$1.$ The problem statement, all variables and given/known data
(Sorry, don't know how to get TeX to work...)
Consider the space of functions $V_{\nu}$ defined on the vertices of a cube. Symmetries of the cube define a representation $D$ in this space.
(a) What is the dimension of the representation $D$?
(b) Decompose the space $V_{\nu}$ (the representation of $D$) into invariant subspaces irreducible with respect to the rotation group $O$ of the cube. Hint: think about elements of various conjugacy classes of $O$ geometrically, do they fix any vertices of the cube? This should give you the characters $D$ without the finding the representations of the matrices.
(c) Decompose the space $V_{\nu}$ into invariant supspaces irreduciable with respect to the full symmetry group $O_{h}$ of the cube.
$2.$ Relevant equations
Projection operators.
$3.$ The attempt at a solution
I would think the dimension in (a) would be 8. For (b), thinking about the hint, there are 5 conjugacy classes. Do they fix any vertices of the cube? I would think yes, but for the life of me I can't figure out which ones. I'm thinking the base is fixed, or at least two diagonal points. I assume I would get the characters from thinking about this but im unsure the direct means by which I would determine them without the reps. Further, from (c), I would think it is an application of b. For reference, I have the character table for O and $O_{h}$, computed in a previous problem.
Thanks