Containment of Bruhat intervals modulo a maximal parabolic subgroup
Susama Agarwala, Colleen Delaney, and Karen Yeats just published a preprint Rado matroids and a graphical calculus for boundaries of Wilson loop diagrams that references some old unpublished notes of mine, done when I was a postdoc of Stéphane Launois at the University of Kent. I’ve uploaded my notes Containment of certain Bruhat intervals modulo a maximal parabolic subgroup in type A in case anyone reading their preprint wants to see them.
The main theorem is as follows. Let \(W\) be the Weyl group of type A and \(W_I\) the maximal parabolic subgroup generated by all standard generators except one. Each coset \(xW_I\) has a unique minimal length coset representative \(\bar{x}\). Then, using \(\leqslant\) for the Bruhat order and \([a, b]\) to denote the Bruhat interval \(\{c \in W : a\leqslant c \leqslant b\}\), for \(x, q, w, y \in W\) such that \(q \leqslant \bar{x}\) and \(y \leqslant \bar{w}\) we have \([q, \bar{x}]W_I \subseteq [y, \bar{w}]W_I\) if and only if there exists \(z \in W_I\) such that \(\bar{x}z \leqslant \bar{w}\) and \(q z \geqslant y\).
Incidentally, you can see the arXiv’s html version of their preprint here. Most of the latex-html conversion works fine, but it doesn’t every try to translate the tikz pictures.