Joseph Bengeloun - Quantum Mechanics of Bipartite Ribbon Graphs...
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2021-02-04

Quantum Mechanics of Bipartite Ribbon Graphs: A Combinatorial Interpretation of the Kronecker Coefficient.


The action of subgroups on a product of symmetric groups allows one to enumerate different families of graphs. In particular, bipartite ribbon graphs (with at most edges) enumerate as the orbits of the adjoint action on two copies of the symmetric group (of order n!). These graphs form a basis of an algebra, which is also a Hilbert space for a certain sesquilinear form. Acting on this Hilbert space, we define operators which are Hermitians. We are therefore in the presence of a quantum mechanical model. We show that the multiplicities of the eigenvalues of these operators are precisely the Kronecker coefficients, well known in representation theory. We then prove that there exists an algorithm that delivers the Kronecker coefficients and allow us to interpret those as the dimension of a sub-lattice of the lattice of the ribbon graphs.Thus, this provides an answer to Murnaghan’s question (Amer. J. Math, 1938) on the combinatorial interpretation of the Kronecker coefficient.