  The d-dimensional Lambda-Fleming-Viot generator acting on functions g(x),
with x being a vector of d allele frequencies, can be written as a
Wright-Fisher generator acting on functions g with a modified random linear
argument of x induced by partitioning occurring in the Lambda-Fleming-Viot
process. The eigenvalues and right polynomial eigenvectors are easy to see from
this representation. The two-dimensional process, which has a one-dimensional
generator, is considered in detail. A non-linear equation is found for the
Green's function. In a model with genic selection a proof is given that there
is a critical selection value such that if the selection coefficient is greater
or equal to the critical value then fixation, when the boundary 1 is hit, has
probability 1 beginning from any non-zero frequency. This is an analytic proof
different from proofs by Der, Epstein and Plotkin (2011) and Foucart (2013).
When fixation is not certain the fixation probability can be computed from an
algorithm in the paper. An application in the infinitely-many-alleles
Lambda-Fleming-Viot process is finding an interesting identity for the
frequency spectrum of alleles that is based on size-biassing. The moment dual
process in the Fleming-Viot process is the usual Lambda-coalescent tree back in
time. The Wright-Fisher representation using a different set of polynomials
g_n(x) as test functions produces a dual death process which has a similarity
to the Kingman coalescent and decreases by units of one. The eigenvalues of the
process are analogous to the Jacobi polynomials when expressed in terms of
g_n(x), playing the role of x^n. E[g_n(X)] under the stationary distribution
when there is mutation is analogous to the n th moment in a Beta distribution.
There is a version g_n(X), when X and n are d-dimensional, and even an
intriguing Ewens' sampling formula analogy when d tends to infinity.
