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

Young-Hun Kim, So-Yeon Lee, Young-Tak Oh

TL;DR

This work develops a comprehensive framework for regular Schur labeled skew shape posets and their associated $0$-Hecke modules. It links Stanley's $P$-partition conjecture to a precise criterion: $K_P$ is symmetric and $\Sigma_L(P)$ is a left weak Bruhat interval for regular Schur skew shapes, enabling a description of $\Sigma_L(P)$ as reading words of standard Young tableaux of shape $sh(\tau_P)$. The authors classify the corresponding $0$-Hecke modules $\mathsf{M}_P$ up to isomorphism, show that regularity is equivalent to dual plactic-closedness of $\Sigma_L(P)$, and construct distinguished filtrations with Schur-basis quotients. They also outline significant open questions on extending the classification beyond $\mathsf{RSP}_n$, the decomposition of $\mathsf{M}_P$, and recovering these modules from generic Hecke algebras via specialization $q\to 0$, establishing a rich interplay between poset combinatorics, symmetric functions, and $0$-Hecke representation theory.

Abstract

Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted $Σ_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_n$. We describe the permutations in $Σ_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $Σ_L(P)$'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 posets whose linear extensions form a dual plactic-closed subset of $\mathfrak{S}_n$. 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.

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

TL;DR

This work develops a comprehensive framework for regular Schur labeled skew shape posets and their associated -Hecke modules. It links Stanley's -partition conjecture to a precise criterion: is symmetric and is a left weak Bruhat interval for regular Schur skew shapes, enabling a description of as reading words of standard Young tableaux of shape . The authors classify the corresponding -Hecke modules up to isomorphism, show that regularity is equivalent to dual plactic-closedness of , and construct distinguished filtrations with Schur-basis quotients. They also outline significant open questions on extending the classification beyond , the decomposition of , and recovering these modules from generic Hecke algebras via specialization , establishing a rich interplay between poset combinatorics, symmetric functions, and -Hecke representation theory.

Abstract

Assuming Stanley's -partition conjecture holds, the regular Schur labeled skew shape posets with underlying set are precisely the posets such that the -partition generating function is symmetric and the set of linear extensions of , denoted , is a left weak Bruhat interval in the symmetric group . We describe the permutations in in terms of reading words of standard Young tableaux when is a regular Schur labeled skew shape poset, and classify 's up to descent-preserving isomorphism as ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the -Hecke modules associated with regular Schur labeled skew shape posets up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the posets whose linear extensions form a dual plactic-closed subset of . Using this characterization, we construct distinguished filtrations of with respect to the Schur basis when is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the -Hecke modules are also discussed.
Paper Structure (22 sections, 27 theorems, 191 equations, 2 figures, 2 tables)

This paper contains 22 sections, 27 theorems, 191 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compositions, Young diagrams, and bijective tableaux
  4. Weak Bruhat orders on the symmetric group
  5. Regular posets and Schur labeled skew shape posets
  6. The $0$-Hecke algebra and the quasisymmetric characteristic
  7. Modules arising from posets and weak Bruhat interval modules
  8. The weak Bruhat interval structure of $\Sigma_L(P)$ for $P \in \mathsf{RSP}_n$
  9. An equivalence relation on $\mathrm{Int}(n)$
  10. The classification of $\mathsf{M}_P$'s for $P \in \mathsf{RSP}_n$
  11. A characterization of regular Schur labeled skew shape posets $P$ and distinguished filtrations of $\mathsf{M}_P$
  12. A characterization of regular Schur labeled skew shape posets
  13. Distinguished filtrations of $\mathsf{M}_P$ for $P \in \mathsf{RSP}_n$
  14. Further avenues
  15. The classification problem
  16. ...and 7 more sections

Key Result

Lemma 2.1

(05BB) For $\sigma,\rho \in \mathfrak{S}_n$ with $\sigma \preceq_R \rho$, the map $[\sigma, \rho]_R \rightarrow [\mathrm{id}, \sigma^{-1}\rho]_R, \gamma \mapsto \sigma^{-1} \gamma$ is a poset isomorphism. Equivalently, for $\sigma,\rho \in \mathfrak{S}_n$ with $\sigma \preceq_L \rho$, the map $[\sig

Figures (2)

  • Figure 4.1: The left weak Bruhat intervals in $C$ on $(\mathfrak{S}_4, \preceq_L)$ and the right weak Bruhat intervals $\overline{\mathrm{min}}(C)$ and $\overline{\mathrm{max}}(C)$ on $(\mathfrak{S}_4, \preceq_R)$ in \ref{['eg: C and min(C) and max(C)']}
  • Figure 6.1: The $H_4(0)$-action on the basis $\Sigma_L(P) = [2134, 4321]_L$ for $\mathsf{M}_P$ and the sets $B'_k$$(1 \leq k \leq 5)$ in \ref{['eg: distinguished filt']}

Theorems & Definitions (68)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • proof
  • ...and 58 more