I am reading "Representations and Characters of Group" by Gordon James and Martin Liebeck. I want to apply this technique (14.27) to find irreducible submodules of symmetry group of tetrahedron.
What we do is as follows:
Consider a tetrahedron. It has 4 vertices, label them as 1,2,3,4. Each vertex has 3 axes along the edges from that vertex. Look here-Tetrahedron. In this way, we get 12 vectors.(From vertex 1, we have 12,13,14 and so on)
The center of the tetrahedron has vectors, $w_1$,$w_2$,$w_3$,$w_4$ pointing towards a vertex i.
V is the vector space over $\mathbb{R}$ has the basis $v_{12}$,$v_{13}$,$v_{14}$,$v_{21}$,$v_{23}$,$v_{24}$,$v_{31}$,$v_{32}$,$v_{34}$,$v_{41}$,$v_{42}$,$v_{43}$. W is the vector space over $\mathbb{R}$ spanned by {$w_1$,$w_2$,$w_3$,$w_4$}. Then, V $\equiv$ $\mathbb{R}^{12}$ and W $\equiv$ $\mathbb{R}^{3}$
My aim is to find the $\mathbb{R}$G submodules of $\mathbb{R}^{15}$ $\equiv$ V $\oplus$ W.
The book attempts to do it by using the characters of V and W and then writing it as a representation of characters of $S_{4}$. (The symmetry group of tetrahedron is isomorphic to $S_{4}.)$
The character of V is here.
The character of W is here.
These can be understood easily. It is the number of unshifted vectors under that symmetry operation. Then, we can write it in terms of irreducible representations.
For V, we get, $\chi$ = $\chi_1$ + $\chi_3$ + 2$\chi_4$ + $\chi_5$.
For W, we get, $\phi$ = $\chi_4$.(The character table for $S_{4}$ is here)
I am stuck as to how to proceed. I am giving a snip of the book also. It is given to use the formula $V(\sum_{g \in G} \chi_{i} (g^{-1})g)$. So for $\chi_{1}$, I am getting V(5), but I am not sure as to how to interpret it.
page 1 (I have attached 14.27 before) page 2
Any suggestions/guidance will be helpful. Thank you.