4

What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?

I know that one can express this as $\operatorname{rk} \left( (\langle e_i, e_j \rangle )_{i,j=1,\dotsc,n}\right)$, the rank of the Gram matrix.

Is there an analog of the hook formula, where one uses “$p$-cores” instead of the usual hooks?

YCor
  • 60,149
Jackson Walters
  • 519
  • 10
    This is a very well known open problem. Good luck :) – Geordie Williamson Nov 11 '23 at 05:11
  • 1
    This sounds fun! Can you produce data of this, in some table? – Per Alexandersson Nov 11 '23 at 17:23
  • 1
    One very special case where the dimension are known is that if $p$ is an odd prime and $r < p$ (else the labelling partition is not $p$-regular) then $\mathrm{dim}\ D^{(p-r,1^r)} = \binom{n-e}{r}$ where $e = 1$ if $p$ does not divide $n$ and $e = 2$ if $p$ divides $n$. This comes from the isomorphism $D^{(p-r,1^r)} \cong \bigwedge^r D^{(n-1,1)}$. The dimensions when $\lambda$ has two parts can be computed recursively using known formulae for the decomposition numbers: I don't know if these is a simple closed formula here. In general, as the expert above me says, it's an open problem. – Mark Wildon Nov 11 '23 at 17:33
  • @GeordieWilliamson Ah :) Thank you, good to know (perhaps, sometimes it helps to not know how hard a problem is before starting). Per: That was probably going to be my first step. I'm pretty comfortable working with Specht modules in Sage, so I will go ahead and code this up. Mark: Very cool! I'm surprised the formula is so simple in that case. It might be enough to just know asymptotics, given that I'm really interested in the possibility of generalizing Moore et. al's papers on Strong Fourier Sampling to the modular setting. – Jackson Walters Nov 11 '23 at 20:20
  • 1
    @JacksonWalters: Please be aware that the only currently supported implementations of Specht modules in Sage are "seminormal", "orthogonal" and "Specht", and (to my knowledge) all of them are written in a way that presupposes at least characteristic $0$. (A pity if you ask me: at least Specht should know better...) – darij grinberg Nov 12 '23 at 01:50
  • @darijgrinberg Thanks, I’m aware that positive characteristic isn’t quite implemented. There is https://doc.sagemath.org/html/en/reference/spkg/symmetrica.html, however it appears the contact person passed away in 2013. Nevertheless, I don’t think you need them to be implemented directly. Isn’t it the case that the Gram matrix just takes inner products of $e_i$, which are basis vectors for the Specht modules, and only afterwards reduces modulo $p$, to then compute the rank as in https://math.mit.edu/~charchan/ModularRepresentationsSymmetricGroupSeminar.pdf? – Jackson Walters Nov 12 '23 at 02:07
  • @JacksonWalters: As long as the basis is a $\mathbb Z$-basis, this is true. Is it for the Sage implementation? I'd hope but I don't know. – darij grinberg Nov 12 '23 at 02:30
  • @darijgrinberg I'm a bit confused as it doesn't even seem like 'specht_module' is implemented here: https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/specht_module.html. It throws 'ModuleNotFoundError' (punny). The representation matrices, however, are: https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/symmetric_group_representations.html – Jackson Walters Nov 12 '23 at 03:30
  • Ah, right, specht_module works over any field (not sure how I had forgotten it, as I was the one reviewing that ticket IIRC). What exactly does not work? – darij grinberg Nov 12 '23 at 03:47
  • Let us continue this discussion in chat. – Jackson Walters Nov 12 '23 at 03:48
  • 1
    Relevant prior MO question: https://mathoverflow.net/questions/138310 – Sam Hopkins Nov 13 '23 at 16:26
  • @SamHopkins Thanks for cluing me into the larger story. Here is Sage code which computes the dimensions using the rank of the Gram matrix: https://github.com/jacksonwalters/mod_repn_symmetric_group. Travis Scrimshaw was a help and has implemented an alternate method using the cellular basis of the Iwahori-Hecke algebra at $q=1$ here: https://github.com/sagemath/sage/pull/36718. Tables for =3,4,5,6 have been computed and are available in the linked repo. The PR for that code to Sage is here: https://github.com/sagemath/sage/pull/36724 – Jackson Walters Nov 15 '23 at 20:05

0 Answers0