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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 formulationin terms of determinants and minors of Toeplitz$\pm$Hankel matrices. Weestablish a number of factorizations and expansions for such integrals, alsowith insertions of irreducible characters. As a specific example, we computeboth at finite and large $N$ the partition functions, Wilson loops and Hopflinks of Chern-Simons theory on $S^{3}$ with the aforementioned symmetrygroups. The identities found for the general models translate in this contextto relations between observables of the theory. Finally, we use characterexpansions to evaluate averages in random matrix ensembles of Chern-Simonstype, describing the spectra of solvable fermionic models with matrix degreesof freedom.

Matrix models for classical groups and Toeplitz$\pm $Hankel minors with applications to Chern-Simons theory and fermionic models