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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 $P_{(2,1)}$.

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  • $\begingroup$ I am not sure I understand your question. Are you asking how $(123)$ acts on a standard $(2,1)$-tableau? $\endgroup$ – David Hill Mar 8 '16 at 16:45
  • $\begingroup$ OR, are you asking: given a standard $\lambda$-tableau $T$, define $\psi:S_n\to P_\lambda$ by $\psi(\sigma)=\sigma.T$. What is the image of $\psi$? $\endgroup$ – David Hill Mar 8 '16 at 16:48
  • $\begingroup$ The map $\psi$ should be defined by $\sigma \mapsto \sigma\cdot e_T$ where $e_T$ is the polytabloid associated to $T$. The analogous map $\phi(\sigma) = \sigma\cdot c_\lambda$ is a projection map because $c_\lambda$ is a central idempotent. Is your map $\psi$ a projection? And if so, how can we see it? $\endgroup$ – AS_Butler Mar 9 '16 at 3:30

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