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The first main result of this note, Theorem 1.2, establishes thedeterminantal identities (7) and (8) for the expectation, under a determinantalpoint process governed by an integrable projection kernel, of scaling limits ofcharacteristic polynomials sampled at several points. The determinantalidentities (7) and (8) can be seen as the scaling limit of the identity ofFyodorov and Strahov for the averages of ratios of products of the values ofthe characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanskiand Strahov derived the determinantal identity of Fyodorov and Strahov from thestability of the Giambelli formula under averaging. In Theorem 1.4 thestability of the Giambelli formula under averaging is established fordeterminantal point process with integrable projection kernels. The proof ofTheorems 1.2 and 1.4 relies on the characterization of conditional measures ofour point processes as orthogonal polynomial ensembles.

A determinantal point process governed by an integrable projection kernel is Giambelli compatible