In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as $e_t = \sum_{\pi \in C_t} \text{sgn}(\pi) e_{\pi \lbrace t \rbrace}$. Here $t$ is a tableau of $\lambda$, $C_t$ is the column stablizing subgroup for $t$ in $S_n$. $\text{sgn}$ is the signature of the permutation $\pi$. Finally {t} is the equivalence class of tableau (called tabloid) represented by $t$, where two tableaux are considered equivalent if they have the same row entries.

My question is, how is the definition of polytabloids above independent of the choice of $t$ in its equivalence class? For instance, if $t$ is the tableau {1,2},{3,4}, then it's equivalent to s={2,1},{3,4}, but $e_t \neq e_s$. So maybe it's not independent of representative. But then there seems to be too many polytabloids. I would also appreciate if someone could help me establish the connection with Fulton and Harris's book on representation theory problem 4.47. I am not sure what is meant by a standard tableau there. Also in the second construction of the irreps of $S_n$ in the same problem, I am not sure how the action of $S_n$ on the polynomials is defined.

10more comments