克莱伯猜想与对称函数的互补积
Kleber's conjecture and complementary products of symmetric functions
AI总结：
研究任意交换环\(R\)上舒尔函数的克莱伯矩形补猜想，通过分量分裂的一般独立性定理证明相关乘积线性无关，还得出小池 - 寺田通用特征积线性无关，以及特征零域上单项式对称函数类似结果和\(\mathbb{Z}\)上整线性无关性。
AI中文摘要：
我们证明了任意交换环\(R\)上关于舒尔函数的克莱伯矩形补猜想，表明对于固定矩形，由无序互补对索引的乘积\(s_\lambda s_{\lambda^\vee}\)在\(\Lambda_R\)中线性无关。证明基于分量分裂的一般独立性定理。\(s_\lambda s_{\lambda^\vee}\)的独立性还得出了在任何域上小池 - 寺田通用特征积的线性无关性，回答了高 - 奥雷洛维茨 - 勇的一个问题。我们还证明了特征零域上单项式对称函数的类似结果以及\(\mathbb{Z}\)上的整线性无关性。
英文摘要：
We prove Kleber's rectangular-complement conjecture for Schur functions over an arbitrary commutative ring $R$, showing that, for a fixed rectangle, the products $s_λs_{λ^\vee}$, indexed by unordered complementary pairs, are linearly independent in $Λ_R$. The proof rests on a general independence theorem for componentwise splittings, which asserts that for every partition $θ$, the products $s_αs_β$ are linearly independent as $\{α,β\}$ ranges over unordered pairs of partitions satisfying $α+β=θ$. The independence of the products $s_λs_{λ^\vee}$ also yields linear independence of the Koike--Terada universal-character products over any field, answering a question of Gao--Orelowitz--Yong. We also prove the analogous result for monomial symmetric functions over fields of characteristic zero, as well as integral linear independence over $\mathbb{Z}$.
