GLS homogenization tilde map

In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Geiß, Leclerc and Schr{ö}er, they defined a special map denoted by ``tilde". This map lifts each element $f$ of the coordinate ring of a Schubert cell uniquely to an element $\tilde{f}$ of the (multi-homogeneous) coordinate ring of the corresponding partial flag variety. The significance of this map appears from its essential role; it lifts the cluster algebra of the coordinate ring of a cell to a cluster algebra living in the coordinate ring of the corresponding partial flag variety. This paper takes a closer look at this map and gives an explicit algorithm to calculate it for the $\mathit{\text{generalized}}\mathit{\text{minors}}$.