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Abstract

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 Pl\"{u}cker 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.