 Permutation representation

For a finite $G$-set $X$, the character of $\mathbb{C}[X]$ at $g$ is $|X^g|$. Its invariant dimension is the number of $G$-orbits.

In the context of group theory, a **permutation representation** is a way of representing a group as a group of permutations. Specifically, if \( G \) is a group, a permutation representation of \( G \) is a homomorphism from \( G \) to the symmetric group \( S_n \), which is the group of all permutations of a set of \( n \) elements.