  A classical identity due to Giambelli in representation theory states that
the character in any representation is expressed as a determinant whose
components are characters in the hook representation constructed from all the
combinations of the arm and leg lengths of the original representation.
Previously it was shown that the identity persists in taking, for each
character, the matrix integration in the super Chern-Simons matrix model in the
grand canonical ensemble. We prove here that this Giambelli compatibility still
holds in the deformation of the fractional-brane background.
