{
  "id": "math/0112131",
  "version": "v1",
  "published": "2001-12-12T21:43:49.000Z",
  "updated": "2001-12-12T21:43:49.000Z",
  "title": "On 321-avoiding permutations in affine Weyl groups",
  "authors": [
    "R. M. Green"
  ],
  "comment": "16 pages, AMSTeX",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "We introduce the notion of 321-avoiding permutations in the affine Weyl group $W$ of type $A_{n-1}$ by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in $W$ coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of $W$ (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations. Using Shi's characterization of the Kazhdan--Lusztig cells in the group $W$, we use our main result to show that the fully commutative elements of $W$ form a union of Kazhdan--Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan--Lusztig basis of the associated Hecke algebra to be computed combinatorially. We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to $GL_n({\\Bbb C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-12-12T21:43:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15"
    ],
    "keywords": [
      "permutations",
      "fully commutative elements",
      "kazhdan-lusztig cells",
      "extended affine weyl group"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....12131G"
    }
  }
}