A decorated flag is a nested sequence of linear subspaces, each equipped with a nonzero volume element. In this talk, we will introduce a permutation action on decorated flags compatible with certain linear operators. We will explore this action through explicit calculations in the three-dimensional case, and explain how it relates to a permutation action constructed by Goncharov and Shen in 2016. Finally, we will briefly introduce a family of braid actions and explore when they recover the permutation actions.