“…4), the choice of a reduced word for each u j then gives rise to a parametrization of O (u 1 ,...,un) by C l(u 1 )+···+l(un) , where l is the length function on W , and we refer to the resulting coordinates on O (u 1 ,...,un) as Bott-Samelson coordinates (see Definition 2.7). Explicit formulas for the Poisson structure π n on O (u 1 ,...,un) in Bott-Samelson coordinates have been studied in [14,15]. By Bott-Samelson coordinates on a double Bruhat cell G u,v we mean the coordinates on G u,v obtained through the open Fomin-Zelevinsky embedding F u,v and the combination of coordinates on T and Bott-Samelson coordinates on O (v −1 ,u) .…”