# Canonical sheaf of Schubert cycles

Suppose we have a smooth subvariety $$X\subset Gr(2,n)$$ of a Grassmannian, that can be expressed as usual as a linear combination of Schubert cycles. I would like to obtain information on the canonical sheaf of $$X$$. Are there formulas for the canonical sheaf of Schubert cycles? Are there examples that I could check of such a computation?