Immanants of blocks from random matrices in some unitary ensembles

The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong–Ou–Mandel effect and the boson sampling problem. In that context, it would be useful to know the distribution of the permanent and other immanants for random matrices, but that seems a difficult problem. We advance this program by calculating the average of the squared modulus of a generic immanant for blocks from random matrices in the unitary group, in the orthogonal group and in the circular orthogonal ensemble. In the case of the permanent in the unitary group, we also compute the variance. Our approach is based on Weingarten functions and factorizations of permutations. In the course of our calculations we are led to two curious conjectures relating dimensions of irreducible representations of the orthogonal and symplectic groups to the value of zonal and symplectic zonal polynomials at the identity. (paper)