{
  "id": "2010.14494",
  "version": "v1",
  "published": "2020-10-27T17:53:41.000Z",
  "updated": "2020-10-27T17:53:41.000Z",
  "title": "Functors between Deligne categories and rings of numbers $\\sum_{n\\in \\mathbb Z_+} a_n\\binom{s}{n}$ where $s\\in \\overline{\\mathbb Q}$ and $a_n \\in \\mathbb Z_+$",
  "authors": [
    "Daniil Kalinov",
    "Andrei Mandelshtam"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "The semiring $R_+(x)$ is defined as a set of all nonnegative integer linear combinations of binomial coefficients $\\binom{x}{n}$ for $n \\in \\mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(s)$ for complex numbers $s$. Particularly interesting is the case of algebraic $s$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\\textrm{Rep}(S_t)$ (and also $\\textrm{Rep}(\\textrm{GL}_t)$) for $t \\in \\mathbb C\\backslash \\mathbb Z_+$. We prove that this semiring is a ring if and only if $s$ is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by $s,$ i.e. all elements of $\\mathcal O_{\\mathbb Q(s)},$ are also contained in this ring. We also give two explicit representations of $R_+(s)$ for both algebraic integers and general algebraic numbers $s.$ One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. Moreover, this leads to a particularly simple description of $R_+(s)$ for both quadratic algebraic numbers and roots of unity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-27T17:53:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "deligne categories",
      "algebraic integers",
      "nonnegative integer linear combinations",
      "quadratic algebraic numbers",
      "general algebraic numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}