  We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon
for the lower global crystal base for the Fock representation of quantum affine
sl_n. We also show that the coefficients of the lower global crystal base
coincide with certain affine Kazhdan-Lusztig polynomials. It is known that the
coefficients of the global crystal base are q-analogues of decomposition
numbers for Specht modules of the Hecke algebra of type A_n, and that the
coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of
decomposition numbers for tilting modules for quantum sl_k. Thus our algorithm
allows fast computation of these decomposition numbers.
