I am continue working with standard $\mathfrak{gl}(N,\mathbb{C})$ representation in $\mbox{Sym}^k\mathbb{C}^N$.
The character of such representation is given by $\chi_k (z_1,\ldots,z_N) = \sum\limits_{k_1,\ldots,k_N \geq 0, ~k_1+\ldots+k_N=k} z_1^{k_1}\ldots z_N^{k_N}$. There exists a statement that $\sum\limits_{k=0}^\infty \chi_k(z_1,\ldots,z_N)\omega^k = \prod\limits_{i=1}^\infty \frac{1}{1-z_i\omega}$, but I really stuck with it and have no idea how to prove it. Could anyone show me the proof or give some useful hints?
$\frac{1}{1-\omega z_i} = \sum\limits_{j=1}^\infty (\omega z_i)^{j-1}$ is an infinite sum of geometric progression with the start value equals to $1$ and common ratio $\omega z_i$.
Now suppose we have two polynomials $f$ and $g$, then their product $h$ looks like $h_k = \sum\limits_{i+j=k}f_i g_j$.
Combining this facts we obtain $\prod\limits_{i=1}^\infty \frac{1}{1-z_i\omega} = \sum\limits_{k=0}^\infty (\sum\limits_{k_1+\ldots+k_N=k}z_1^{k_1}\ldots z_N^{k_N})\omega^k = \sum\limits_{k=0}^\infty \chi_k(z_1,\ldots,z_N)\omega^k$.