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{
"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"
}
}
}