AI 中文总结
该研究扩展了杰克情形下的谱理论，将NS算子推广为带辅助变量多项式系数对称函数的Lax算子，得到麦克唐纳小伍德-理查森系数的双参数盒和恒等式及相关线性关系。
AI 中文摘要
我们研究纳扎罗夫和斯克莱宁于2015年引入的麦克唐纳（Macdonald）对称函数的Lax算子，扩展了米克勒-莫尔（Mickler-Moll）在杰克（Jack）情形下发展的谱理论。我们将NS算子扩展为作用于带辅助变量多项式系数的对称函数的Lax算子，证明其具有由分划及其可加角标引的正交本征基，角标内容为本征值，首项由麦克唐纳对称函数给出。我们得到作者早期关于小伍德-理查森（Littlewood-Richardson）系数的杰克盒和恒等式的麦克唐纳类似式，随后利用瓦尔纳（Warnaar）的有限秩乘积公式将该盒和恒等式强化为双参数公式，其留数给出麦克唐纳小伍德-理查森系数间的显式线性关系。
英文摘要
We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.
Comments32 pages
