{
  "id": "1307.6808",
  "version": "v2",
  "published": "2013-07-25T16:31:28.000Z",
  "updated": "2019-06-17T13:25:40.000Z",
  "title": "Fusion procedure for the Yang-Baxter equation and Schur-Weyl duality",
  "authors": [
    "L. Poulain d'Andecy"
  ],
  "comment": "36 pages",
  "journal": "Algebr. Represent. Theory 20 (2017), no. 6, 1379--1414",
  "categories": [
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang-Baxter equation on irreducible representations of $\\mathfrak{gl}_N$, $\\mathfrak{gl}_{N|M}$, $U_q(\\mathfrak{gl}_N)$ and $U_q(\\mathfrak{gl}_{N|M})$. The solutions are obtained via the fusion procedure for the Yang--Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur--Weyl duality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-25T16:31:28.000Z",
      "abstract": "We first review the fusion procedure for an arbitrary solution of the Yang-Baxter equation and the study of distinguished invariant subspaces for the fused solutions. Then we apply these general results to four particular solutions: the Yang solution, its standard deformation and their generalizations for super vector spaces. For the Yang solution, respectively, its \"super\" generalization, we explain how, using the fusion formula for the symmetric group together with the (super) Schur-Weyl duality, the fusion procedure allows to construct a family of fused solutions of the Yang-Baxter equation acting on irreducible representations of the general linear Lie algebra, respectively, of the general linear Lie superalgebra. For the deformations of the two previous solutions, we use the fusion formula for the Hecke algebra together with the (super) quantum Schur--Weyl duality to obtain fused solutions acting on irreducible representations of the quantum groups associated to the general linear Lie (super)algebras.",
      "comment": "35 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2019-06-17T13:25:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "yang-baxter equation",
      "schur-weyl duality",
      "fusion procedure",
      "fused solutions",
      "fusion formula"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Represent. Theory"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6808P"
    }
  }
}