Kazhdan–Lusztig R-polynomials, combinatorial invariance, and hypercube decompositions

We show how the R-polynomials of the symmetric groups can be computed, in a poset-theoretic way, from canonical hypercube decompositions. This involves a new combinatorial concept, which we call a shortcut. We conjecture that the same formula holds for a certain class of combinatorially defined hypercube decompositions. We also study the behavior of these concepts under the operation of taking the direct product of two Bruhat intervals, and characterize the shortcuts of the canonical hypercube decompositions. Our main conjecture implies the Combinatorial Invariance Conjecture.