.25&0.25 & 0.25 & 0 \\ .&.&.&.&.&. \\.&.&.&.&.&.\\.&.&.&.&.&.\\.&.&.&.&.&.\\.&.&.&.&.&. \end{bmatrix}\begin{bmatrix} x_{1,n}\\ x_{2,n} \\ x_{3,n}\\x_{4,n}\\x_{5,n}\\x_{6,n} \end{bmatrix}$
You can then find out what happens to the matrix when you put it to a large power...
