 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 $\chi (g^2)=5$, henceThe representations are not isomorphic.