I am studying the irreps of Sn. I will use the following example tableau:
$$|1|2|\\ |3|4| $$
From what I understand, one approach to constructing the irreducible representations is as follows:
For a given standard tableau $T$, we can construct the idempotent Y(T) = Q(T)P(T), where Q(T) is the signed sum of the elements in the tableau's column stabiliser and P(T) is the sum of the elements in the row stabiliser. Y(T) generates an ideal in the group algebra and furnishes a subrepresentation of the regular representation. For our example:
$$Y(T) = (e - (13) - (24) + (13)(24))(e + (12) + (34) + (12)(34))
$$
It turns out in this way we can completely reduce the regular representation, and the reps constructed in this way are irreducible. Further, reps corresponding to tableaux with the same Young diagram are equivalent.
I am also learning about Specht modules, and the approach seems to have a lot of similarities.
The action of an element $g$ on a tableau is defined:
$$\quad\quad\quad|1|2| \space\quad|2|3|\\
(123) * |3|4| = |1|4|
$$
Now consider $Y(T)*T$. The result is the same for row equivalent tableaux and this defines an equivalence class called a polytabloid. Given a Young diagram $\lambda$, the Specht module $S^{(\lambda)}$ is the module spanned by the polytabloids.
We can show that under the action of Sn, the Specht modules provide an irreducible representation. Again for different $\lambda$ we get different irreps and so we obtain all irreps. The polytabloids corresponding to the standard tableaux of $\lambda$ form a basis of $S^{(\lambda)}$
My question:
Are these two approaches the same ? For example, can the fact the standard tableaux of $\lambda$ form a basis of $S^{(\lambda)}$ be used to show that idempotents $Y(T)$ for tableaux corresponding to the same diagram $\lambda$ furnish equivalent representations ?
In trying to link the two approaches, my main problem is that the action of $g$ seems to be different. In the first instance, $g$ acts by left translation. In the second instance, $g$ appears to act by conjugation: if we take
$$|1|2|\\
\qquad\qquad\quad|3|4| = (12)(34)
$$
Then here
$$(123)*(12)(34) = (14)(23) = (123)*(12)(34)*(123)^{-1}
$$
Tabloid is an equivalence class of tableaux under row equivalence. In this way, both approaches are one-to-one. In practice, it's easier to work with tabloids.