Diagrammatics for Kazhdan-Lusztig (R)over-tilde-polynomials

Let (W, S) be an arbitrary Coxeter system. We introduce a family of polynomials, {(R) over tilde (u,(v) under bar)(t)}, indexed by pairs (u, (v) under bar) formed by an element u is an element of W and a (non-necessarily reduced) word (v) under bar in the alphabet S. The polynomial (R) over tilde (u,(v) under bar)(t) is obtained by considering a certain subset of Libedinsky's light leaves associated to the pair (u, (v) under bar). Given a reduced expression (v) under bar of an element v is an element of W, we show that (R) over tilde (u,(v) under bar)(t) coincides with the Kazhdan-Lusztig (R) over tilde -polynomial (R) over tilde (u,(v) under bar)(t). Using the diagrammatic approach, we obtain some closed formulas for R-polynomials. (C) 2019 Elsevier Ltd. All rights reserved.