发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造显式斜对称化种子，证明对称平方plethysm环的簇代数结构，并用theta基计数最高权权重，给出Jacobi-Trudi恒等式。
AI 中文摘要
设 $\Bbbk$ 为特征零的代数闭域，令 $V=\Bbbk^\ell$，$W=\Bbbk^m$，并设 \\[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}。\\] 我们通过限制并折叠带旗标的 $m$-箭 Kronecker 箭图的行列式种子，构造了一个显式的斜对称化种子 $\Sigma_{\ell,m}$。对于所有 $\ell,m\ge2$，我们有 \\[ \mathcal R_{\ell,m}=\mathcal U(\Sigma_{\ell,m})，\\] 其中冻结系数为多项式，且 $\Sigma_{\ell,m}$ 具有红化序列。theta 基恰好对参数位于有理多面体锥 $\mathscr C_{\ell,m}$ 中时跨越冻结边界。其权纤维计数 $\mathcal R_{\ell,m}$ 的多重分次最高权重数，而 Jacobi--Trudi 恒等式将对称平方 plethysm 系数表示为这些计数的有限交错和。优化的冻结变量给出了 $\mathscr C_{\ell,m}$ 的显式有限不等式组。
英文摘要
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 determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell,m\ge2$, we have \[ \mathcal R_{\ell,m}=\mathcal U(Σ_{\ell,m}), \] with polynomial frozen coefficients, and $Σ_{\ell,m}$ 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 $\mathscr C_{\ell,m}$.
Comments44 pages, comments are welcome; v2. corrected a mutation sequence in Appendix D
