I am looking at some of my old Representation Theory work in advance of a visiting research fellow project in the area I am seeking to undertake in the coming weeks.
The lecturer wrote that "a permutation representation is a representation of the form $\mathbb{C}[X]$ for a $G$-set $X$, and a stable permutation representation is a representation $V$ so that $V \oplus \mathbb{C}[Y]$ is a permutation for some $G$-set $Y$." They then left the following as an exercise, and I am stumped by it.
- Give an example of a stable permutation representation that is not a permutation representation.
My previous self wrote the hint: "construct a $G$-set from disjoint unions of copies of $G=C_2$ and the fixed-point set of $G$ acting on the $G$-set X" but I feel I have misunderstood the definition to the point that this is almost meaningless to me.
Thanks in advance!