$g$-vectors and $DT$-$F$-polynomials for Grassmannians

We review $\mathrm{H}\mathrm{o}\mathrm{m}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen–King–Su’s Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of the Plücker coordinates with respect to the triangular initial seed and 2) we express the $F$-polynomials associated with the Donaldson–Thomas transformation in terms of $3$-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.