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

초록

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, & mldr;, |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 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 MP 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 S-|P|. Using this characterization, we construct distinguished filtrations of 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 M-P are also discussed.

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