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Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeledskew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely theposets $P$ such that the $P$-partition generating function is symmetric and theset of linear extensions of $P$, denoted $\Sigma_L(P)$, is a left weak Bruhatinterval in the symmetric group $\mathfrak{S}_n$. We describe the permutationsin $\Sigma_L(P)$ 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 todescent-preserving isomorphism as $P$ ranges over regular Schur labeled skewshape posets. The results obtained are then applied to classify the $0$-Heckemodules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets$P$ up to isomorphism. Then we characterize regular Schur labeled skew shapeposets as the posets whose linear extensions form a dual plactic-closed subsetof $\mathfrak{S}_n$. Using this characterization, we construct distinguishedfiltrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is aregular Schur labeled skew shape poset. Further issues concerned with theclassification and decomposition of the $0$-Hecke modules $\mathsf{M}_P$ arealso discussed.

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