  Duality plays an important role in population genetics. It can relate results
from forwards-in-time models of allele frequency evolution with those of
backwards-in-time genealogical models; a well known example is the duality
between the Wright-Fisher diffusion for genetic drift and its genealogical
counterpart, the coalescent. There have been a number of articles extending
this relationship to include other evolutionary processes such as mutation and
selection, but little has been explored for models also incorporating crossover
recombination. Here, we derive from first principles a new genealogical process
which is dual to a Wright-Fisher diffusion model of drift, mutation, and
recombination. Our approach is based on expressing a putative duality
relationship between two models via their infinitesimal generators, and then
seeking an appropriate test function to ensure the validity of the duality
equation. This approach is quite general, and we use it to find dualities for
several important variants, including both a discrete L-locus model of a gene
and a continuous model in which mutation and recombination events are scattered
along the gene according to continuous distributions. As an application of our
results, we derive a series expansion for the transition function of the
diffusion. Finally, we study in further detail the case in which mutation is
absent. Then the dual process describes the dispersal of ancestral genetic
material across the ancestors of a sample. The stationary distribution of this
process is of particular interest; we show how duality relates this
distribution to haplotype fixation probabilities. We develop an efficient
method for computing such probabilities in multilocus models.
