I am trying to understand the proof of the lemma A.28 in the representation theory book by Fulton and Harris. Let $x_1,\ldots, x_k$ and $y_1,\ldots, y_k$ two distinct indipendent variables. Denote $P_j(x)=\sum_{i=1}^kx_i^j$ the usual Newton function. In the first part of the proof the authors say
$\log\left(\displaystyle\frac{1}{\prod_{i,j}(1-x_iy_j)}\right)=\displaystyle\sum_{j=1}^\infty\frac{1}{j}P_j(x)P_j(y)$.
But I can't see why. I tried using the taylor expansion series for $\log(1+x)$ but it doesn't work. Any help? Thanks.
After expanding the first term you will get $-\sum_{i,j}^{}\log (1 - x_iy_j) = \sum_{i,j} \sum_{k=1}^{\infty} \frac{1}{k}x_i^ky_j^k $ which after changing the order of summation becomes $\sum_{k=1}^{\infty} \frac{1}{k} \sum_{i,j} x_i^ky_j^k $ The inner sum is nothing but $P_k(x)P_k(y)$.