Graded cellularity and the Monotonicity Conjecture

TL;DR

This paper offers a new proof of the monotonicity conjecture for Kazhdan-Lusztig polynomials by leveraging the graded cellularity of Libedinsky Double Leaves, connecting algebraic structures to combinatorial properties.

Contribution

It introduces a novel approach using graded cellularity to prove the monotonicity conjecture for Kazhdan-Lusztig polynomials across all Coxeter systems.

Findings

Proof of the monotonicity conjecture for Kazhdan-Lusztig polynomials

Connection between graded cellularity and polynomial properties

New perspective on Kazhdan-Lusztig theory

Abstract

The graded cellularity of Libedinsky Double Leaves, which form a basis for the endomorphism ring of the Bott_Samelson_Soergel bimodules, allows us to view the Kazhdan_Lusztig polynomials as graded decomposition numbers. Using this point of view, I provide in this paper a new proof of the monotonicity conjecture for the Kazhdan_Lusztig polynomials of any Coxeter system.