In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Vanden Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and ${\mathrm{\Theta }}_k$. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators. Even though we deal specifically with the relation between the $D_k$ and ${\mathrm{\Theta }}_k$ operators, our work introduces a method for finding relations between ${\mathrm{\Theta }}_k$ and other plethystic operators which are important in this area of study.

中文翻译：

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θ 算子猜想的证明

在（广义）Delta 猜想及其组合形式的上下文中，D'Adderio、Iraci 和 Vanden Wyngaerd 最近陈述了一个关于两个对称函数算子的猜想， $D_{克}$ 和 ${\mathrm{\Theta }}_{克}$. 我们证明了这个 Theta 算子猜想，发现它是 Mellit 和 Garsia 的五项关系的结果。结果，我们发现了令人惊讶的书写方式$D_{克}$运营商。尽管我们专门处理了两者之间的关系$D_{克}$ 和 ${\mathrm{\Theta }}_{克}$ 运算符，我们的工作介绍了一种查找之间关系的方法 ${\mathrm{\Theta }}_{克}$ 和其他在这个研究领域很重要的 plethystic 算子。