some confusion about the explicit construction of irreducible representations of $S_n$

Question

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.

Answer 1

1. Yes, equivalent tableaux $t$ may yield different $e_t$'s. However, equivalent tableaux $t$ yield the equivalent $\pi t$'s for any permutation $\pi$, so that the notation $\pi\left\lbrace t\right\rbrace$ on page 132 is justified. Nobody is claiming that $e_t$ depends on the tabloid $\left\lbrace t\right\rbrace$ only.

2. The Specht module $S^{\lambda}$ is defined as the vector space generated by $e_t$ for all Young tableaux $t$ corresponding to the partition $\lambda$. Now it turns out that there is a lot of redundancy in these $e_t$; that is, they are linearly dependent. One very nice basis of $S^{\lambda}$ is $\left\lbrace e_t \mid t\text{ is a standard tableau}\right\rbrace$, where a tableau is called standard if the rows are strictly monotonically increasing and the columns are strictly monotonically increasing (the word "strictly" is not of much importance here, because the numbers in our tableaus are pairwise distinct, but sometimes one also considers tableaux where the entries may be equal, and then it matters).

3. Concerning Fulton-Harris' problem 4.47, the first part (about the $E_T$) is exactly the definition of the Specht module that Diaconis gives. As for the second part (about the $F_T$), you have to show that there is an $S_d$-equivariant isomorphism $V_{\lambda}\to W_{\lambda}$, where $V_{\lambda}$ is the Specht module defined by means of the $E_T$'s, and $W_{\lambda}$ is the $k\left[S_d\right]$-submodule ($k\left[S_d\right]$ is what Fulton-Harris denotes by $\mathbb C\left[\mathfrak{S}_d\right]$) of $k\left[x_1,x_2,...,x_d\right]$ spanned by the polynomials $F_T=\prod\limits_{i < j;\ i\text{ and }j\text{ lie in the same column of }T}\left(x_i-x_j\right)$. To construct this isomorphism, let $\Psi$ be the vector space homomorphism $U_{\lambda}\to k\left[x_1,x_2,...,x_d\right]$ (where $U_{\lambda}$ is the representation of $S_d$ with basis the tabloids for the Young diagram $\lambda$) defined by $\Psi\left(\left\lbrace T\right\rbrace\right) = \prod\limits_{i=1}^{d}x_i^{\left(\text{number of the row in which }i\text{ lies in the tableau }T\right)-1}$ for every tableau $T$ (this is well-defined since the product on the right hand side depends only the equivalence class $\left\lbrace T\right\rbrace$ of $T$). Besides, $\Phi$ is easily seen to be $S_d$-equivariant and injective. Now, $\Psi\left(V_{\lambda}\right)=W_{\lambda}$, because every tableau $T$ satisfies $\Psi\left(E_T\right)=F_T$ or $\Psi\left(E_T\right)=-F_T$ (by Vandermonde's determinant, applied to the entries in every column of $T$), and thus the restriction of this homomorphism $\Psi$ to the subspace $V_{\lambda}$ of $U_{\lambda}$ is a $G$-equivariant bijective homomorphism $V_{\lambda}\to W_{\lambda}$. Thus, $V_{\lambda}\cong W_{\lambda}$ as representations of $S_d$.

4. My first actual source for the representation theory of $S_n$ were Etingof's lecture notes, but beware: they are very compressed and don't have much on $S_n$ (that's not the point of them either). Then, there is Fulton-Harris with a whole chapter on $S_n$ (but the proofs are mostly exiled into the exercises, which means that you often get hints rather than proofs). "The Representation Theory of the Symmetric Group" by James and Kerber looks very good as a comprehensive reference. There are also typewriter-style lecture notes by James (LNM 682: "The Representation Theory of the Symmetric Groups") which have the advantage of being just 136 pages long. I don't have any experience with them, however.