  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$\pm$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.
