We discuss generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual pk variables are accompanied by anti-commuting Grassmann variables θk. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These polynomials have similar properties to canonical Macdonald polynomials – they respect two different orderings in the set of (super)-Young diagrams simultaneously.

中文翻译：

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超级分区的麦克唐纳多项式


我们讨论了著名的麦克唐纳多项式对于超杨图情况的推广，其中半箱与全箱处于同等地位。这些超麦克唐纳多项式是扩展变量集的多项式：通常的 pk 变量伴随着反交换格拉斯曼变量 θk。从最近定义的超舒尔多项式开始，并利用三角分解的正交关系，我们能够完全确定超麦克唐纳多项式。这些多项式与规范麦克唐纳多项式具有相似的属性 - 它们同时尊重（超）杨图集中的两种不同的排序。