My friend and academic brother Néstor Díaz and I started talking about working on a project together probably a bit over a year ago. I pitched this idea on proving shellability of Bruhat order for some of the symmetric varieties I was familiar with, figuring that it would be a fun way to learn more about poset topology, and that it wouldn´t be terribly difficult to get some results building off of past work of Incitti, Can, Cherniavsky, Twelbeck, Wyser and others.
The results we set out for turned out to have some fight in them! We thought we had something several times and then found problems and counterexamples in the course of writing things more carefully. Eventually, we shifted gears slightly to working with the sects of the (p,q)-clans. The idea here was that (1) the sects are slightly simpler and smaller than arbitrary intervals of clans, as they group “like” clans together, and (2) we know a nice bijection of sects and collections of rook placements. This problem was also not without challenges and subtleties, but after working out a way to associate a partial permutation to a clan in a way so that covering moves could be coherently labelled, the argument came together.
We had pretty much worked this out by the time we met up at the Schubert Summer School at UIUC in June, but for various reasons it took us a while to write up the details. We’re excited that we were finally able to post a pre-print to arXiv.org last week just in time for the holidays. Enjoy!