On the generating poset of Schubert cycles and the characterization of   Gorenstein property
  The homogeneous coordinate ring of a Schubert variety (a Schubert cycle for short) is an algebra with straightening law generated by a distributive lattice. This paper gives a simple method to study the set of all the join-irreducible elements of this distributive lattice, and gives a simple proof of the criterion of the Gorenstein property of Schubert cycles. 
Mitsuhiro Miyazaki