  Characterizing time-evolution of allele frequencies in a population is a
fundamental problem in population genetics. In the Wright-Fisher diffusion,
such dynamics is captured by the transition density function, which satisfies
well-known partial differential equations. For a multi-allelic model with
general diploid selection, various theoretical results exist on representations
of the transition density, but finding an explicit formula has remained a
difficult problem. In this paper, a technique recently developed for a
diallelic model is extended to find an explicit transition density for an
arbitrary number of alleles, under a general diploid selection model with
recurrent parent-independent mutation. Specifically, the method finds the
eigenvalues and eigenfunctions of the generator associated with the
multi-allelic diffusion, thus yielding an accurate spectral representation of
the transition density. Furthermore, this approach allows for efficient,
accurate computation of various other quantities of interest, including the
normalizing constant of the stationary distribution and the rate of convergence
to this distribution.
