Kazhdan--Lusztig cells and the Murphy basis

Let $H$ be the Iwahori--Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan--Lusztig basis and the Murphy basis. While the former admits a deep geometric interpretation, the latter leads to a purely combinatorial construction of the representations of $H$, including the Dipper--James theory of Specht modules. In this paper, we establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan--Lusztig basis and Lusztig's results on the $a$-function.