Consider a given partition $\lambda=(\lambda_1,\lambda_2,...,\lambda_N)$ and start with the monomial
$$z_1^{\lambda_1}z_2^{\lambda_2}...z_N^{\lambda_N}$$
in $N$ variables $z_1,z_2,...,z_N$. Now we want to symmetrize monomials as in eq. (3.2) of this paper, by performing:
$$m_\lambda=\sum_{s\in W}z_1^{s(\lambda_1)}z_2^{s(\lambda_2)}...z_N^{s(\lambda_N)}$$
where the sum is over the elements of the Weyl group $W$. I am not familiar with the Weyl group. The above is reminiscent of summation over permutation group $S_N$ with elements given by all permutations $\sigma$ of the original order. But I assume the Weyl group is something different, since it has its own name? Could someone explain to me what the elements of the Weyl group are in this context and how the $s(x)$ acts on the partition elements? It would be great if you could show an example for i.e. $N=2$. Thanks for any suggestion!