The anti-spherical category

TL;DR

This paper introduces a diagrammatic categorification of the anti-spherical module for Coxeter groups, proving non-negativity of certain polynomials and confirming a monotonicity conjecture, with implications for the p-canonical basis.

Contribution

It develops a new diagrammatic framework for the anti-spherical module, enabling proofs of polynomial non-negativity and monotonicity, and provides tools for computing elements of the p-canonical basis.

Findings

Deodhar's parabolic Kazhdan-Lusztig polynomials have non-negative coefficients.

A monotonicity conjecture of Brenti's is validated.

A basis of morphisms called 'light leaves' is constructed.

Abstract

We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.