Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with fixed number of edges. The is also an associative algebra, where the product derived from permutation group products. existence and structure this algebra has consequences. product, which can be expressed in terms integer graph reconnection coefficients, used to Hamiltonians eigenvalues normalized characters symmetric elements degeneracies given Kronecker are tensor multiplicities representations. square coefficient for triple Young diagrams shown equal dimension sub-lattice lattice graphs. This leads answer long-standing question combinatorial interpretation coefficients. As avenues future research, we discuss applications mechanics algorithms computation. describe membrane these systems.