Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials

TL;DR

The paper proves the combinatorial invariance of the coefficient of $q$ in Kazhdan–Lusztig polynomials for arbitrary Coxeter groups by introducing the invariant $d_{u,v}$ from $R_{u,v}$ and showing $d_{u,v}=g_{u,v}$, where $g_{u,v}$ is a poset-invariant diamond-generating set size. This yields combinatorial invariance for Bruhat intervals of length at most $6$ and establishes invariance for the corresponding coefficients in $R_{u,v}$ and $\ ilde{R}_{u,v}$. It further confirms the Gabber–Joseph conjecture for the coefficient of $q^{\\ell(u,v)-1}$ in the Ext-series, giving $\,\dim \mathrm{Ext}^{\\ell(u,v)-1}_{\mathcal{O}}(M_u,M_v)=d_{u,v}$ and hence a poset-invariant Ext-dimension. The approach fuses divergences of increasing paths with Richardson-variety geometry to connect KL-polynomial data with representation-theoretic invariants, with remarks on cluster structures in crystallographic types. Overall, the work unifies combinatorial, geometric, and representation-theoretic aspects of Bruhat intervals and their associated invariants.

Abstract

We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan-Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture for Bruhat intervals of length at most $6$. We also prove the Gabber-Joseph conjecture for the second-highest $\mathrm{Ext}$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group.

Figures (3)

• Figure 1: A Bruhat graph $\Gamma_{u,v}$ and a minimum-size diamond-generating set (shown in red).
• Figure 2: An example of a supporting chain $\mathcal{C}$. The longest path $\gamma_0$ is shown in black, the second-length path $\gamma$ is shown in orange, and $\mathcal{C}$ is shown in red. The cover relations in $[y,z]$ are shown with dotted lines.
• Figure 3: The inductive construction of a supporting chain. The action of $s$ is indicated with blue edges. The second-length path for $[su,sv]$ and supporting chain $\mathcal{C}'=\{\textcolor{purple}{sx_1,\circ}\}$ are shown with dashed orange and purple lines, respectively. The supporting chain for $[u,v]$ is $\mathcal{C}=\{\textcolor{red}{x_1,\bullet,sy}\}$, shown in red.

Theorems & Definitions (40)

• Conjecture 1.1: Combinatorial Invariance Conjecture
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 2.1: Kazhdan--Lusztig Kazhdan-Lusztig-1
• Theorem 2.2: Kazhdan--Lusztig Kazhdan-Lusztig-1
• Definition 2.3
• Definition 2.4