{
  "id": "2104.12336",
  "version": "v1",
  "published": "2021-04-26T03:53:41.000Z",
  "updated": "2021-04-26T03:53:41.000Z",
  "title": "$\\imath$Hall algebra of Jordan quiver and $\\imath$Hall-Littlewood functions",
  "authors": [
    "Ming Lu",
    "Shiquan Ruan",
    "Weiqiang Wang"
  ],
  "comment": "40 pages",
  "categories": [
    "math.RT",
    "math.CO",
    "math.QA"
  ],
  "abstract": "We show that the $\\imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $\\imath$Hall algebra to the ring of symmetric functions in two parameters $t, \\theta$, which maps the $\\imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($\\imath$HL) functions. The (modified) $\\imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $\\imath$HL functions. The modified $\\imath$HL functions specialize at $\\theta=0$ to the modified HL functions; they specialize at $\\theta=1$ to the deformed universal characters of type C, which further specialize at $(t=0, \\theta =1)$ to the universal characters of type C.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-26T03:53:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hall algebra",
      "jordan quiver",
      "hall-littlewood functions",
      "hl functions admit",
      "transition relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}