  In this paper, we compute the expectation of traces of powers of the
hermitian matrix Jacobi process for a large enough but fixed size. To proceed,
we first derive the semi-group density of its eigenvalues process as a bilinear
series of symmetric Jacobi polynomials. Next, we use the expansion of power
sums in the Schur polynomial basis and the integral Cauchy-Binet formula in
order to determine the partitions having non zero contributions after
integration. It turns out that these are hooks of bounded weight and the sought
expectation results from the integral of a product of two Schur functions with
respect to a generalized Beta distribution. For special values of the
parameters on which the matrix Jacobi process depends, the last integral
reduces to the Cauchy determinant and we close the paper with the investigation
of the asymptotic behavior of the resulting formula as the matrix size tends to
infinity.
