Regular Schur labeled skew shape posets and their 0-Hecke modules

Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$ , is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$ . We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$ ’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$ -Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$ . Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$ -Hecke modules $\mathsf {M}_P$ are also discussed.