{
  "id": "2406.13728",
  "version": "v1",
  "published": "2024-06-19T17:48:19.000Z",
  "updated": "2024-06-19T17:48:19.000Z",
  "title": "A Combinatorial Perspective on the Noncommutative Symmetric Functions",
  "authors": [
    "Angela Hicks",
    "Robert McCloskey"
  ],
  "comment": "34 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The noncommutative symmetric functions $\\textbf{NSym}$ were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants $\\{\\boldsymbol{e}_n\\}_{n\\in \\mathbb{N}}$ that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions $\\Lambda$. Giving noncommutative analogues of generating function relations for other bases of $\\Lambda$ allowed Gelfand et al. to define additional bases of $\\textbf{NSym}$ and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in $\\Lambda$: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of $\\Lambda$ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by E\\u{g}ecio\\u{g}lu and Remmel, 1990.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T17:48:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "noncommutative symmetric functions",
      "combinatorial perspective",
      "noncommutative analogue",
      "traditional symmetric functions",
      "determine change-of-basis formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}