  We study the algebra of invariant representative functions over the N-fold
Cartesian product of copies of a compact Lie group G modulo the action of
conjugation by the diagonal subgroup. We construct a basis of invariant
representative functions referred to as quasicharacters. The form of the
quasicharacters depends on the choice of a reduction scheme. We determine the
multiplication law of quasicharacters and express their structure constants in
terms of recoupling coefficients. Via this link, the choice of the reduction
scheme acquires an interpretation in terms of binary trees. We show explicitly
that the structure constants decompose into products over primitive elements of
9j symbol type. For SU(2), everything boils down to the combinatorics of
angular momentum theory. In the final part, we show that the above calculus
enables us to calculate the matrix elements of bi-invariant operators occuring
in quantum lattice gauge theory. In particular, both the quantum Hamiltonian
and the orbit type relations may be dealt with in this way, thus, reducing both
the construction of the costratification and the study of the spectral problem
to numerical problems in linear algebra. We spell out the spectral problem for
G=SU(2) and we present sample calculations of matrix elements of orbit type
relations for the gauge groups SU(2) and SU(3). The methods developed may be
useful in the study of virtually all quantum models with polynomial constraints
related to some symmetry.
