{
  "id": "1903.03998",
  "version": "v1",
  "published": "2019-03-10T14:34:38.000Z",
  "updated": "2019-03-10T14:34:38.000Z",
  "title": "LLT polynomials, elementary symmetric functions and melting lollipops",
  "authors": [
    "Per Alexandersson"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Stanley--Stembridge conjecture and previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary basis in for the class of graphs called ``melting lollipops'' previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-10T14:34:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elementary symmetric functions",
      "melting lollipops",
      "unicellular llt polynomials implies",
      "elementary symmetric basis",
      "abelian area sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}