I am very familiar with the representation theory of the symmetric group (on a field of characteristic 0) using tabloids and polytabloids. However, I have some difficulties to translate these notions in term of the group algebra $\mathbb{C}[\mathfrak{S}_n]$.
Let $\lambda$ be a partition of $n$ and $t:\lambda \longrightarrow \{1,...,n\}$ be a fixed bijective tableau. Then, any other such tableau $\tilde{t}$ can be written as $\tilde{t} = t \cdot \pi$ for $\pi \in \mathfrak{S}_n$. This gives a bijection between tableaux and $\mathfrak{S}_n$.
Now, let $R(t)$ be the row stabilizer of $t$ and $a_{\lambda, t} = \displaystyle \sum_{\sigma \in R(t)} \sigma$. Then, as an element of $\mathbb{C}[\mathfrak{S}_n]$, the tabloid $\{ t \}$ corresponds to $a_{\lambda, t}$, and for $\tilde{t} = t \cdot \pi$, we have $a_{\lambda, \tilde{t}} = \pi a_{\lambda, t} \pi^{-1}$.
Let $M^{\lambda}$ be the permutation module generated by the tabloids of shape $\lambda$. How can we see it as a submodule of $\mathbb{C}[\mathfrak{S}_n]$? It is tempting to say that $M^{\lambda}$ is isomorphic to $\mathbb{C}[\mathfrak{S}_n] \cdot a_{\lambda, t}$, but the action of $\pi$ doesn't seem to be defined the same way.
More generally, I am looking for a reference that explains all the representation theory of the symmetric group in terms of the group algebra. I have seen that the Specht module $S^{\lambda}$ is isomorphic to $\mathbb{C}[\mathfrak{S}_n] \cdot c_{\lambda}$ (where $c_{\lambda}$ is the Young symmetrizer), but for the same reason, I do not see how the two actions are compatible, so I would like a detailed reference explaining the links between the algebra and the tableaux constructions.
Thanks in advance!