Character expansions, Itzykson-Zuber integrals, and the QCD partition function

A combinatorial formula to generate $U\left(N\right)\mathrm{}$ character expansions is presented. It is shown that the resulting character expansion formulas greatly simplify a number of problems where integrals over the group manifolds need to be calculated. Several examples are given, including direct and very quick calculations of the Itzykson-Zuber integral and the finite volume effective partition function of QCD in the sector with a given topological charge.