{
  "id": "1609.07575",
  "version": "v1",
  "published": "2016-09-24T06:03:43.000Z",
  "updated": "2016-09-24T06:03:43.000Z",
  "title": "Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture",
  "authors": [
    "James Haglund",
    "Brendon Rhoades",
    "Mark Shimozono"
  ],
  "comment": "45 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The symmetric group $\\mathfrak{S}_n$ acts on the polynomial ring $\\mathbb{Q}[\\mathbf{x}_n] = \\mathbb{Q}[x_1, \\dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\\mathfrak{S}_n$-invariant polynomials with vanishing constant term. The quotient $R_n = \\frac{\\mathbb{Q}[\\mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} \\subseteq \\mathbb{Q}[\\mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $k \\leq n$. The corresponding quotient $R_{n,k} := \\frac{\\mathbb{Q}[\\mathbf{x}_n]}{I_{n,k}}$ carries a graded action of $\\mathfrak{S}_n$ and specializes to $R_n$ when $k = n$. We generalize many of the nice properties of $R_n$ to $R_{n,k}$. In particular, we describe the Hilbert series of $R_{n,k}$, give extensions of the Artin and Garsia-Stanton monomial bases of $R_n$ to $R_{n,k}$, determine the reduced Gr\\\"obner basis for $I_{n,k}$ with respect to the lexicographic monomial order, and describe the graded Frobenius series of $R_{n,k}$. Just as the combinatorics of $R_n$ are controlled by permutations in $\\mathfrak{S}_n$, we will show that the combinatorics of $R_{n,k}$ are controlled by ordered set partitions of $\\{1, 2, \\dots, n\\}$ with $k$ blocks. The {\\em Delta Conjecture} of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of $R_{n,k}$ is (up to a minor twist) the $t = 0$ specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded $\\mathfrak{S}_n$-module $V_{n,k}$ whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module $R_{n,k}$ solves this problem in the specialization $t = 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-24T06:03:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordered set partitions",
      "delta conjecture",
      "generalized coinvariant algebras",
      "graded frobenius series",
      "lexicographic monomial order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}