Simplification for the Laplace-Beltrami Operator

Shigekazu Nakagawa $\hspace{-1pt}^{*}$ , Hiroki Hashiguchi $\hspace{-1pt}^{**}$ , Naoto Niki $\hspace{-1pt}^{**}$

$\hspace{-1pt}^{*}$Kurashiki University of Science and the Arts
$\hspace{-1pt}^{**}$Science University of Tokyo

Abstract:

The zonal polynomials are symmetric polynomial functions each of which have to be an eigenfunction of a differential operator called the Laplace-Beltrami operator. They appear in the expansions in many distributions of statistics based on normal populations.

Formally speaking, the definition of the zonal polynomials is as follows. Let $P_{n,\ell}$ be the set of all partitions of the non-negative integer n no longer than $\ell > 0$,

$$P_{n,\ell} =\{\lambda =(\lambda_{1}, \lambda_{2}, \ldots , \l... ...{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{\ell} \geq 0 \},$$

and $D_{\ell}$ be the Laplace-Beltrami operator

$$D_{\ell} = \sum_{i=1}^{\ell}x_{i}^{2}\frac{\partial^{2}}{\par... ...rac{x_{i}^{2}}{x_{i} - x_{j}} \frac{\partial}{\partial x_{i}}.$$

Then, for given $\lambda \in P_{n,\ell}$the zonal polynomial $z(\lambda)$in variables $x_{1}, x_{2}, \ldots , x_{\ell}$ is not only a homogeneous degree n symmetric polynomial but also an eigenfunction of $D_{\ell}$,

$$D_{\ell}z(\lambda) = d(\lambda)z(\lambda),$$

where $d(\lambda)$ is the corresponding eigenvalue

$$d(\lambda) = \sum_{i=1}^{\ell}\lambda_{i}(\lambda_{i} + \ell - i -1).$$

For a fixed basis of symmetric polynomials, $z(\lambda)$ is uniquely expressed under some regular conditions. For example, if $\lambda = (\lambda_{1}, \lambda_{2}, \ldots , \lambda_{\ell})$runs through all partitions in $P_{n,\ell}$then $\{e(\lambda) = e_{1}^{\lambda_{1}}e_{2}^{\lambda_{2}}\cdots e_{\ell}^{\lambda_{\ell}} \mid \lambda \in P_{n,\ell}\}$forms a basis where $e_{1}, e_{2}, \cdots , e_{\ell}$ are fundamental symmetric polynomials $e_{1} = x_{1} + x_{2} + \cdots + x_{\ell}, e_{2} = x_{1}x_{2} + x_{1}x_{3} \cdots + x_{\ell -1}x_{\ell}, \cdots , e_{\ell} = x_{1}x_{2}\cdots x_{\ell}$. If we take such $\{e(\lambda) \mid \lambda \in P_{n,\ell}\}$as a basis then $z(\lambda)$ can be expressed

$$z(\lambda) = \sum_{\mu \in P_{n,\ell}} c[\lambda, \mu]e(\mu^{\prime}),$$

(where $\mu^{\prime}$ means the conjugate of $\mu$) and the coefficients $c[\lambda, \mu]$ are uniquely determined.

The problem is to find $c[\lambda, \mu]$ for given $\lambda$and develop a symbolic algorithm to determine them. The key is simplifying $D_{\ell}$, that is, finding a simple expression of $D_{\ell}$ in terms of $e_{i}\,(1 \leq i \leq \ell)$. The relevant problem is for the case $\ell \geq 4$while a symbolic algorithm based on a recurrence formula is well-known for the case $\ell \leq 3$.

In this paper, we give a simplification for the Laplace-Beltrami operator for any $\ell$. Moreover, for $\ell = 4$ we give a symbolic algorithm including sufficient conditions that the coefficients vanish to determine the coefficients of zonal polynomials.