Monomial integrals on the classical groups

This paper presents a powerful method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group by one of the authors, Gorin [J. Math. Phys., 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows us to compute a given monomial integral very efficiently. The result is always a rational function of the matrix dimension. Zapotitlán 2008 American Institute of Physics