Pircon kernels and up-down symmetry

TL;DR

This paper introduces the concept of up-down symmetry in pircons, demonstrating its implications for Kazhdan--Lusztig polynomials and extending Deodhar's duality within this framework.

Contribution

It establishes that up-down symmetry ensures Kazhdan--Lusztig R-polynomials form a P-kernel and extends Deodhar's duality to pircons.

Findings

Up-down symmetry implies R-polynomials are P-kernels.

The property holds in classical cases.

Extended Deodhar duality to pircons.

Abstract

We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan--Lusztig $R^x$-polynomials of a pircon $P$ are a $P$-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan--Lusztig theory.