Algebraic and combinatorial structures in integrable systems

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The PI is proposing to study questions in combinatorial representation theory which arise from problems in mathematical physics, in particular, exactly solvable two-dimensional models in statistical mechanics and conformal field theory. Representations studied inlcude finite-dimensional and integrable modules of affine Lie algebras, loop algebras, and quantum affine algebras. Questions addressed include the Feigin Loktev conjecture for fusion product, formulas for refined (generalizations of) Littlewood-Richardson coefficients, fermionic character formulas for integrable modules of affine algebras, refinements of the Feigin-Stoyanovsky construction, and semi-infinite wedge products. The combinatorial questions include identities for certain fermionic sum formulas for multiplicity coefficients of KR-modules and their generalizations. A component of the project is a representation-theoretical construction of Baxter's matrices for generalized vertex models and the associated functional equations.Integrable models in statistical mechanics and quantum field theory arise in various contexts in physics and mathematics. Most recently, in the study of SLE, the fractional quantum Hall effect, models for entanglement in quantum mechanics and string theory. These models, which gave rise to the invention of quantum groups, have remarkable combinatorial properties and have been a fertile ground for studying properties of representations of Lie algebras and their deformations, as well as combinatorial and algebraic identities. For example the fermionic character formulas mentioned above are intimately related to fractional statistics or anyons.