= Solution
Its character is the number of fixed points, and averaging it gives the number of orbits. Complementation $\Sigma\mapsto X\setminus\Sigma$ is a $G$-equivariant bijection $\Gamma_n\to\Gamma_{d-n}$.
The exterior-power assertion is false. For $n=2$ and $g=(12)(34)\in A_5$, $g$ fixes two $2$-subsets, so $\chi_{\mathbb C[\Gamma_2]}(g)=2$. The natural permutation character has $\chi(g)=1$ and $\chi(g^2)=5$, hence
$$
\chi_{\wedge^2\mathbb C[X]}(g)=\tfrac12(\chi(g)^2-\chi(g^2))=-2.
$$
The representations are not isomorphic.
Back to article page
