Let $\mathbb{C}[\mathfrak{S}_n]$ be the regular representation of the symmetric group and $S^{\lambda}$ the Specht module associated to the partition $\lambda \vdash n$. We know that we have the decomposition $$ \mathbb{C}[\mathfrak{S}_n] \cong \displaystyle \bigoplus_{\lambda \vdash n} f_{\lambda}S^{\lambda}, $$ where $f_{\lambda}$ is the number of standard tableaux of shape $\lambda$. In perticular, there exists morphisms of representations $S^{\lambda} \hookrightarrow \mathbb{C}[\mathfrak{S}_n]$ and $\mathbb{C}[\mathfrak{S}_n] \twoheadrightarrow S^{\lambda}$.
Can we construct those morphisms in a way that the composition is the identity (on $S^{\lambda}$) and that the projection is defined straightforwardly (i.e. described for each element of $\mathfrak{S}_n$ and extended linearly, instead of decomposing $\mathbb{C}[\mathfrak{S}_n]$) ?