Augmentation subrepresentation of a permutation representation
= Augmentation subrepresentation of a permutation representation
For a finite $G$-set $X$, the coefficient-sum map $\varepsilon:\mathbb C[X]\to\mathbb C$ is equivariant. Its kernel
$$
\mathbb C[X]_0
=\left\{\sum_{x\in X}a_xx:\sum_{x\in X}a_x=0\right\}
$$
is the augmentation subrepresentation. If $X$ is nonempty, then
$$
\mathbb C[X]
=\mathbb C\!\left(\sum_{x\in X}x\right)
\oplus\mathbb C[X]_0.
$$