{
  "id": "enwiki-001377203472",
  "slug": "jack-function",
  "title": "Jack function",
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  "lead": "In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.",
  "body": "In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials. ==Definition== The Jack function J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m) of an integer partition \\kappa , parameter \\alpha , and arguments x_1,x_2,\\ldots,x_m can be recursively defined as follows: For m=1 : J_{k}^{(\\alpha )}(x_1)=x_1^k(1+\\alpha)\\cdots (1+(k-1)\\alpha) For m>1: J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m)=\\sum_\\mu J_\\mu^{(\\alpha )}(x_1,x_2,\\ldots,x_{m-1}) x_m^ \\beta_{\\kappa \\mu}, where the summation is over all partitions \\mu such that the skew partition \\kappa/\\mu is a horizontal strip, namely \\kappa_1\\ge\\mu_1\\ge\\kappa_2\\ge\\mu_2\\ge\\cdots\\ge\\kappa_{n-1}\\ge\\mu_{n-1}\\ge\\kappa_n ( \\mu_n must be zero or otherwise J_\\mu(x_1,\\ldots,x_{n-1})=0 ) and \\beta_{\\kappa\\mu}=\\frac{ \\prod_{(i,j)\\in \\kappa} B_{\\kappa\\mu}^\\kappa(i,j) }{ \\prod_{(i,j)\\in \\mu} B_{\\kappa\\mu}^\\mu(i,j) }, where B_{\\kappa\\mu}^\\nu(i,j) equals \\kappa_j'-i+\\alpha(\\kappa_i-j+1) if \\kappa_j'=\\mu_j' and \\kappa_j'-i+1+\\alpha(\\kappa_i-j) otherwise. The expressions \\kappa' and \\mu' refer to the conjugate partitions of \\kappa and \\mu , respectively. The notation (i,j)\\in\\kappa means that the product is taken over all coordinates (i,j) of boxes in the Young diagram of the partition \\kappa . ===Combinatorial formula=== In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials J_\\mu^{(\\alpha )} in n variables: J_\\mu^{(\\alpha )} = \\sum_{T} d_T(\\alpha) \\prod_{s \\in T} x_{T(s)}. The sum is taken over all admissible tableaux of shape \\lambda, and d_T(\\alpha) = \\prod_{s \\in T \\text{ critical}} d_\\lambda(\\alpha)(s) with d_\\lambda(\\alpha)(s) = \\alpha(a_\\lambda(s) +1) + (l_\\lambda(s) + 1). An admissible tableau of shape \\lambda is a filling of the Young diagram \\lambda with numbers 1,2,...,n such that for any box (i,j) in the tableau, T(i,j) \\neq T(i',j) whenever i'>i. T(i,j) \\neq T(i,j-1) whenever j>1 and i' A box s = (i,j) \\in \\lambda is critical for the tableau T if j > 1 and T(i,j)=T(i,j-1). This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials. ==C normalization== The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product: \\langle f,g\\rangle = \\int_{[0,2\\pi]^n} f \\left (e^{i\\theta_1},\\ldots,e^{i\\theta_n} \\right ) \\overline{g \\left (e^{i\\theta_1},\\ldots,e^{i\\theta_n} \\right )} \\prod_{1\\le j This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as C_\\kappa^{(\\alpha)}(x_1,\\ldots,x_n) = \\frac{\\alpha^ (|\\kappa|)!}{j_\\kappa} J_\\kappa^{(\\alpha)}(x_1,\\ldots,x_n), where j_\\kappa=\\prod_{(i,j)\\in \\kappa} \\left (\\kappa_j'-i+\\alpha \\left (\\kappa_i-j+1 \\right ) \\right ) \\left (\\kappa_j'-i+1+\\alpha \\left (\\kappa_i-j \\right ) \\right ). For \\alpha=2, C_\\kappa^{(2)}(x_1,\\ldots,x_n) is often denoted by C_\\kappa(x_1,\\ldots,x_n) and called the Zonal polynomial. ==P normalization== The P normalization is given by the identity J_\\lambda = H'_\\lambda P_\\lambda , where H'_\\lambda = \\prod_{s\\in \\lambda} (\\alpha a_\\lambda(s) + l_\\lambda(s) + 1) where a_\\lambda and l_\\lambda denotes the arm and leg length respectively. Therefore, for \\alpha=1, P_\\lambda is the usual Schur function. Similar to Schur polynomials, P_\\lambda can be expressed as a sum over Young tableaux. However, one need to add an extra weight to each tableau that depends on the parameter \\alpha . Thus, a formula for the Jack function P_\\lambda is given by P_\\lambda = \\sum_{T} \\psi_T(\\alpha) \\prod_{s \\in \\lambda} x_{T(s)} where the sum is taken over all tableaux of shape \\lambda , and T(s) denotes the entry in box s of T. The weight \\psi_T(\\alpha) can be defined in the following fashion: Each tableau T of shape \\lambda can be interpreted as a sequence of partitions \\emptyset = \\nu_1 \\to \\nu_2 \\to \\dots \\to \\nu_n = \\lambda where \\nu_{i+1}/\\nu_i defines the skew shape with content i in T. Then \\psi_T(\\alpha) = \\prod_i \\psi_{\\nu_{i+1}/\\nu_i}(\\alpha) where \\psi_{\\lambda/\\mu}(\\alpha) = \\prod_{s \\in R_{\\lambda/\\mu}-C_{\\lambda/\\mu} } \\frac{(\\alpha a_\\mu(s) + l_\\mu(s) +1)}{(\\alpha a_\\mu(s) + l_\\mu(s) + \\alpha)} \\frac{(\\alpha a_\\lambda(s) + l_\\lambda(s) + \\alpha)}{(\\alpha a_\\lambda(s) + l_\\lambda(s) +1)} and the product is taken only over all boxes s in \\lambda such that s has a box from \\lambda/\\mu in the same row, but not in the same column. ==Connection with the Schur polynomial== When \\alpha=1 the Jack function is a scalar multiple of the Schur polynomial J^{(1)}_\\kappa(x_1,x_2,\\ldots,x_n) = H_\\kappa s_\\kappa(x_1,x_2,\\ldots,x_n), where H_\\kappa=\\prod_{(i,j)\\in\\kappa} h_\\kappa(i,j)= \\prod_{(i,j)\\in\\kappa} (\\kappa_i+\\kappa_j'-i-j+1) is the product of all hook lengths of \\kappa . ==Properties== If the partition has more parts than the number of variables, then the Jack function is 0: J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m)=0, \\mbox{ if }\\kappa_{m+1}>0. ==Matrix argument== In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If X is a matrix with eigenvalues x_1,x_2,\\ldots,x_m , then J_\\kappa^{(\\alpha )}(X)=J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m). ==Bibliography== . . . ==External links== Software for computing the Jack function by Plamen Koev and Alan Edelman. MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package) SAGE documentation for Jack Symmetric Functions",
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    "title": "Jack function",
    "author": [
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        "family": "Anticloud FZ LLE",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/jack-function",
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