{
  "id": "1812.05863",
  "version": "v1",
  "published": "2018-12-14T11:31:15.000Z",
  "updated": "2018-12-14T11:31:15.000Z",
  "title": "Exponents associated with $Y$-systems and their relationship with $q$-series",
  "authors": [
    "Yuma Mizuno"
  ],
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "Let $X_r$ be a finite type Dynkin diagram, and $\\ell$ be a positive integer greater than or equal to two. The $Y$-system of type $X_r$ with level $\\ell$ is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair $(X_r, \\ell)$, we define an integer sequence called exponents using formulation of the $Y$-system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type $X_r$, and prove this conjecture for $(A_1 ,\\ell)$ and $(A_r, 2)$ cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from $q$-series identities for this relationship.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-14T11:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "17B22",
      "81R10"
    ],
    "keywords": [
      "relationship",
      "finite type dynkin diagram",
      "affine lie algebra",
      "integrable highest weight module",
      "series identities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}