Séminaire Lotharingien de Combinatoire, 91B.27 (2024), 12 pp.
Young-Hun Kim, So-Yeon Lee and Young-Tak Oh
Regular Schur Labeled Skew Shape Posets and Their 0-Hecke Modules
Abstract.
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,...,|P|} such that the P-partition
generating function is symmetric and the set of linear extensions of
P, denoted ΣL(P), is a left
weak Bruhat interval in the
symmetric group S|P|. We describe the permutations in
ΣL(P) in terms of reading words of
standard Young tableaux
when P is a regular Schur labeled skew shape poset, and classify
ΣL(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
MP with respect to the Schur basis when P is a regular
Schur labeled skew shape poset.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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