We study matrix integration over the classical Lie groups U (N ), Sp(2N ), O(2N ) and O(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz±Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern-Simons theory on S 3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.

We express minors of Toeplitz matrices of finite and large dimension, including the case of symbols with Fisher-Hartwig singularities, in terms of specializations of symmetric functions. By comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.