Affine algebra representations and discrete integrability

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The research project exploits relations among (1) discrete integrable models in statistical mechanics, (2) representation theory of affine and quantum affine algebras, and (3) the combinatorics of dynamical systems such as cluster algebras. From the physical point of view, the project has applications to wall crossing formulas in string theory and the wave functions of quasi-particles responsible for the quantum Hall effect in condensed matter physics. The problems investigated include fermionic constructions of affine Lie algebra modules and their fusion products; Applications of methods from statistical physics to give explicit solutions for the cluster variables in cluster algebras related to discrete integrable systems such as T-systems or Q-systems, thereby giving proofs of the relevant positivity conjectures; and non-commutative generalizations of these systems, related to the Kontsevich non-commutative wall-crossing formula, or the quantum cluster algebras which describe quantum discrete Liouville or Hirota equations.This research is at the boundary between mathematics and physics. It seeks to apply techniques from statistical mechanics to solving problems in combinatorics and representation theory of affine Lie algebras and their quantization. The prime characteristic of these problems is integrability, that is, they arise from systems with a high degree of symmetry. Often this symmetry allows for finding explicit solutions in the form of physical partition functions, which are manifestly positive sums over configurations of a system. This positivity property is frequently a conjectured property of the underlying mathematical objects.