Laplace-Beltrami operator on the set of symmetric homogeneous polynomials

Question

I've read in a paper that the Laplace-Beltrami operator on the set of symmetric homogeneous polynomials is defined by $$ \frac{\alpha}{2}\sum_{i=1}^m\frac{\partial^2}{\partial y_i^2} + \sum_{1\leq i \neq j \leq m} \frac{1}{y_i-y_j}\frac{\partial}{\partial y_i}. $$ Maybe I'm missing something obvious, but I don't understand the division by $y_i - y_j$. Why $\frac{\partial}{\partial y_i} P(y_1, \ldots, y_m)$ would be divisible by $y_i - y_j$? Am I missing something obvious? Or am I totally misunderstand?