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{
"Title": "Hall\u2013Littlewood polynomials $P_\\lambda(x_1,\\dots,x_n;t)$",
"Definition": "The Hall\u2013Littlewood polynomial $P_\\lambda(x_1,\\dots,x_n;t)$ CITE{Wiki} is Macdonald's $P$-normalised Hall\u2013Littlewood symmetric polynomial CITE{Macdonald}, given by CITE{formula-symmetrization} for a partition $\\lambda$ with at most $n$ parts.",
"Keywords": [
"Hall-Littlewood symmetric functions",
"Hall Littlewood polynomials",
"Macdonald Hall-Littlewood polynomials"
],
"Tags": [
"polynomial",
"combinatorics",
"algebra",
"bases for symmetric polynomials"
],
"Parameters": {
"n": {
"type": "Z",
"title": "number of variables",
"display": "$n$",
"constraints": "a positive integer"
},
"lambda": {
"type": "Symbolic",
"display": "$\\lambda$",
"constraints": "a partition with at most $n$ parts",
"title": "shape"
}
},
"Comments": {
"notation": "The partition is written as its parts, largest first, separated by commas: $2,1,1$ is the partition $(2,1,1)$ of $4$.",
"convention": "$t$ is an indeterminate, and $\\lambda$ is padded with zeros to length $n$ in CITE{formula-symmetrization}. In this $P$ normalisation, the coefficient of HREF{Monomial_symmetric_polynomials}[the monomial symmetric polynomial] $m_\\lambda$ is $1$, and the other monomial terms are indexed by partitions $\\mu$ lower than $\\lambda$ in dominance order. The $Q_\\lambda$ normalisation multiplies $P_\\lambda$ by a factor depending on $\\lambda$ and $t$. The transformed Hall\u2013Littlewood polynomial $Q'_\\lambda(x;t)=Q_\\lambda[x/(1-t);t]$ is not a scalar multiple of $P_\\lambda$ in general."
},
"Formulas": {
"formula-symmetrization": "$P_\\lambda(x_1,\\dots,x_n;t)=v_\\lambda(t)^{-1}\\sum_{w\\in S_n} w\\left(x_1^{\\lambda_1}\\cdots x_n^{\\lambda_n} \\prod_{1\\leq i<j\\leq n}\\frac{x_i-tx_j}{x_i-x_j}\\right)$, where $v_\\lambda(t)=\\prod_{r\\geq0}\\prod_{j=1}^{m_r(\\lambda)}(1-t^j)/(1-t)$, $m_r(\\lambda)$ is the number of parts of $\\lambda$ equal to $r$, $\\ell(\\lambda)$ is the number of nonzero parts, and $m_0(\\lambda)=n-\\ell(\\lambda)$ counts the zeros added by padding.",
"formula-schur": "$P_\\lambda(x;0)=s_\\lambda(x)$, HREF{Schur_polynomials}[the Schur polynomial].",
"formula-monomial": "$P_\\lambda(x;1)=m_\\lambda(x)$, HREF{Monomial_symmetric_polynomials}[the monomial symmetric polynomial].",
"formula-transition": "$s_\\lambda(x)=\\sum_\\mu K_{\\lambda\\mu}(t)P_\\mu(x;t)$, where the sum is over partitions $\\mu$ of $|\\lambda|$ with at most $n$ parts and $K_{\\lambda\\mu}(t)$ is HREF{T259}[the Kostka-Foulkes polynomial]."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "from sage.all import QQ, PolynomialRing, SymmetricFunctions\n\nT = PolynomialRing(QQ, 't')\nP = SymmetricFunctions(T.fraction_field()).hall_littlewood().P()\ngrouped = P([3, 2]).expand(4, alphabet=['x1', 'x2', 'x3', 'x4'])\n\n# Sage groups coefficients in t; this writes the same polynomial\n# expanded over QQ[x1, x2, x3, x4, t], as the table does.\nR = PolynomialRing(QQ, ['x1', 'x2', 'x3', 'x4', 't'])\nprint(R(str(grouped)))"
}
},
"Similar tables": [
{
"table": "HREF{Schur_polynomials}[Schur polynomials]",
"relation": "are the $t=0$ specialization"
},
{
"table": "HREF{Monomial_symmetric_polynomials}[monomial symmetric polynomials]",
"relation": "are the $t=1$ specialization"
},
{
"table": "HREF{T259}[Kostka-Foulkes polynomials]",
"relation": "are the transition coefficients from Schur polynomials to this Hall\u2013Littlewood $P$ basis"
}
],
"Links": {
"Wiki": {
"title": "Wikipedia: Hall-Littlewood polynomials",
"url": "https://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials"
},
"SageHallLittlewood": {
"title": "SageMath: Hall-Littlewood symmetric functions",
"url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/hall_littlewood.html"
}
},
"References": {
"Macdonald": {
"bib": "I. G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995.",
"mr": "1354144"
}
},
"Data properties": {
"type": "Z[]",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every $P_\\lambda$ with $1\\leq n\\leq4$, $1\\leq|\\lambda|\\leq4$, and $\\ell(\\lambda)\\leq n$; extending the same rectangle to $n=5$ would make $P_{(4)}$ 2493 characters long, while extending to $|\\lambda|=5$ at $n=4$ would make $P_{(5)}$ 2114 characters long",
"rigour details": "Every entry is an exact polynomial with integer coefficients. The generator computes Macdonald's symmetrization formula in the fraction field of $\\mathbb{Z}[x_1,\\dots,x_n,t]$ and refuses an entry unless the denominator cancels.\nBefore the values were written, every entry was compared with Sage's Hall-Littlewood $P$ basis CITE{SageHallLittlewood}. The same values were checked at $t=0$ against HREF{Schur_polynomials}[the Schur polynomials], at $t=1$ against HREF{Monomial_symmetric_polynomials}[the monomial symmetric polynomials], and in the transition identity with HREF{T259}[the Kostka-Foulkes polynomials]."
