Weighted Grassmannians and their explicit description

We propose an explicit construction of a weighted generalised Grassmannian. For a weighted Grassmannian (i.e., for series A) we obtain an effective parametrisation of possible $\mathbb{Z}$-gradings on Plücker coordinates, and provide the explicit formulae for its dualising sheaf and Hilbert series in terms of this parametrisation. Our approach can be generalised to other irreducible root systems.