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Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
Ben Geloun, Joseph1 ; Ramgoolam, Sanjaye2
1 Laboratoire d’Informatique de Paris Nord UMR CNRS 7030 Université Sorbonne Paris Nord, 99, avenue J.-B. Clement, 93430 Villetaneuse, France. International Chair in Mathematical Physics and Applications, ICMPA–UNESCO Chair, 072 B.P. 50 Cotonou, Benin.
2 School of Physics and Astronomy Centre for Research in String Theory Queen Mary University of London London E1 4NS United Kingdom. School of Physics and Mandelstam Institute for Theoretical Physics, University of Witwatersrand, Wits, 2050, South Africa.
Algebraic Combinatorics, Tome 6 (2023) no. 2, pp. 547-594

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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 combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.

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DOI : 10.5802/alco.254
Classification : 05E10, 05C85, 05A19, 06B99, 22D20
Keywords: Belyi maps, Kronecker coefficients, quantum physics, Ribbon graphs

Ben Geloun, Joseph 1 ; Ramgoolam, Sanjaye 2

1 Laboratoire d’Informatique de Paris Nord UMR CNRS 7030 Université Sorbonne Paris Nord, 99, avenue J.-B. Clement, 93430 Villetaneuse, France. International Chair in Mathematical Physics and Applications, ICMPA–UNESCO Chair, 072 B.P. 50 Cotonou, Benin.
2 School of Physics and Astronomy Centre for Research in String Theory Queen Mary University of London London E1 4NS United Kingdom. School of Physics and Mandelstam Institute for Theoretical Physics, University of Witwatersrand, Wits, 2050, South Africa.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Quantum mechanics of bipartite ribbon graphs: {Integrality,} {Lattices} and {Kronecker} coefficients},
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Ben Geloun, Joseph; Ramgoolam, Sanjaye. Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients. Algebraic Combinatorics, Tome 6 (2023) no. 2, pp. 547-594. doi: 10.5802/alco.254

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