Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows my attempt (which seems to fail):

The character table of $A_5$, which will be needed:

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There is a 9 dimensional subrepresentation obtained as follows. There is a 10-dimensional sub representation, call this $\mathbb{C}^{10}$, by assigning values to pairs of opposite faces of the icosahedron. Then with the condition that summing of all these values will give zero this sub representation of the sub representation $\mathbb{C}^{10}$ has dimension 9, which I will denote by $\mathbb{C}^9$. Now I want to decompose this representation using the character table of $A_5$, since $\chi_{\mathbb{C^{10}}}(g)=\chi_{\mathbb{C^{1}}}(g)+\chi_{\mathbb{C^{9}}}(g)=1+\chi_{\mathbb{C^{9}}}(g)$.

For $g=(123)$, which the rotation of order three on the axis through opposite faces, we see that only one pair of faces is fixed, so $\chi_{\mathbb{C^{10}}}((123))=1$, so we see that $\chi_{\mathbb{C^{9}}}((123))=0$. This can be obtained if it were the case that $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$. Of course we must try some more elements to guarantee that this is the case.

For $g=(12345)$, which is the rotation of order 5 obtained by rotating on the axis through a pair of vertices, we have that no pair of faces is fixed, so $\chi_{\mathbb{C^{10}}}((12345))=0$, which implies $\chi_{\mathbb{C^{9}}}((12345))=-1$ which again is obtained if $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$.

Now if $g=(12)(34)$ I get problems. This element is obtained by rotation of order 2 on the axis through opposite edges. This does also not fix any pair of faces, so $\chi_{\mathbb{C^{10}}}((12)(34))=0$ which implies again $\chi_{\mathbb{C^{9}}}((12)(34))=-1$, but $\chi_{\mathbb{C^{4}}}((12)(34))+\chi_{\mathbb{C^{5}}}((12)(34))=1$ so it cannot be the case that $\mathbb{C}^9=\mathbb{C}^4\oplus\mathbb{C}^5$.

I dont understand what is going wrong here, because using the other irreducible representations I cannot make it consistent with the values of the characters. I would really appreciate if someone could clarify where I am mistaken. Thank you.

  • $\begingroup$ There is an inner product on the vector space of all class functions on a given group $G$, and the irreducible representations are an orthonormal basis for this vector space. The coefficients of any representation with respect to this basis can be obtained by calculating the inner product of the representation with each of the irreducible representations in turn. $\endgroup$ – Gerry Myerson Nov 15 '14 at 22:47
  • $\begingroup$ @Gerry Myerson, for this I should now what the value is of $\chi_{\mathbb{C}^{10}}((12)(34))$, which it seem I calculated wrong. $\endgroup$ – Badshah Nov 15 '14 at 22:52

(12)(34) stabilizes two pairs of opposite faces (because, in each case, it swaps them).

  • $\begingroup$ I am sorry, but I dont know exactly what stabilizes mean in this context. Could you elaborate? $\endgroup$ – Badshah Nov 15 '14 at 22:26
  • $\begingroup$ What I mean is that in two cases it fixes the pair of faces, as a pair, because it interchanges them. $\endgroup$ – Hugh Thomas Nov 16 '14 at 3:13

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