Séminaire du LPTMS: Christian Hagendorf *** séminaire exceptionnel ***


11:00 - 12:00

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405

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The spin one XXZ chain and symmetry classes of alternating sign matrices

Christian Hagendorf (Université Catholique de Louvain, Belgique)

The spin 1/2 XXZ Heisenberg chain is a prototype model in the statistical mechanics of magnetism. It describes the interaction of localised spins on a one-dimensional chain with anisotropy. In 2000, Razumov and Stroganov observed for a particular anisotropy a surprising appearance of integer sequences in the spin-chain ground state. These sequences enumerate so-called alternating sign matrices. They are well known to combinatorialists through rather different problems such as determinant calculus or the enumeration of plane partitions. In this talk, I give an overview of the relation between spin chains, models of statistical mechanics and alternating sign matrices. Furthermore, I discuss the spin one XXZ chain whose ground state displays for any anisotropy connections to enumeration problems of alternating sign matrices with symmetries. The proof of these connections uses tools from quantum integrability, in particular the algebraic Bethe ansatz and the quantum separation of variables method. (Joint work with Alexi Morin-Duchèsne)

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