Math 6320 • Algebra II

Let \(F\) be a field of characteristic \(p\). Let \(S_n\) be the group of permutations of \(X = \{1,2,\ldots ,n\}\). Let \(V = F[X]\) be the space of \(F\)-valued functions. It is a linear representation of \(S_n\), where \(\sigma \in S_n\) acts on \(f \in F[X]\cong F^n\) by \((\pi (\sigma )f)(x)=f(\sigma ^{-1}x)\). We have an intertwining map \(P:F[X]\to F\), where on \(F\) we have the trivial action of \(S_n\), given by \(f\mapsto f(1)+f(2)+\cdots +f(n)\). Prove that this map splits, that is, there exists an intertwining map \(Q\) in the opposite direction such that \(P\circ Q=1\), if and only if \(p\) does not divide \(n\).