UW Combinatorics and Geometry Seminar: Tien Le

Given a parabolic subgroup $W_I$of a Coxeter system $(W,S)$, the set of left cosets $w{W}_I$, for $I\subseteq S,w\in W$, is well-studied. However, the set of double cosets $I,J\subseteq S$for $I,J\subseteq S$is a less studied object. I want to analyze the double cosets, in particular, to investigate the conditions on and so that the set of double cosets carries some certain structure. The main result that I have found is that: Let $G$denote $SL(n)$and let $X$be a double flag variety $G/P_1\times G/P_2$. If $c_G(X)\le 1$, then the inclusion poset of $G$-orbit closures in $X$is a graded lattice.