Theta 猜想
The Theta Conjecture
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- Scuola Normale Superiore（高等师范学院）
- Università Pegaso（佩加索大学）
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中文总结 AI 辅助
本文提出双变量Theta猜想，通过定义带标记Dyck路径上的对角反转统计量，细化并支持已证明的单变量Theta猜想，并证明其Schröder情形。
中文摘要 AI 辅助
我们定义了带标记的 Dyck 路径上的对角反转统计量，该路径同时携带标记的上升和标记的可收缩谷。这为 D'Adderio、Iraci 和 Vanden Wyngaerd 提出的单变量 Theta 猜想（最近由 D'Adderio、Pagaria 及本工作作者证明）提供了一个双变量细化的显式候选。通过令 $q=1$，可以从我们的双变量版本恢复单变量猜想。新猜想在两个装饰参数之一消失时，分别恢复了 Delta 猜想的上升版本和谷版本，并为对称函数 $\Theta_{e_l} \Theta_{e_k}\nabla e_{n-k-l}$ 提供了组合解释，该函数也被猜想为具有两组交换变量和两组反交换变量的对角余不变量的某个分次模的 Frobenius 特征标。为了支持 Theta 猜想及其接触细化，我们证明了其 Schröder 情形，即对称函数侧与 $e_{n-d} h_d$ 的内积与限制在 Schröder 路径上的组合侧相匹配。证明使用了比先前已知情形更精细的组合论证，利用了两种高斯乘积恒等式。
英文摘要
We define a diagonal-inversion statistic on labeled Dyck paths carrying both decorated rises and decorated contractible valleys. This gives an explicit candidate for a bivariate refinement of the univariate Theta conjecture of D'Adderio, Iraci, and Vanden Wyngaerd, recently proved by D'Adderio, Pagaria, and the authors of this work. The univariate conjecture can be recovered from our bivariate version by setting $q=1$. The new conjecture recovers the rise and valley versions of the Delta conjecture when either decoration parameter vanishes, and provides a combinatorial interpretation of the symmetric function $Θ_{e_l} Θ_{e_k}\nabla e_{n-k-l}$, which is also conjectured to be the Frobenius characteristic of a certain graded module of diagonal coinvariants with two sets of commuting variables and two sets of anticommuting variables. In support of the Theta conjecture and its touching refinement, we prove its Schröder case, that is, the scalar product of the symmetric function side with $e_{n-d} h_d$ matches the combinatorial side restricted to Schröder paths. The proof uses a finer combinatorial argument than the previously known cases, leveraging two Gaussian product identities.
