Relations between Kazhdan-Lusztig bases, Murphy bases and seminormal bases

Let (Formula presented.) be the Robinson-Schensted correspondence between the symmetric group (Formula presented.) and the set of pairs of standard tableaux with the same shapes. We show that each Kazhdan-Lusztig basis (KL basis for short) element (Formula presented.) can be expressed as a linear combination of some (Formula presented.) which satisfies that (Formula presented.), where “ (Formula presented.) ” is the dominance (partial) order between standard tableaux, (Formula presented.) denotes the conjugate of (Formula presented.) for each standard tableau (Formula presented.), (Formula presented.) is the seminormal basis of the Iwahori-Hecke algebra associated to (Formula presented.). As a result, we generalize an earlier result of Geck on the relation between the KL basis and the Murphy basis. Similar relations between the twisted KL basis, the dual seminormal basis and the dual Murphy basis are obtained.