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.

  • $\begingroup$ Try writing \lbrace and \rbrace for the left and right curly brackets, respectively. $\endgroup$ Aug 15, 2010 at 1:15
  • $\begingroup$ @darij: do you mean one needs to fix a tableau representative t in each tabloid? Is the canonical choice the ones with increasing row entries? I wonder why these things are not spelled out in the text. What's the clearest exposition in this subject? $\endgroup$
    – John Jiang
    Aug 15, 2010 at 1:33
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    $\begingroup$ @John Jiang : For the things that confuse you, I think that James/Kerber might be too much. A very clear introduction can be found in the first 2 chapters of Sagan's book "The symmetric group". $\endgroup$ Aug 15, 2010 at 2:40
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    $\begingroup$ @Andy: thank you for the nice reference. It looks exactly like the kind of exposition I wanted, with examples and diagrams to check understanding. $\endgroup$
    – John Jiang
    Aug 16, 2010 at 4:40
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    $\begingroup$ @John Jiang: As darij grinberg says, Goldschmidt's lectures won't be helpful to you. You should also be aware that the approach to representations of symmetric groups in Diaconis and in Sagan derives from the lecture notes of James. But none of these sources seems likely to point you toward the eigenvalues of representing matrices. $\endgroup$ Aug 16, 2010 at 17:31

1 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.

  • $\begingroup$ While the general outline is classical and well agreed on, treatments of symmetric group representations/characters vary quite a lot depending on other connections people want to make. For myself, the Springer LNM 682 by Gordon James works well, partly because it's set up to facilitate reduction mod a prime where everything is far more intricate (see Kleshchev's recent AMS Bulletin article). The characters are unambiguous, but the actual representations can be constructed in different ways with different preferred bases. And notation varies too. $\endgroup$ Aug 15, 2010 at 14:09
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    $\begingroup$ P.S. The term "tabloid" was (I think) introduced by Gordon James, who has been one of the pioneers in re-inventing the theory and interpreting for modern users some of the esoteric multi-volume work of Young. Though I haven't spent much time with it, Bruce Sagan's modern textbook (in a combinatorial spirit) is reasonable. Lecture notes or book chapters can be too sketchy for a beginner. $\endgroup$ Aug 15, 2010 at 16:18
  • $\begingroup$ @Darij: thanks for the nice answers. You have convinced me further that the explicit form of ierreducible representations are pretty difficult for eigenvalue analysis, because it is not straightforward to figure out the representation in the given canonical basis. I will post another research-level question about the possibility of complete diagonalization for fourier transforms of measures evaluated at irreducible representations. This is of sublime interest in finite state markov chain theory. $\endgroup$
    – John Jiang
    Aug 16, 2010 at 4:46

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