We distinguish a class of random point processes which we call Giambelli
compatible point processes. Our definition was partly inspired by determinantal
identities for averages of products and ratios of characteristic polynomials
for random matrices found earlier by Fyodorov and Strahov. It is closely
related to the classical Giambelli formula for Schur symmetric functions.
We show that orthogonal polynomial ensembles, z-measures on partitions, and
spectral measures of characters of generalized regular representations of the
infinite symmetric group generate Giambelli compatible point processes. In
particular, we prove determinantal identities for averages of analogs of
characteristic polynomials for partitions.
Our approach provides a direct derivation of determinantal formulas for
correlation functions.