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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 aWright-Fisher generator acting on functions g with a modified random linearargument of x induced by partitioning occurring in the Lambda-Fleming-Viotprocess. The eigenvalues and right polynomial eigenvectors are easy to see fromthis representation. The two-dimensional process, which has a one-dimensionalgenerator, is considered in detail. A non-linear equation is found for theGreen's function. In a model with genic selection a proof is given that thereis a critical selection value such that if the selection coefficient is greateror equal to the critical value then fixation, when the boundary 1 is hit, hasprobability 1 beginning from any non-zero frequency. This is an analytic proofdifferent from proofs by Der, Epstein and Plotkin (2011) and Foucart (2013).When fixation is not certain the fixation probability can be computed from analgorithm in the paper. An application in the infinitely-many-allelesLambda-Fleming-Viot process is finding an interesting identity for thefrequency spectrum of alleles that is based on size-biassing. The moment dualprocess in the Fleming-Viot process is the usual Lambda-coalescent tree back intime. The Wright-Fisher representation using a different set of polynomialsg_n(x) as test functions produces a dual death process which has a similarityto the Kingman coalescent and decreases by units of one. The eigenvalues of theprocess are analogous to the Jacobi polynomials when expressed in terms ofg_n(x), playing the role of x^n. E[g_n(X)] under the stationary distributionwhen 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 anintriguing Ewens' sampling formula analogy when d tends to infinity.

The Lambda-Fleming-Viot process and a connection with Wright-Fisher diffusion