Diagonal Specht ideals and their varieties

We study the diagonal Specht ideals $I_λ\subseteq\mathcal{R}_{m,n}:=\mathbb{C}[\mathbf{x}_1,\dots,\mathbf{x}_m]$, $\mathbf{x}_r=(x_{r,1},\dots,x_{r,n})$, generated by the $λ$-isotypic component of $\mathcal{R}_{m,n}$ for the diagonal action of $S_n$. We characterize the set of zeros of $I_λ$ as $V_λ