{
  "id": "2412.12141",
  "version": "v1",
  "published": "2024-12-09T15:20:28.000Z",
  "updated": "2024-12-09T15:20:28.000Z",
  "title": "Young diagrams, Borel subalgebras and Cayley graphs",
  "authors": [
    "Ian M. Musson"
  ],
  "comment": "Major revision to the first part of arXiv:2312.11046. Comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathtt{k}$ be an algebraically closed field of characteristic zero and $n, m$ coprime positive integers. Let ${\\stackrel{{\\rm o}}{\\mathfrak{g}}}$ be the Lie superalgebra ${\\mathfrak{sl}}(n|m)$ and let $\\mathfrak T_{iso}$ be the groupoid introduced by Sergeev and Veselov \\cite{SV2} with base the set of odd roots of ${\\stackrel{{\\rm o}}{\\mathfrak{g}}}$. We show the Cayley graphs for three actions of $\\mathfrak T_{iso}$ are isomorphic, These actions originate in quite different ways. Consider the set $X$ of Young diagrams contained in a rectangle with $n$ rows and $m$ columns. By adding or deleting rows and columns from certain diagrams and keeping track of the total number of boxes added or deleted, we obtain an equivalence relation on $X\\times {\\mathbb Z}$ such that $\\mathfrak T_{iso}$ acts on the set of equivalence classes $[X\\times {\\mathbb Z}]$. We compare the action on $[X\\times {\\mathbb Z}]$ to an action on Borel subalgebras of the affinization ${\\widehat{L}(\\stackrel{{\\rm _o}}{{\\mathfrak{g}}})}$ of ${\\stackrel{{\\rm o}}{\\mathfrak{g}}}$ which are related by odd reflections. The third action comes from an action of $\\mathfrak T_{iso}$ on $\\mathtt{k}^{n|m}$ defined by Sergeev and Veselov, motivated by deformed quantum Calogero-Moser problems \\cite{SV1}. This action will be considered in \\cite{M24}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-09T15:20:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65"
    ],
    "keywords": [
      "young diagrams",
      "borel subalgebras",
      "cayley graphs",
      "third action comes",
      "deformed quantum calogero-moser problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}