{
  "id": "math/0609659",
  "version": "v2",
  "published": "2006-09-23T02:04:30.000Z",
  "updated": "2007-07-10T07:17:05.000Z",
  "title": "On the affine Schur algebra of type A",
  "authors": [
    "Dong Yang"
  ],
  "comment": "20pages, also available at http://learn.tsinghua.edu.cn:8080/2002315664/dyang.htm",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "The affine Schur algebra $\\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\\mathcal{U}(\\hat{\\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\\widetilde{GL}_{n,a}$ (defined in Section \\ref{S:semigroups}) action on the tensor space and it equals to the 'dual' of a certain formal coalgebra related to this semigroup. By these approaches we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur-Weyl duality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-10T07:17:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G15"
    ],
    "keywords": [
      "affine schur algebra",
      "affine schur-weyl duality",
      "tensor space",
      "representation map",
      "extended affine weyl group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9659Y"
    }
  }
}