[Paper Review] Nested quantum Dyck paths and nabla(s_lambda)

Nicholas A. Loehr, Gregory S. Warrington|ArXiv.org|May 31, 2007

Advanced Combinatorial Mathematics22 references3 citations

TL;DR

This paper proposes a unified combinatorial formula for the monomial expansion of the nabla operator applied to any Schur function $\nabla(s_\lambda)$, using nested labelled Dyck paths weighted by area and diagonal inversion statistics. The key contribution is a conjecture that unifies prior $q,t$-combinatorial conjectures and reduces the proof of all such conjectures to constructing a sign-reversing involution on signed, weighted combinatorial objects.

ABSTRACT

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal inversion" statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q=1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.

Motivation & Objective

• To provide a combinatorial formula for the monomial expansion of $\nabla(s_\lambda)$ for any partition $\lambda$, generalizing and unifying prior conjectures.
• To extend the scope of $q,t$-combinatorial interpretations of the nabla operator beyond specific cases like $\nabla(e_n)$ or $\nabla(p_n)$.
• To establish a framework that reduces proving all extant nabla conjectures to constructing a sign-reversing, weight-preserving involution on explicit signed, weighted combinatorial objects.
• To provide a proof of the conjecture when $q=1$ using the combinatorics of the inverse Kostka matrix $K^{-1}$.

Proposed method

• The conjecture models $\nabla(s_\lambda)$ as a sum over nested labelled Dyck paths weighted by area and diagonal inversion statistics.
• The formula is derived using the inverse Kostka matrix to express $\nabla(m_\lambda)$ in terms of $\nabla(s_\rho)$, enabling monomial expansion.
• A master identity is formulated as a matrix equation $TA = ALB$, where $T$ is diagonal, $A$ encodes monomial coefficients of $\tilde{H}_\mu$, $L$ is the inverse Kostka matrix, and $B$ encodes the conjectured $\nabla(s_\lambda)$ expansion.
• The method relies on constructing a sign-reversing, weight-preserving involution on tuples of Haglund fillings, special rim hook tabloids, and Dyck path pairs to cancel negative terms.
• The proof strategy reduces to finding a bijection between fixed points of the involution and Haglund fillings of shape $\mu$ with content $x_\nu$, preserving $q$- and $t$-weights.
• The approach leverages known combinatorial interpretations of modified Macdonald polynomials and the Schur-positivity of $\tilde{H}_\mu$ to guide cancellation.

Experimental results

Research questions

• RQ1 Can a single combinatorial model describe the monomial expansion of $\nabla(s_\lambda)$ for all partitions $\lambda$?
• RQ2 How can the full set of extant $q,t$-combinatorial conjectures for nabla be reduced to a single structural problem?
• RQ3 What is the role of the inverse Kostka matrix in proving the $q=1$ case of the conjecture?
• RQ4 Is there a sign-reversing, weight-preserving involution on the set of signed, weighted combinatorial objects that realizes the monomial expansion of $\nabla(s_\lambda)$?
• RQ5 Can the matrix identity $TA = ALB$ be verified combinatorially via fixed-point analysis of the involution?

Key findings

• The conjecture provides a unified framework for the monomial expansion of $\nabla(s_\lambda)$ using nested labelled Dyck paths with area and diagonal inversion statistics.
• When $q=1$, the conjecture is proven using the combinatorics of the inverse Kostka matrix $K^{-1}$, which connects Schur and monomial bases.
• The formula for $\nabla(m_\lambda)$ is expressed as a signed sum over Haglund fillings, special rim hook tabloids, and Dyck path pairs with content matching $x_\nu$.
• Computer evidence suggests all entries in the matrix ${}_{(m_\xi)}[\nabla]_{(m_\nu)}$ have coefficients of like sign, implying cancellation via sign-reversing involution is possible.
• The master identity $T_{\mu}A(\nu,\mu) = \sum_{\lambda} A(\lambda,\mu) \sum_{\rho} \sum_{T \in \mathrm{SRHT}(\lambda,\rho)} \sum_{(G,R) \in LNDP_\rho : x_R = x_\nu} \operatorname{sgn}(T)\operatorname{sgn}(\rho)t^{\operatorname{area}(G,R)}q^{\operatorname{dinv}(G,R)}$ is necessary and sufficient for the full conjecture.
• The existence of a sign-reversing, weight-preserving involution on the combinatorial objects would imply the conjecture, reducing all extant nabla conjectures to this single problem.