Jack function

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.

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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