Trivial Kazhdan-Lusztig polynomials and cubulation of the Bruhat graph

TL;DR

This work establishes a deep link between trivial Kazhdan–Lusztig polynomials and cubulations of Bruhat intervals in Coxeter groups. It proves that a cubulation of [1,y] implies palindromic Poincaré polynomials and, via Carrell–Peterson and Elias–Williamson, the triviality P_{x,y}=1 for all x≤y; it then investigates the converse, giving positive results in types A and B/C for y=w_0 and in tilde A_2 for infinite families, while exhibiting counterexamples in certain exceptional types. The paper develops a graph-theoretic and growth-theoretic framework, including cubical lattices, normal form forests, and power-series methods, to characterize when the converse can hold and to identify the affine tilde A_n obstruction in the growth/girth sense. It provides explicit constructions of cubulations in type tilde A_2 and establishes a broader “poison subsystem” principle: a failure of the converse in a subsystem propagates to larger systems. Overall, the results illuminate when Bruhat intervals admit cubical substructures and how this constrains KL-polynomials, with implications for affine and exceptional Coxeter groups.

Abstract

For $(W,S)$ an arbitrary Coxeter system and any $y \in W$, we investigate the relationship between the condition that the Kazhdan-Lusztig polynomial $P_{x,y}$ is trivial for all $x \leq y$, and the condition that the Bruhat graph for the interval $[1,y]$ can be cubulated, meaning roughly that this graph can be spanned by a product of subintervals of $\mathbb{Z}$. In one direction, we combine results of Carrell-Peterson and Elias-Williamson to prove that if $[1,y]$ can be cubulated, then $P_{x,y} = 1$ for all $x \leq y$. We then investigate the converse of this statement. For $(W,S)$ finite and $w_0$ the longest element in $W$, so that $P_{x,w_0} = 1$ for all $x \in W$, we construct cubulations of $[1,w_0]$ in types $A$ and $B/C$. However, in some exceptional types, we determine elements $y \in W$ such that $P_{1,y} = 1$ but $[1,y]$ cannot be cubulated. We then prove that if there are infinitely many $y \in W$ such that $[1,y]$ can be cubulated, then $(W,S)$ must be of type $\tilde{A}_n$ for some $n \geq 1$. Finally, for $(W,S)$ of type $\tilde{A}_2$, we exhibit a cubulation of $[1,y]$ for each of the infinitely many $y \in W$ such that $P_{x,y} = 1$ for all $x \leq y$.

Figures (10)

• Figure 1: The Hasse diagram for the interval $[1 , s_1 s_0 s_2 s_1 s_0 s_2 s_0]$ in type $\tilde{A}_2$ is depicted on the left. This graph is spanned by the cubical lattice $\mathcal{C}(2,2,3)$ on the right, and hence the Bruhat graph for this interval can be cubulated.
• Figure 2: The normal form forest for type $A_3$ obtained by labeling nodes of the Dynkin diagram from left to right.
• Figure 3: The Bruhat graph for type $A_2$, with all edges oriented upwards. The Hasse diagram for the Bruhat order consists of all edges except for the vertical red one, and the subgraph consisting of the black edges is the Cayley graph in type $A_2$.
• Figure 4: From left to right, we depict a cubulation for standard parabolic Coxeter elements of lengths $2$ and $3$, and for dihedral elements of length $3$.
• Figure 5: On the left (respectively, right), we depict the inductive construction in the proof of Proposition \ref{['prop:NFcubical']} in type $A_n$ for $n = 2$ (respectively, $n = 3$).

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1: Proposition 3.4.2 of BjoernerBrenti
• Example 2.2
• Definition 3.1: Induced and spanning subgraphs
• Definition 3.2: Cubical lattice
• Lemma 3.3