{
  "id": "2209.11734",
  "version": "v1",
  "published": "2022-09-23T17:30:13.000Z",
  "updated": "2022-09-23T17:30:13.000Z",
  "title": "Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories",
  "authors": [
    "Emily Barnard",
    "Eric J. Hanson"
  ],
  "comment": "37 pages, 7 figures",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $\\Lambda$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $\\tau$-exceptional sequences can be used to give a \"brick labeling\" of a certain poset of wide subcategories of finitely-generated $\\Lambda$-modules. When $\\Lambda$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or simply the \"$\\kappa$-map\". Generalizing known results for finite semidistributive lattices, we show that the $\\kappa$-map determines exactly when a set of completely join-irreducible elements forms a \"canonical join representation\". A consequence is that the corresponding \"canonical join complex\" is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes of finite-dimensional algebras. Finally, in the case of lattices of torsion classes of finite-dimensional algebras, we demonstrate how Jasso's $\\tau$-tilting reduction can be encoded using the $\\kappa$-map. We use this to define $\\kappa^d$-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to $\\tau$-exceptional sequences in the algebra setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-23T17:30:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "06A07",
      "06D75",
      "16G20",
      "18E40"
    ],
    "keywords": [
      "exceptional sequences",
      "wide subcategories",
      "poset topology",
      "finite semidistributive lattices",
      "finite-dimensional algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}