},
"Display properties": {
"number-header": "$P_\\lambda(x_1,\\dots,x_n;t)$"
},
"Numbers": {
"1": {
"1": "x1",
"2": "x1^2",
"3": "x1^3",
"4": "x1^4"
},
"2": {
"1": "x1 + x2",
"2": "-x1*x2*t + x1^2 + x1*x2 + x2^2",
"1,1": "x1*x2",
"3": "-x1^2*x2*t - x1*x2^2*t + x1^3 + x1^2*x2 + x1*x2^2 + x2^3",
"2,1": "x1^2*x2 + x1*x2^2",
"4": "-x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t + x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4",
"3,1": "-x1^2*x2^2*t + x1^3*x2 + x1^2*x2^2 + x1*x2^3",
"2,2": "x1^2*x2^2"
},
"3": {
"1": "x1 + x2 + x3",
"2": "-x1*x2*t - x1*x3*t - x2*x3*t + x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2",
"1,1": "x1*x2 + x1*x3 + x2*x3",
"3": "x1*x2*x3*t^2 - x1^2*x2*t - x1*x2^2*t - x1^2*x3*t - 2*x1*x2*x3*t - x2^2*x3*t - x1*x3^2*t - x2*x3^2*t + 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*x2*x3*t^2 - x1*x2*x3*t + 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": "x1^2*x2*x3*t^2 + x1*x2^2*x3*t^2 + x1*x2*x3^2*t^2 - x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t - x1^3*x3*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x2^3*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - x1*x3^3*t - x2*x3^3*t + 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": "-x1^2*x2^2*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t + 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*x3*t - x1*x2^2*x3*t - x1*x2*x3^2*t + 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*x2*t - x1*x3*t - x2*x3*t - x1*x4*t - x2*x4*t - x3*x4*t + 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*x2*x3*t^2 + x1*x2*x4*t^2 + x1*x3*x4*t^2 + x2*x3*x4*t^2 - x1^2*x2*t - x1*x2^2*t - x1^2*x3*t - 2*x1*x2*x3*t - x2^2*x3*t - x1*x3^2*t - x2*x3^2*t - x1^2*x4*t - 2*x1*x2*x4*t - x2^2*x4*t - 2*x1*x3*x4*t - 2*x2*x3*x4*t - x3^2*x4*t - x1*x4^2*t - x2*x4^2*t - x3*x4^2*t + 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*x2*x3*t^2 - x1*x2*x4*t^2 - x1*x3*x4*t^2 - x2*x3*x4*t^2 - x1*x2*x3*t - x1*x2*x4*t - x1*x3*x4*t - x2*x3*x4*t + 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*x2*x3*x4*t^3 + x1^2*x2*x3*t^2 + x1*x2^2*x3*t^2 + x1*x2*x3^2*t^2 + x1^2*x2*x4*t^2 + x1*x2^2*x4*t^2 + x1^2*x3*x4*t^2 + 3*x1*x2*x3*x4*t^2 + x2^2*x3*x4*t^2 + x1*x3^2*x4*t^2 + x2*x3^2*x4*t^2 + x1*x2*x4^2*t^2 + x1*x3*x4^2*t^2 + x2*x3*x4^2*t^2 - x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t - x1^3*x3*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x2^3*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - x1*x3^3*t - x2*x3^3*t - x1^3*x4*t - 2*x1^2*x2*x4*t - 2*x1*x2^2*x4*t - x2^3*x4*t - 2*x1^2*x3*x4*t - 3*x1*x2*x3*x4*t - 2*x2^2*x3*x4*t - 2*x1*x3^2*x4*t - 2*x2*x3^2*x4*t - x3^3*x4*t - x1^2*x4^2*t - 2*x1*x2*x4^2*t - x2^2*x4^2*t - 2*x1*x3*x4^2*t - 2*x2*x3*x4^2*t - x3^2*x4^2*t - x1*x4^3*t - x2*x4^3*t - x3*x4^3*t + 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*x2*x3*x4*t^3 + x1*x2*x3*x4*t^2 - x1^2*x2^2*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - 2*x1^2*x2*x4*t - 2*x1*x2^2*x4*t - 2*x1^2*x3*x4*t - 5*x1*x2*x3*x4*t - 2*x2^2*x3*x4*t - 2*x1*x3^2*x4*t - 2*x2*x3^2*x4*t - x1^2*x4^2*t - 2*x1*x2*x4^2*t - x2^2*x4^2*t - 2*x1*x3*x4^2*t - 2*x2*x3*x4^2*t - x3^2*x4^2*t + 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*x2*x3*x4*t^3 - x1^2*x2*x3*t - x1*x2^2*x3*t - x1*x2*x3^2*t - x1^2*x2*x4*t - x1*x2^2*x4*t - x1^2*x3*x4*t - 3*x1*x2*x3*x4*t - x2^2*x3*x4*t - x1*x3^2*x4*t - x2*x3^2*x4*t - x1*x2*x4^2*t - x1*x3*x4^2*t - x2*x3*x4^2*t + 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*x2*x3*x4*t^3 - x1*x2*x3*x4*t^2 - x1*x2*x3*x4*t + 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"
}
}
}