Cluster Algebras for Bosonic Plethysm

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $V=\Bbbk^\ell$ and $W=\Bbbk^m$, and set \[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}. \] We construct an explicit skew-symmetrizable seed $Σ_{\ell,m}$ by restricting and folding the determinan