  Jack characters are a generalization of the characters of the symmetric
groups; a generalization that is related to Jack symmetric functions. We
investigate the structure coefficients for Jack characters; they are a
generalization of the connection coefficients for the symmetric groups. More
specifically, we study the cumulants which measure the discrepancy between
these structure coefficients and the simplistic structure coefficients related
to the disjoint product. We show that Jack characters satisfy approximate
factorization property: their cumulants are of very small degree and the
character related to a given partition is approximately equal to the product of
the characters related to its parts. This result will play a key role in the
proof of Gaussianity of fluctuations for a class of random Young diagrams
related to the Jack polynomials.
