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 $\Sigma_{\ell,m}$ by restricting and folding the determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell,m\ge2$, we have $$\mathcal R_{\ell,m}=\mathcal U(\Sigma_{\ell,m}),$$ with polynomial frozen coefficients, and $\Sigma_{\ell,m}$ admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone $\mathscr C_{\ell,m}$. Its weight fibers count the multigraded highest-weight multiplicities of $\mathcal R_{\ell,m}$, and the Jacobi--Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for $\mathscr C_{\ell,m}$.