{
  "id": "2310.20571",
  "version": "v1",
  "published": "2023-10-31T16:01:18.000Z",
  "updated": "2023-10-31T16:01:18.000Z",
  "title": "Regular Schur labeled skew shape posets and their 0-Hecke modules",
  "authors": [
    "Young-Hun Kim",
    "So-Yeon Lee",
    "Young-Tak Oh"
  ],
  "comment": "44 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "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 $\\Sigma_L(P)$, is a left weak Bruhat interval in the symmetric group $\\mathfrak{S}_n$. 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 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-31T16:01:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "06A07",
      "05E10",
      "05E05"
    ],
    "keywords": [
      "schur labeled skew shape poset",
      "regular schur labeled skew shape",
      "hecke modules",
      "left weak bruhat interval",
      "linear extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}