This talk will look at the Kazhdan--Lusztig basis of the Hecke algebra of type A from two points of view. From one point of view the basis is an example of an IC basis, or "canonical basis". It often happens that IC bases have remarkable properties, and it may be easy to prove that IC bases exist, but very hard to construct them explicitly. The other point of view regards the Kazhdan--Lusztig basis as an example of a cellular basis, which is very useful from the point of view of representation theory.
