{
  "id": "2010.05602",
  "version": "v1",
  "published": "2020-10-12T11:13:24.000Z",
  "updated": "2020-10-12T11:13:24.000Z",
  "title": "A symmetric group action on the irreducible components of the Shi variety associated to $W(\\widetilde{A}_n)$",
  "authors": [
    "Nathan Chapelier-Laget"
  ],
  "comment": "18 pages, 6 figures, 1 table",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $W_a$ be an affine Weyl group with corresponding finite root system $\\Phi$. In the article \"Alcoves corresponding to an affine Weyl group\" Jian-Yi Shi characterized each element $w \\in W_a$ by a $ \\Phi^+$-tuple of integers $(k(w,\\alpha))_{\\alpha \\in \\Phi^+}$ subject to certain conditions. In the article \"Shi variety corresponding to an affine Weyl group\" a new interpretation of the coefficients $k(w,\\alpha)$ is given. This description led us to define an affine variety $\\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. It turns out that this variety has more than one irreducible component, and the set of these components, denoted $H^0(\\widehat{X}_{W_a})$, admits many interesting properties. In particular the group $W_a$ acts on it. In this article we show that the irreducible components of $\\widehat{X}_{W(\\widetilde{A}_n)}$ are in bijection with the circular permutations of $W(A_n) = S_{n+1}$. We also compute the action of $W(A_n)$ on $H^0(\\widehat{X}_{W(\\widetilde{A}_n)})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-12T11:13:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shi variety",
      "symmetric group action",
      "irreducible component",
      "affine weyl group",
      "corresponding finite root system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}