Friday, 22 June 2012

Definition of Zonal Polynomial

Let $Y$ be $m \times m $ Symmetric Matrix with Latent Roots(Eigen values) $y_1 \cdots y_m$.

Let $$\kappa=\left(k_1 \cdots k_m \right)$$

be the Partition of an Integer $k$ with Partition Size not more than $m$.

The Zonal Polynomial Corresponding to $\kappa$ Denoted by $C_{\kappa}(Y)$ is a Symmetric , Homogeneous Polynomial of Degree $k$ in Latent Roots $y_1 \cdots y_m$ Such that:

$$ C_{\kappa}(Y)= d_k\,y_1^{k_1} \cdots y_m^{k_m}+ \text{Terms of Lower Weight}$$, $d_k$ being a constant

$C_{\kappa}(Y)$ is an Eigen Function of Differential Operator $\Delta_{Y}$ given by:

$$ \Delta_Y=\sum_{i=1}^{m}y_i^2 \frac{\partial^2 }{\partial y_i^2}+\sum_{i=1}^m \sum_{j=1,j \neq i}^m \frac{y_i^2}{y_i-y_j}\frac{\partial }{\partial y_i}$$

As $\kappa$ varies over all Partitions of $k$,The Zonal Polynomials have Unit Coefficients in the Expansion of $Tr(Y)^k$ i.e.,

$$ Tr(Y)^k= \left(y_1+ \cdots +y_m \right)^k= \sum_{\kappa} C_{\kappa}(Y) $$

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