{
  "Title": "Schur polynomials $s_\\lambda$",
  "Definition": "For a partition $\\lambda$ with at most $n$ parts, the Schur polynomial is $s_\\lambda(x_1, \\dots, x_n) = \\frac{\\det\\left(x_i^{\\lambda_j + n - j}\\right)} {\\det\\left(x_i^{n-j}\\right)}$, the ratio of two determinants, the denominator being the Vandermonde determinant. It is a polynomial despite the division.",
  "Parameters": {
    "n": {
      "type": "Z",
      "constraints": "$1 \\leq n \\leq 4$",
      "display": "$n$"
    },
    "lambda": {
      "type": "Symbolic",
      "constraints": "a partition of at most $4$, with at most $n$ parts",
      "display": "$\\lambda$"
    }
  },
  "Data properties": {
    "type": "Z[]",
    "complete": "false",
    "rigour": "exact"
  },
  "Formulas": {
    "formula-jacobi-trudi": "$s_\\lambda = \\det\\left(h_{\\lambda_i - i + j}\\right)_{1 \\leq i,j \\leq \\ell(\\lambda)}$, where $h_d$ are the HREF{Complete_homogeneous_symmetric_polynomials}[complete homogeneous symmetric polynomials] and $h_0 = 1$, $h_d = 0$ for $d < 0$ (the Jacobi-Trudi identity). This is how the table is computed.",
    "formula-edges": "$s_{(k)} = h_k$ and $s_{(1^k)} = e_k$, the complete homogeneous and HREF{Elementary_symmetric_polynomials}[elementary symmetric polynomials].",
    "formula-tableaux": "$s_\\lambda = \\sum_T x^{T}$, summed over semistandard Young tableaux of shape $\\lambda$ with entries in $\\{1, \\dots, n\\}$, where $x^{T}$ records how often each entry occurs.",
    "formula-kostka": "$s_\\lambda = \\sum_\\mu K_{\\lambda\\mu}\\, m_\\mu$, the HREF{Monomial_symmetric_polynomials}[monomial symmetric polynomials] with the Kostka numbers as coefficients."
  },
  "Comments": {
    "comment-notation": "The partition is written as its parts, largest first, separated by commas: $2,1,1$ is the partition $(2,1,1)$ of $4$.",
    "comment-range": "Partitions of at most $4$ in at most $4$ variables, so that the longest entry is 387 characters. These grow faster than the monomial polynomials they expand into, which is why this table stops one variable sooner.",
    "comment-representations": "$s_\\lambda$ in $n$ variables is the character of an irreducible polynomial representation of $GL_n$, evaluated at a diagonal matrix with entries $x_1, \\dots, x_n$; the partitions with at most $n$ parts index those representations."
  },
  "Links": {
    "Wiki": {
      "title": "Wikipedia: Schur polynomial",
      "url": "https://en.wikipedia.org/wiki/Schur_polynomial"
    }
  },
  "Tags": [
    "polynomial",
    "combinatorics",
    "algebra",
    "bases for symmetric polynomials"
  ],
  "Programs": {
    "program-sage": {
      "language": "Sage",
      "code": "def schur(n, lam):\n    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])\n    h = lambda d: (R.one() if d == 0 else R.zero() if d < 0 else\n                   sum(prod(c) for c in\n                       Combinations(list(R.gens())*d, d).list()))\n    f = SymmetricFunctions(QQ).s()[lam]\n    return R(f.expand(n, alphabet=[str(g) for g in R.gens()]))\n\nschur(4, [3, 2])                 # a partition of 5, past this table"
    }
  },
  "Numbers": {
    "1": {
      "1": "x1",
      "2": "x1^2",
      "3": "x1^3",
      "4": "x1^4"
    },
    "2": {
      "1": "x1 + x2",
      "2": "x1^2 + x1*x2 + x2^2",
      "1,1": "x1*x2",
      "3": "x1^3 + x1^2*x2 + x1*x2^2 + x2^3",
      "2,1": "x1^2*x2 + x1*x2^2",
      "4": "x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4",
      "3,1": "x1^3*x2 + x1^2*x2^2 + x1*x2^3",
      "2,2": "x1^2*x2^2"
    },
    "3": {
      "1": "x1 + x2 + x3",
      "2": "x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2",
      "1,1": "x1*x2 + x1*x3 + x2*x3",
      "3": "x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3",
      "2,1": "x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2",
      "1,1,1": "x1*x2*x3",
      "4": "x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4",
      "3,1": "x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3",
      "2,2": "x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2",
      "2,1,1": "x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2"
    },
    "4": {
      "1": "x1 + x2 + x3 + x4",
      "2": "x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2",
      "1,1": "x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4",
      "3": "x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3",
      "2,1": "x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + 2*x1*x2*x4 + x2^2*x4 + 2*x1*x3*x4 + 2*x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2",
      "1,1,1": "x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4",
      "4": "x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4 + x1^3*x4 + x1^2*x2*x4 + x1*x2^2*x4 + x2^3*x4 + x1^2*x3*x4 + x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x4^4",
      "3,1": "x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x1^3*x4 + 2*x1^2*x2*x4 + 2*x1*x2^2*x4 + x2^3*x4 + 2*x1^2*x3*x4 + 3*x1*x2*x3*x4 + 2*x2^2*x3*x4 + 2*x1*x3^2*x4 + 2*x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + 2*x1*x2*x4^2 + x2^2*x4^2 + 2*x1*x3*x4^2 + 2*x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3",
      "2,2": "x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 2*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x3^2*x4^2",
      "2,1,1": "x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 3*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2",
      "1,1,1,1": "x1*x2*x3*x4"
    }
  }
}