{
  "id": "2111.12426",
  "version": "v3",
  "published": "2021-11-24T11:13:04.000Z",
  "updated": "2023-09-22T06:49:00.000Z",
  "title": "Skew Howe duality and limit shapes of Young diagrams",
  "authors": [
    "Anton Nazarov",
    "Olga Postnova",
    "Travis Scrimshaw"
  ],
  "comment": "57 pages, 15 figures, 2 tables; v3 fixed typos, added comparison to Biane's result, updated references, fixed typos; v2 fixed typos in Theorem 4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of Conjecture 4.17 in v1",
  "journal": "J. Lond. Math. Soc., 2023",
  "doi": "10.1112/jlms.12813",
  "categories": [
    "math.RT",
    "math.CO",
    "math.PR"
  ],
  "abstract": "We consider the skew Howe duality for the action of certain dual pairs of Lie groups $(G_1, G_2)$ on the exterior algebra $\\bigwedge(\\mathbb{C}^{n} \\otimes \\mathbb{C}^{k})$ as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe for the pairs $(\\mathrm{GL}_{n}, \\mathrm{GL}_{k})$, $(\\mathrm{SO}_{2n+1}, \\mathrm{Pin}_{2k})$, $(\\mathrm{Sp}_{2n}, \\mathrm{Sp}_{2k})$, and $(\\mathrm{Or}_{2n}, \\mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The $G_1$-representation multiplicity is given as a determinant formula using the Lindstr\\\"om-Gessel-Viennot lemma and as a product formula using Dodgson condensation. These admit natural $q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up to an overall factor of $q$), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at $q =1$), we take the infinite rank limit and prove the diagrams converge uniformly to the limit shape.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2023-09-22T06:49:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "60C05",
      "22E46",
      "60G55",
      "05E05"
    ],
    "keywords": [
      "young diagrams",
      "limit shape",
      "combinatorial skew howe duality",
      "product formula",
      "infinite rank limit"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}