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So I just learn about Jucys-Murphy elements. They are elements of $\mathbb{C}[\mathfrak{S}_n]$, the group algebra of the symmetric group, defined as:

$$ X_i = \displaystyle \sum_{k=1}^{i-1} (k,i) $$

for $1\leq i \leq n$. Using them, we can construct the irreducible representations $S^{\lambda}$ of $\mathfrak{S}_n$. This approach is due to Vershik and Okounkov. They construct a basis of $\mathbb{C}[\mathfrak{S}_n]$ consisting of eigenvectors for the $X_i$’s, with eigenvalues allowing us to distinguish the different irreducible representations. This way, the branching rule is natural, and standard tableaux a corollary of the branching rule; it is the “opposite” point of view compared to the classical construction of Specht modules (as in Fulton’s book). Note that both approaches give that the dimension of $S^{\lambda}$ is $|\text{SYT}(\lambda)|$, where $\text{SYT}(\lambda)$ is the set of standard tableaux of shape $\lambda$.

As far as I understand (their paper is hard to read, at least for me), every representation of $\mathfrak{S}_n$ has a (unique up to scaling?) basis consisting of eigenvectors for all $X_i$’s.

My question is about how we can use them for permutation modules. A permutation module $M^{\lambda}$ can be seen as the vector space generated by tabloids of shape $\lambda$. A tabloid of shape $\lambda$ is an equivalence class on tableaux of shape $\lambda$, where two tableaux are in the same equivalence class if they only differ by a permutation of entries in the same row.

It is well-known that

$$ M^{\lambda} \cong \displaystyle \bigoplus_{\mu} (S^{\lambda})^{|\text{SSYT}(\mu,\lambda)|}, $$

where $\text{SSYT}(\mu,\lambda)$ is the set of semistandard tableaux of shape $\mu$ and content $\lambda$.

In his book, James describes a way to decompose $M^{\lambda}$ as this direct sum by using what he calls semistandard homomorphisms. Thus, we have one vector of $M^{\lambda}$ per pair $(T,t)$, where $T \in \text{SSYT}(\mu,\lambda)$ and $t\in \text{SYT}(\mu)$.

Problem is, this basis of $M^{\lambda}$ does not consist of eigenvectors for the Jucys-Murphy elements $X_i$’s.

Question : Is there a known decomposition of $M^{\lambda}$ in a basis such that its elements are eigenvectors for the action of the $X_i$’s? And if yes, is there a link with James’ semistandard homomorphisms?

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    $\begingroup$ "Unique up to scaling?" Yes if the representation is multiplicity-free; not otherwise. $\endgroup$
    – darij grinberg
    Commented Dec 11, 2022 at 4:50
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    $\begingroup$ "Hard to read" Yes, and imprecise at many places. I think the Ceccherini-Silberstein et al book is much better. $\endgroup$
    – darij grinberg
    Commented Dec 11, 2022 at 4:51
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    $\begingroup$ Nice question. My quick suspicion is that Murphy's papers might contain an answer, although perhaps not the most explicit one. Note that the whole symmetric group algebra is an $M^\lambda$, and the basis you're looking for in this case is Young's seminormal or orthogonal basis (up to scaling they are the same). The way Murphy constructs it, by killing all but the right eigenvalue each time, should work for every module. $\endgroup$
    – darij grinberg
    Commented Dec 11, 2022 at 4:54
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    $\begingroup$ I don't think anyone has written down seminormal bases for the permutation modules. Murphy constructed "cellular bases" for these modules, but this does not directly give seminormal bases. James and I gave related bases for the q-Schur algebras that may be useful in connecting with James' homomorphisms; see "A q-analogue of the Jantzen-Schaper theorem", Proc. London Math. Soc. (3) 74 (1997), no. 2, 241–274. $\endgroup$
    – Andrew
    Commented Dec 11, 2022 at 13:38

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