{
  "id": "2406.19715",
  "version": "v1",
  "published": "2024-06-28T08:00:18.000Z",
  "updated": "2024-06-28T08:00:18.000Z",
  "title": "A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring",
  "authors": [
    "John Lentfer"
  ],
  "comment": "31 pages, 8 figures, 4 tables, comments welcome!",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $S_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim-Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2023) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-28T08:00:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05A15",
      "05E10",
      "05E16"
    ],
    "keywords": [
      "bosonic-fermionic coinvariant ring",
      "conjectural basis",
      "monomial basis",
      "conjecture",
      "conjectural frobenius series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}