  We present a combinatorial procedure (based on the W-graph of the Coxeter
group) which shows that the characters of many intersection cohomology
complexes on low rank complex flag varieties with coefficients in an arbitrary
field are given by Kazhdan-Lusztig basis elements. Our procedure exploits the
existence and uniqueness of parity sheaves. In particular we are able to show
that the characters of all intersection cohomology complexes with coefficients
in a field on the flag variety of type A_n for n < 7 are given by
Kazhdan-Lusztig basis elements. By results of Soergel, this implies a part of
Lusztig's conjecture for SL(n) with n \le 7. We also give examples where our
techniques fail.
  In the appendix by Tom Braden examples are given of intersection cohomology
complexes on the flag varities for SL(8) and SO(8) which have torsion in their
stalks or costalks.
