For a finite irreducible Coxeter group, what’s the largest set of pairwise-mutually incomparable elements with respect to the weak order?

Question

Given a finite irreducible Coxeter group $W$, what’s the largest subset $K\subseteq W$ such that for all $u,v \in K$, it is not true that

$u <_R v$ (nor $v <_R u$)

where $<_R$ denotes the weak right order?

It would be interesting to know specific examples of such maximal sets and whether a nice general answer exists (possibly for all Coxeter groups and the strong Bruhat order).

I wouldn’t be surprised if a general theorem for finding the largest incomparable subsets of suitably nice posets exists but I haven’t found it.

Answer 1

The paper https://epubs.siam.org/doi/abs/10.1137/0601021 of Stanley shows that for any Weyl group, the (strong) Bruhat order is Sperner, i.e., the largest antichain is the largest rank. (An antichain is a set of incomparable elements, i.e., exactly what you are asking about.)

The paper https://www.ams.org/journals/proc/2020-148-01/S0002-9939-2019-14655-4/ (arXiv: https://arxiv.org/abs/1811.05501 ) by Gaetz and Gao establishes that the weak order on the symmetric group is Sperner.

It is probably still open whether the weak order for all other finite Coxeter groups is Sperner.