{
  "id": "2105.00099",
  "version": "v1",
  "published": "2021-04-30T21:34:49.000Z",
  "updated": "2021-04-30T21:34:49.000Z",
  "title": "On the annihilator ideal in the $bt$-algebra of tensor space",
  "authors": [
    "Steen Ryom-Hansen"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the representation theory of the braids and ties algebra, or the $bt$-algebra, $ \\cal E$. Using the cellular basis $\\{m_{{\\mathfrak s} {\\mathfrak t}} \\}$ for $ \\cal E$ obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules $M(\\lambda)$ and $ M(\\Lambda) $ for $\\cal E$. We show that the tensor product module $V^{\\otimes n}$ for $\\cal E$ is a direct sum of $ M(\\lambda)$'s. We introduce the dual cellular basis $\\{n_{{\\mathfrak s} {\\mathfrak t}} \\}$ for $ \\cal E $ and study its action on $ M(\\lambda) $ and $ M(\\Lambda ) $. We show that the annihilator ideal $ \\cal I $ in $ \\cal E $ of $ V^{\\otimes n } $ enjoys a nice compatibility property with respect to $\\{n_{{\\mathfrak s} {\\mathfrak t}} \\}$. We finally study the quotient algebra $ {\\cal E}/{\\cal I} $, showing in particular that it is a simultaneous generalization of H\\\"arterich's 'generalized Temperley-Lieb algebra' and Juyumaya's 'partition Temperley-Lieb algebra'.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-30T21:34:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "annihilator ideal",
      "tensor space",
      "juyumayas partition temperley-lieb algebra",
      "nice compatibility property",
      "dual cellular basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}