  We define solvable quantum mechanical systems on a Hilbert space spanned by
bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also
an associative algebra, where the product is derived from permutation group
products. The existence and structure of this Hilbert space algebra has a
number of consequences. The algebra product, which can be expressed in terms of
integer ribbon graph reconnection coefficients, is used to define solvable
Hamiltonians with eigenvalues expressed in terms of normalized characters of
symmetric group elements and degeneracies given in terms of Kronecker
coefficients, which are tensor product multiplicities of symmetric group
representations. The square of the Kronecker coefficient for a triple of Young
diagrams is shown to be equal to the dimension of a sub-lattice in the lattice
of ribbon graphs. This leads to an answer to the long-standing question of a
combinatoric interpretation of the Kronecker coefficients. As an avenue to
explore quantum supremacy and its implications for computational complexity
theory, we outline experiments to detect non-vanishing Kronecker coefficients
for hypothetical quantum realizations/simulations of these quantum systems. The
correspondence between ribbon graphs and Belyi maps leads to an interpretation
of these quantum mechanical systems in terms of quantum membrane world-volumes
interpolating between string geometries.
