It is known that the group algebra of the symmetric group decomposes into a direct sum of its irreducible representations \begin{equation}K[S_n] = \bigoplus_{\lambda\vdash n} P_\lambda^{\oplus m_\lambda} \tag{$\ast$} \end{equation} where $P_\lambda$ is the irreducible $S_n$-module corresponding to the partition $\lambda$, and $m_\lambda = \dim(P_\lambda)$. One way to define the modules $P_\lambda$ is as the projection from the group algebra by the young symmetrizer $c_\lambda$, i.e., $$P_\lambda = K[S_n]\cdot c_\lambda$$ With this description it is easy to see how to project an element of $S_n$ (viewed as living in the group algebra $K[S_n]$) to its irreducible components as per equation ($\ast$), namely, you just multiply on the right by the young symmetrizer, i.e., \begin{align} \phi: S_n \subset K[S_n] &\rightarrow P_\lambda \\ \sigma &\mapsto \sigma\cdot c_\lambda \end{align} There is a different description of the irreducible representations of $S_n$ where $P_\lambda$ has basis given by standard $\lambda$-tableau.
Question. What is the analog of the map $\phi$ written in this basis?
As a concrete example, how does one project $(123) \in S_3 \subset K[S_3]$ to $P_{(2,1)}$ using standard $(2,1)$-tableau as a basis for $P_{(2,1)}$.