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We adapt the hypercube decompositions introduced byBlundell-Buesing-Davies-Veli\v{c}kovi\'{c}-Williamson to prove theCombinatorial Invariance Conjecture for Kazhdan-Lusztig $R$-polynomials in thecase of elementary intervals in $S_n$. This significantly generalizes the mainpreviously-known case of the conjecture, that of lower intervals.

Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals