  We employ general parabolic recursion methods to demonstrate the recently
devised hypercube formula for Kazhdan-Lusztig polynomials of $S_n$, and
establish its generalization to the full setting of a finite Coxeter system
through algebraic proof. We introduce procedures for positive decompositions of
$q$-derived Kazhdan-Lusztig polynomials within this setting, that utilize
classical Hecke algebra positivity phenomena of Dyer-Lehrer and
Grojnowski-Haiman. This leads to a distinct algorithmic approach to the
subject, based on induction from a parabolic subgroup. We propose suitable weak
variants of the combinatorial invariance conjecture and verify their validity
for permutation groups.

Representation Theory;Combinatorics