[Paper Review] The Delta square conjecture

Michele D’Adderio, Alessandro Iraci+1 more|arXiv (Cornell University)|Aug 1, 2018

Advanced Combinatorial Mathematics15 references4 citations

TL;DR

This paper proposes the generalized Delta square conjecture, a combinatorial formula for the symmetric function $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of decorated partially labelled square paths. It extends the square conjecture of Loehr and Warrington and the generalized Delta conjecture, proving special cases including the $q=0$ specialization and the Schr"oder case, which generalizes the $q,t$-square theorem.

ABSTRACT

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}Δ_{h_m}Δ_{e_{n-k}}ω(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by proving the specialization $m=q=0$, reducing it to the same case of the Delta conjecture, and the Schröder case, i.e. the case $\langle \cdot ,e_{n-d}h_d angle$. The latter provides a broad generalization of the $q,t$-square theorem of Can and Loehr (Can, Loehr 2006). We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed $m$ and $n$. Finally, in the appendix, we give a new proof of the Delta conjecture at $q=0$.

Motivation & Objective

• To extend the square conjecture of Loehr and Warrington to a broader combinatorial framework involving symmetric functions and decorated square paths.
• To generalize the generalized Delta conjecture of Haglund, Remmel, and Wilson to the square path setting.
• To resolve an open question posed in [Haglund-Remmel-Wilson-2015] regarding the combinatorics of the generalized Delta conjecture in the square path model.
• To establish a linear relation among conjectures via a new combinatorial involution on decorated square paths.
• To provide a new proof of the $q=0$ case of the Delta conjecture, enhancing self-containment and potential independent interest.

Proposed method

• Proposes a new symmetric function formula involving $\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ scaled by $\frac{[n-k]_t}{[n]_t}$, conjectured to equal a generating function over decorated partially labelled square paths.
• Introduces combinatorial objects: decorated partially labelled square paths, with statistics tracking area, dinv, and labelled parts.
• Uses symmetric function identities involving Hall-Littlewood polynomials $\widetilde{H}_\mu$, power sum $p_n$, and the involution $\omega$, to relate the symmetric function side to path statistics.
• Applies the $q=0$ specialization to reduce the generalized Delta square conjecture to the known $q=0$ case of the Delta conjecture, leveraging known results from [Garsia-Haglund-Remmel-Yoo-2017].
• Proves the Schr"oder case $\langle \cdot, e_{n-d}h_d \rangle$ by generalizing the $q,t$-square theorem of Can and Loehr, using path decomposition and generating function identities.
• Introduces a combinatorial involution on decorated square paths that preserves key statistics and establishes a linear relation among conjectures with fixed $m$ and $n$, linking them to symmetric function identities.

Experimental results

Research questions

• RQ1 Can the generalized Delta conjecture be extended to the square path model, unifying the Dyck path and square path combinatorics?
• RQ2 Does the generalized Delta square conjecture reduce to the known square conjecture when $m=k=0$?
• RQ3 Is the Schr"oder case of the generalized Delta square conjecture a broad generalization of the $q,t$-square theorem of Can and Loehr?
• RQ4 Can the $q=0$ specialization of the generalized Delta square conjecture be proven by reducing it to the $q=0$ case of the Delta conjecture?
• RQ5 What combinatorial structure underlies the linear relation among generalized Delta square conjectures with fixed $m$ and $n$?

Key findings

• The generalized Delta square conjecture reduces to the square conjecture of Loehr and Warrington when $m=k=0$, confirming consistency with a known theorem.
• The $q=0$ specialization of the generalized Delta square conjecture is proven by reducing it to the $q=0$ case of the Delta conjecture, which is already established.
• The Schr"oder case $\langle \cdot, e_{n-d}h_d \rangle$ of the generalized Delta square conjecture is proven, providing a broad generalization of the $q,t$-square theorem of Can and Loehr.
• A new combinatorial involution is constructed on decorated partially labelled square paths, which establishes a linear relation among conjectures with fixed $m$ and $n$, linking them to symmetric function identities.
• A new proof is given for the $q=0$ case of the Delta conjecture, using symmetric function identities and path statistics, which is self-contained and potentially of independent interest.
• The proof of the Schr"oder case involves a detailed analysis of generating functions over paths, using $t$-binomial identities and properties of Hall-Littlewood polynomials, yielding explicit $t$-analogs of binomial coefficients.