Inclusion map between subset permutation modules
= Inclusion map between subset permutation modules
For $r\leq n/2$, the equivariant map from $(r-1)$-subsets to $r$-subsets that sends each subset to the sum of the $r$-subsets containing it is injective over $\mathbb C$. Hence $\pi_r-\pi_{r-1}$ is the character of a representation.