Jack function

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In mathematics, the Jack function, introduced by Henry Jack, 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[edit]

The Jack function J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m) of integer partition \kappa, parameter \alpha and arguments x_1,x_2,\ldots, 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^{|\kappa /\mu|}\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[edit]

In 1997, F. Knop and S. Sahi [1] 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) ≠ T( i',j) whenever i' > i.
  • T(i,j) ≠ T( i',j-1) whenever j>1 and i' < 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[edit]

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(e^{i\theta_1},\cdots,e^{i\theta_n})\overline{g(e^{i\theta_1},\cdots,e^{i\theta_n})}\prod_{1\le j<k\le n}|e^{i\theta_j}-e^{i\theta_k}|^{2/\alpha}d\theta_1\cdots d\theta_n

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,x_2,\ldots,x_n)
=
\frac{\alpha^{|\kappa|}(|\kappa|)!}
{j_\kappa}
J_\kappa^{(\alpha)}(x_1,x_2,\ldots,x_n),

where


j_\kappa=\prod_{(i,j)\in \kappa}
(\kappa_j'-i+\alpha(\kappa_i-j+1))(\kappa_j'-i+1+\alpha(\kappa_i-j)).

For \alpha=2,\; C_\kappa^{(2)}(x_1,x_2,\ldots,x_n) denoted often as just C_\kappa(x_1,x_2,\ldots,x_n) is known as the Zonal polynomial.


P normalization[edit]

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) and 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 [2] 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[edit]

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[edit]

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[edit]

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).

References[edit]

External links[edit]