Weighted Grassmannians and their explicit description

TL;DR

This paper introduces an explicit construction of weighted Grassmannians, providing parametrizations, formulas for dualizing sheaves, and Hilbert series, with potential generalizations to other root systems.

Contribution

It offers a new explicit method for constructing weighted Grassmannians and computing key invariants, extending to other root systems.

Findings

Effective parametrization of $ ext{Z}$-gradings on Plücker coordinates

Explicit formulas for dualizing sheaf and Hilbert series

Potential generalization to other irreducible root systems

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.