离开霍尔：Neguţ算子的显式公式

中文摘要 AI 辅助

q,t-组合学领域近期的重大突破包括Carlsson和Mellit引入的Dyck路径代数$\boldsymbol{\reals}_{q,t}$，以及Blasiak等人提出的Catalanimals，二者均促成了对Bergeron等人提出的有理洗牌猜想不同扩展形式的独立证明。本文的首个主要贡献是在代数$\boldsymbol{\reals}_{q,t}$内部给出了Neguţ算子的简洁显式公式，建立了有理洗牌猜想的原始算子与对应Catalanimals之间直接、初等的联系。该公式绕过了椭圆霍尔代数，使这些算子成为透明、可操作的工具，我们可在任意对称函数上精确且高效地计算其作用，而非仅作用于常数。第二个主要贡献是给出了一系列将Neguţ算子与D'Adderio等人引入的Theta算子关联起来的显式公式。为证明这些公式，我们将前述Theta算子扩展至整个代数$\boldsymbol{\reals}_{q,t}$，由此获得一系列新的组合学结果。该扩展背后的代数计算已在Lean中形式化。为展示我们结果的威力，我们还给出了D'Adderio等人于2019年首次提出的Theta猜想的证明，该证明也部分在Lean中形式化。

英文摘要

Recent major breakthroughs in $q,t$-combinatorics include the introduction of the Dyck path algebra $\mathbb{A}_{q,t}$ by Carlsson and Mellit and of the Catalanimals by Blasiak et al., both of which led, among other things, to independent proofs of different extensions of the rational shuffle conjecture of Bergeron et al. The first main contribution of this paper is a simple, explicit formula inside the algebra $\mathbb{A}_{q,t}$ for the Negut operators, yielding a direct, elementary connection between the original operators of the rational shuffle conjecture and the corresponding Catalanimals. Our formula bypasses the elliptic Hall algebra, turning these operators into transparent, workable tools whose action we can compute exactly and efficiently on any symmetric function, not just constants. Our second main contribution consists of a series of explicit formulas relating the Negut operators to the Theta operators introduced by D'Adderio et al. To prove these formulas, we provide an extension of the aforementioned Theta operators to the entire algebra $\mathbb{A}_{q,t}$, allowing us to obtain a series of new combinatorial results. The algebraic computations underlying this extension have been formalized in Lean. To showcase the power of our results, we give a proof, also partially formalized in Lean, of the Theta conjecture of D'Adderio et al., first stated in 2019.

发表机构

• Università di Pisa（比萨大学）
• Scuola Normale Superiore（意大利高等师范学院）
• Università Pegaso（佩加索大学）
• Università di Bologna（博洛尼亚大学）

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