四阶的一般子群永久优势猜想
The General Subgroup Permanental-Dominance Conjecture in Order Four
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中文总结 AI 辅助
本文证明了一般子群永久优势猜想的四阶情形，覆盖$S_4$子群的全部37种不可约特征标，其中35种由相关不等式推导，2种非实$A_4$特征标通过平方和等方法解决。
中文摘要 AI 辅助
一般子群永久优势猜想此前仅在矩阵阶数为3时为人所知。本文证明了其完整的四阶情形：对每个子群$H\leq S_4$、$H$的每个不可约复特征标$\chi$以及每个$4\times4$半正定埃尔米特矩阵$A$，均有$d_\chi^H(A)/\chi(1)\leq \operatorname{per} A$。与通常的不变量特化不同，该结果覆盖了来自$S_4$的11类共轭子群的37种不可约特征标情形。其中35种情形由一般主子式、矩量及块收缩不等式推导得出；两种非实$A_4$特征标被归约为$3\times3$半正定格拉姆矩阵锥上的多项式非负性问题，通过秩1平方和恒等式、精确正定内部证书、有理格拉姆证书及格拉姆锥的闭性得以解决。
英文摘要
The general subgroup permanental-dominance conjecture was previously known only through matrix order three. This paper proves its complete order-four case: for every subgroup $H\leq S_4$, every irreducible complex character $χ$ of $H$, and every $4\times 4$ Hermitian positive-semidefinite matrix $A$, it establishes $d_χ^H(A)/χ(1)\leq \operatorname{per} A$. Unlike the usual immanant specialization, the result covers all thirty-seven irreducible-character cases arising from the eleven conjugacy classes of subgroups of $S_4$. Thirty-five cases follow from general principal-minor, moment, and block-contraction inequalities. The two non-real $A_4$ characters are reduced to polynomial nonnegativity on the cone of $3\times 3$ positive-semidefinite Gram matrices and are resolved by a rank-one sum-of-squares identity, an exact positive-definite interior certificate, rational Gram certificates, and closure of the Gram cone.
