  This paper is concerned with integrals which integrands are the monomials of
matrix elements of irreducible representations of classical groups. Based on
analysis on Young tableaux, we discuss some related duality theorems and
compute the asymptotics of the group integrals when the signatures of the
irreducible representations are fixed, as the rank of the classical groups go
to infinity. These group integrals have physical origins in quantum mechanics,
quantum information theory, and lattice Gauge theory.
