A Sufficient Condition for a Hibi Ring to Be Level and Levelness of Schubert Cycles

Abstract

Let K be a field, D a finite distributive lattice and P the set of all join-irreducible elements of D. We show that if {y ∈ P | y ≥ x} is pure for any x ∈ P, then the Hibi ring ℛ K (D) is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level.