Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / b / i

The vectoris fixed by every element of $G$, so $\mathbb{C}{1}_X\cong {\mathbb{C}}_G$ is a trivial representation. The coefficient-sum subspaceis also $G$-invariant, and every vector has a unique decomposition into a constant vector and an element of $V$. HenceHere $V$ is the augmentation subrepresentation of a permutation representation.