A sufficient condition for a Hibi ring to be level and levelness of Schubert cycles

摘要

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 is an element of P y >= x} is pure for any x is an element of 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.