Flag-symmetric and locally rank-symmetric partially ordered sets

Abstract

For every finite graded poset P with 0 and 1 we associate a certain formal power series FP(x) = FP(x1, x2....) which encodes the flag f-vector (or flag h-vector) of P. A relative version FP/Γ is also defined, where Γ is a subcomplex of the order complex of P. We are interested in the situation where FP or FP/Γ is a symmetric function of x1,x 2..... When FP or FP/Γ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called g-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power q, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples (P, Γ) are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.