Polytopes and the Isotropic Schubert Calculus

Abstract:

   Combinatorial invariants of ranked posets count rank-selected
chains in the poset, which generalize (flag) $f$-vectors of posets
Likewise, known multiplication formulas in the Schubert calculus may
be expressed combinatorially as a sum over chains in the Bruhat
order satisfying certain conditions.  Both cases also possess some
Hopf-algebra structures.

   We describe a theory of Pieri Operators on posets which unifies
these two examples, explaining their common Hopf-algebraic structure.
This unified theory also shows a direct link between the Isotropic
Schubert calculus and the invariants of polytopes---through certain
Eulerian Pieri operators.