Flag-symmetric and locally rank-symmetric partially ordered sets

Richard P. Stanley

The electronic journal of combinatorics, The Foata Festschrift volume, Tome 3 (1996) no. 2

Résumé

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-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 $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

Détail

DOI : 10.37236/1264

Classification : 05E05, 06A07, 55U99
Mots-clés : poset, symmetric function, flag-symmetric poset, distributive lattice, chains, Cohen-Macaulay, Schur-positive, Schur functions, rank positive, modular lattices, reconstruction, Kostka polynomials, lexicographic shellability