  We give an exact solution to the Kolmogorov equation describing genetic drift
for an arbitrary number of alleles at a given locus. This is achieved by
finding a change of variable which makes the equation separable, and therefore
reduces the problem with an arbitrary number of alleles to the solution of a
set of equations that are essentially no more complicated than that found in
the two-allele case. The same change of variable also renders the Kolmogorov
equation with the effect of mutations added separable, as long as the mutation
matrix has equal entries in each row. Thus this case can also be solved exactly
for an arbitrary number of alleles. The general solution, which is in the form
of a probability distribution, is in agreement with the previously known
results--which were for the cases of two and three alleles only. Results are
also given for a wide range of other quantities of interest, such as the
probabilities of extinction of various numbers of alleles, mean times to these
extinctions, and the means and variances of the allele frequencies. To aid
dissemination, these results are presented in two stages: first of all they are
given without derivations and too much mathematical detail, and then
subsequently derivations and a more technical discussion are provided.
