Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory

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. Finally, we apply this theory to the construction of the Hilbert space costratification of (finite) lattice quantum gauge theory. The methods developed in this paper may be useful in the study of virtually all quantum models with polynomial constraints related to some symmetry.