Toward Butler's conjecture

TL;DR

The paper advances Butler's conjecture by constructing a positive, combinatorial F-expansion for the Macdonald intersection polynomial $\mathrm{I}_{\lambda,\mu}[X;q,t]$ via a new column-exchange rule and Butler permutations, yielding a fundamental-quasisymmetric expansion with explicit statistics. It then leverages LLT theory to prove Schur positivity in targeted cases, deriving explicit Schur-coefficient formulas for neighboring Hook shapes and obtaining $q,t$-Kostka–style insights. The authors connect generalized modified Macdonald polynomials to LLT polynomials, show positivity under LLT equivalences, and provide a framework to pursue combinatorial formulas for $(q,t)$-Kostka polynomials, including Specializations at $t=1$ or $q=1$. Overall, the work blends filled-diagram generalizations, bijective proofs, and LLT machinery to push toward a complete combinatorial understanding of Butler’s conjecture and related Kostka polynomials.

Abstract

For a partition $ν$, let $λ,μ\subseteq ν$ be two distinct partitions such that $|ν/λ|=|ν/μ|=1$. Butler conjectured that the divided difference $\operatorname{I}_{λ,μ}[X;q,t]=(T_λ\widetilde{H}_μ[X;q,t]-T_μ\widetilde{H}_λ[X;q,t])/(T_λ-T_μ)$ of modified Macdonald polynomials of two partitions $λ$ and $μ$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{λ,μ}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.

Figures (10)

• Figure 1: The left figure shows the Young diagram in the French notation for a partition $(5,4,3,1)$. The right figure shows the computation of the arm and the leg for the red cell $u$.
• Figure 2: The left figure shows an example of a filled diagram $(D,f)$, and the right figure shows the total order $N_{D}$ on each cell of $D$.
• Figure 3: Applying the operator $\mathop{\mathrm{cyc}}\nolimits$.
• Figure 4: A diagram and its subdiagram.
• Figure 5: Any $(\mu,f_{\mu})\in\mathcal{V}(n,m)$ is of the form in the left figure for suitable $a_i$'s, $b_i$'s and $\alpha$ in $\mathbb{F}$. The right figure shows the corresponding $S(\mu,f_{\mu})$.

Theorems & Definitions (62)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 3.1
• Example 3.2
• Definition 3.3
• Remark 3.4
• Example 3.5