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$1.$ 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.


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TeX is (for the most part) included as normal, i.e. between dollar signs. Here and here are some quick intros using TeX on the site. – Zev Chonoles May 12 '12 at 23:36

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