Monotonicity of inverse Kazhdan-Lusztig polynomials

TL;DR

This paper proves a monotonicity property for inverse Kazhdan-Lusztig polynomials across all Coxeter systems, leveraging Soergel's conjecture and Rouquier complexes, and extends this to parabolic cases.

Contribution

It establishes the monotonicity of inverse Kazhdan-Lusztig polynomials for all Coxeter systems and generalizes this to parabolic polynomials, based on conjectural and categorical frameworks.

Findings

Inverse Kazhdan-Lusztig polynomials are monotonic for all Coxeter systems.

The proof relies on Soergel's conjecture and Rouquier complexes.

Monotonicity is extended to parabolic Kazhdan-Lusztig polynomials.

Abstract

For arbitrary Coxeter systems, we prove that inverse Kazhdan-Lusztig polynomials satisfy a monotonicity property. This follows from the validity of Soergel's conjecture and the existence of injective morphisms between Rouquier complexes in the mixed perverse Hecke category. The monotonicity property is generalised to parabolic Kazhdan-Lusztig polynomials.