When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion?
Let's assume that I'm going to try to do this using the theory of $p$-Kazhdan-Lusztig polynomials and that I want to do this in the language of tilting modules.
Then my question becomes, given $\lambda$ a partition of $n$ with at most $h$ columns: when can I combinatorially understand both $T(\lambda)$ and $T(\mu)$ for all partitions $\mu$ of $n$ which are less dominant than $\lambda$ (in the dominance ordering on the block).
I believe it's true to say that we cannot hope to understand the $p$-canonical basis in a combinatorial fashion in full generality. I believe that would contain (as a subproblem) certain number theoretic questions which are not expected to have combinatorial solutions. But this was just word of mouth, does anyone have a reference for this?
On the other hand, Lusztig--Williamson have a conjecture for tilting modules from the first $p^3$-alcove for $\rm GL_3$ in terms of the combinatorics of ``billiards". If true, this means we can understand ${\rm Dim}_k(D(\lambda))$ providing $\lambda$ has 3 columns and is a partition of weight at most $p^2$(ish).
Now for the general case. It seems fair (given the above conjecture of Lusztig--Williamson and assuming the word-of-mouth citation above is correct) to say that at the moment, the only general class of $p$-Kazhdan--Lusztig polynomials we can hope to combinatorially understand is those which coincide with the actual good old fashioned Kazhdan--Lusztig polynomials.
So.... when does this happen? Is it correct to say that a necessary condition for this is that we are in the first $p^2$-alcove (weight of $\lambda$ is less than $p$)? Now, my reading of Riche--Williamson's result is that each alcove has its own bound on the prime. But then, is there a sensible way of talking about my question -- which involved not only understanding a given partition $\lambda$ in an alcove $A_\lambda$ but all partitions $\mu \vdash n$ such that $\mu<\lambda$ in alcoves $A_\mu$? I'm not necessarily looking for a technical condition, but more of a feeling for ``how deep into the first $p^2$-alcove can we go?".
In particular, we can't go as far as the whole of the first $p^2$-alcove because that would prove Andersen's conjecture (which Riche--Williamson explicitly state that they do not do). But can one think somehow that as $p\rightarrow \infty$ that we get most of the the first $p^2$-alcove in some sense and that it's just a few partitions ``near" the $p^2$-wall for which the result fails?
