Cluster Algebras for Bosonic Plethysm

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero, let $V={\mathbb{k}}^{\ell }$ and $W={\mathbb{k}}^m$, and set \[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}. \] We construct an explicit skew-symmetrizable seed $\hat{I}{\pounds }_{\ell ,m}$ by restricting and folding the determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell ,m\ge 2$, we have \[ \mathcal R_{\ell,m}=\mathcal U(Σ_{\ell,m}), \] with polynomial frozen coefficients, and $\hat{I}{\pounds }_{\ell ,m}$ admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone ${\mathcal{C}}_{\ell ,m}$. 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 ${\mathcal{C}}_{\ell ,m}$.

44 pages, comments are welcome