We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to $q,t$-analogs of the dimensions of the irreducible representations of $C_n \sim S_d$, the wreath product of the cyclic group $C_n$ with the symmetric group.

中文翻译：

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多麦克唐纳多项式

我们介绍了由分区的 ​​$n$-元组索引并具有某些正交性和三角形关系的麦克唐纳多项式。我们证明它们可以明确地作为依赖于特殊字母表的普通麦克唐纳多项式的乘积给出。有了这个因式分解，我们建立了它们最基本的属性，例如它们的范数平方、评估和再现核的显式公式。此外，我们表明与多麦克唐纳多项式相关的 $q,t$-Kostka 系数是正的，并且对应于 $c_n \sim S_d$ 的不可约表示的维度的 $q,t$-analogs，花圈循环群 $C_n$ 与对称群的乘积。