{
  "id": "1709.07751",
  "version": "v1",
  "published": "2017-09-20T20:05:38.000Z",
  "updated": "2017-09-20T20:05:38.000Z",
  "title": "Partition Algebras and the Invariant Theory of the Symmetric Group",
  "authors": [
    "Georgia Benkart",
    "Tom Halverson"
  ],
  "comment": "36 pages. arXiv admin note: text overlap with arXiv:1707.01410",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "The symmetric group $\\mathsf{S}_n$ and the partition algebra $\\mathsf{P}_k(n)$ centralize one another in their actions on the $k$-fold tensor power $\\mathsf{M}_n^{\\otimes k}$ of the $n$-dimensional permutation module $\\mathsf{M}_n$ of $\\mathsf{S}_n$. The duality afforded by the commuting actions determines an algebra homomorphism $\\Phi_{k,n}: \\mathsf{P}_k(n) \\to \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ from the partition algebra to the centralizer algebra $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$, which is a surjection for all $k, n \\in \\mathbb{Z}_{\\ge 1}$, and an isomorphism when $n \\ge 2k$. We present results that can be derived from the duality between $\\mathsf{S}_n$ and $\\mathsf{P}_k(n)$; for example, (i) expressions for the multiplicities of the irreducible $\\mathsf{S}_n$-summands of $\\mathsf{M}_n^{\\otimes k}$, (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$, (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra $\\mathsf{P}_k(n)$. When $2k >n$, the map $\\Phi_{k,n}$ has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of $\\Phi_{k,n}$ in terms of the orbit basis of $\\mathsf{P}_k(n)$ and explain how the surjection $\\Phi_{k,n}$ can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-20T20:05:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C30"
    ],
    "keywords": [
      "partition algebra",
      "symmetric group",
      "invariant theory",
      "centralizer algebra",
      "fold tensor power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}