Kazhdan-Lusztig and R-polynomials from a combinatorial point of view

Abstract Kazhdan-Lusztig and R-polynomials have applications to algebra, topology, and representation theory. Although they were originally defined in terms of Hecke algebras, there are purely combinatorial ways to compute them. In this paper we survey two of them, as well as the main combinatorial properties of the Kazhdan-Lusztig and R-polynomials, and we present some combinatorial conjectures and open problems about them. We also provide several examples, and references for further reading.

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