Conjectures in the representation theory of the symmetric group

Just a few things that come to my mind; most are about the base ring $\mathbb{Z}$, but a good answer would tell us something interesting about $\mathbb{Q}$ as well.

1. Specht modules can be defined not just for partitions and for skew partitions, but for any shape, i.e., any finite set of "boxes" in $\mathbb{Z}^2$. For example, you can define it as the span of the polytabloids in the module of tabloids, or you can (equivalently) define it as the left ideal of the symmetric group algebra generated by the Young symmetrizer $b_T a_T$, where $T$ is any tableau of the given shape, $a_T$ is its row-symmetrizer and $b_T$ is its column-antisymmetrizer. For details, see §1.2 of Ricky Ini Liu, Specht modules and Schubert varieties for general diagrams, thesis 2010. How does such a general Specht module decompose into irreducibles? Even more basically, what is a basis of this module? Can its dimension depend on the characteristic of the base field? Various particular cases have been studied, such as the %-avoiding shapes (Reiner/Shimozono 1998).

2. The Gelfand-Zetlin subalgebra of the group algebra $\mathbb Z\left[S_n\right]$ is the subring (= $\mathbb Z$-subalgebra) generated by the Young-Jucys-Murphy elements $m_k := t_{1,k} + t_{2,k} + \cdots + t_{k-1,k}$, where $t_{i,j}$ is the transposition swapping $i$ with $j$. What is a basis of this subalgebra as a $\mathbb{Z}$-module? Is it a saturated $\mathbb{Z}$-submodule (i.e., a direct summand of $\mathbb Z\left[S_n\right]$ as a $\mathbb{Z}$-module)? Note that over $\mathbb{Q}$, it has a well-known basis, namely the diagonal part of the seminormal basis of $\mathbb Q\left[S_n\right]$ (that is, the family $\left(e_{T,T}\right)$ with $T$ ranging over the standard tableaux with $n$ cells).

EDIT: See https://www.cip.ifi.lmu.de/~grinberg/algebra/nfe2024.pdf for more details on these problems (and a third one on the group ring of $S_n$, albeit not representation-theoretical).

Here are some of my favorites, some were mentioned in comments already. I'm not going to be too picky about the distinction between conjectures vs open questions.

The Saxl Conjecture was already mentioned, but there is also the closely related Tensor Square Conjecture which states that for all sufficiently large $n$ there is some irreducible representation $\chi_\lambda$ such that the tensor square $\chi_\lambda^{\otimes 2}$ contains every irreducible representation with positive multiplicity. I'd go a step further to make what might be called the Strong Tensor Square Conjecture that if you choose an irreducible representation at random from the Plancharel measure, then it satisfies the tensor square property as a representation of the alternating group with probability tending to $1$ as $n \to \infty$. The alternating group modification is needed to deal with the sign representation.

There is also the Strengthened Saxl conjecture due to Bessenrodt, Bowman, and Sutton, which says that over a field of characteristic 2, the tensor square of the Specht module corresponding to a staircase partition contains every simple $2$-modular representation as a submodule (or quotient, equivalently). I'll note that we already know it they all appear as subquotients rather than submodules, and that if true this conjecture would imply the ordinary Saxl conjecture via a lifting to Witt vectors argument.

The Foulkes Conjecture relates the plethysms $s_a[s_b]$ and $s_b[s_a]$. In terms of symmetric group representations these correspond to the permutation representations of $S_{ab}$ acting on the set of (unordered) set-partitions of $[ab]$ into $a$ parts of size $b$, or $b$ parts of size $a$. There is a natural candidate map $\psi_{a,b}$ between them that sends a set-partition of one type to the sum of all "transversal" set partitions of the other type. By transversal I mean that the intersection of any part from the first set-partition with a part from the second set-partition has size $1$. A strengthening of the Foulkes conjecture, sometimes called the Foulkes-Howe Conjecture is that this map $\psi_{a,b}$ has full rank whenever $a \ne b$.

Alex Miller has conjectured that the number of zero entries in the character table of $S_n$ divided by the total number of entries approaches a positive value as $n \to \infty$, it is stated as a question in the paper but he has publicly stated it as a conjecture. However another expert I've spoken to has conjectured the opposite, that the proportion of zero entries tends to $0$ as $n \to \infty$ (but slowly). I won't say who the second person is since I don't know if they want their name publicly associated to that conjecture, but I've heard convincing sounding heuristics for both directions.

In a similar vein to the Kronecker coefficient and plethysm questions, there is The Restriction Problem which asks for a combinatorial description of the branching rule from polynomial representations of $GL_n$ to $S_n$, sitting inside it as permutation matrices. I'll note that a combinatorial interpretation for plethysm coefficients would solve this as well, but there is reason to suspect this might be easier than the general case.

There are the questions of understanding Decomposition Numbers and Characters Formulas for simple $p$-modular representations. There are now some conjectures and theorems relating them to some pretty subtle things like p-Kazhdan-Lusztig bases, but we are very far away from a combinatorial interpretation. Still though, a wildly optimistic hope would be to show the dimension of an irreducible $D^\lambda$ is counted by standard Young tableaux of shape $\lambda$, subject to certain mod-p vanishing or non-vanishing conditions on the contents, hook lengths, or other combinatorial statistics.

There are several conjectures in this area that pertain to computational complexity. Most well known are those that arise from geometric complexity theory à la Mulmuley and Sohoni. The original versions of some of these conjectures, involving so-called "occurrence obstructions," have been disproved, but more nuanced versions, involving multiplicity obstructions, remain open.

In a comment, Richard Stanley mentions the problem of finding a combinatorial interpretation of Kronecker coefficients. In the same breath, one might also mention the problem of finding a combinatorial interpretation of plethysm coefficients. Igor Pak has boldly conjectured that in both cases, no such combinatorial interpretation exists. First he argues that if there exists a combinatorial interpretation, then the problem of computing the coefficients would lie in the complexity class $\mathsf{\#P}$. (This part of the argument is partly linguistic, since it turns on one's understanding of what a "combinatorial interpretation" is.) Then he conjectures that computing Kronecker/plethysm coefficients is not in $\mathsf{\#P}$. It is probably unreasonable to hope for an unconditional proof of this latter claim, but one could hope to prove that the problem is complete for some counting class that is believed to strictly contain $\mathsf{\#P}$.