Symmetric functions and vertex operator algebras

The project will explore connections between the EPSRC research areas of representation theory (algebra), combinatorics and geometry as well as mathematical physics. The central object of study will be the ring of symmetric functions, an important tool in all of the 3 mentioned areas. We will use the boson-fermion correspondence from mathematical physics, which maps configurations of fermions on an infinite lattice to symmetric functions describing bosons, in order to explore combinatorial aspects of the theory and to define so-called vertex operators which represent the main objects in a quantum field theory.

Aims and Objectives: Generalise the boson-fermion correspondence to new classes of symmetric functions and for finite lattice with the aim of constructing new vertex operators.

Novelty of the research methodology: We will employ models from statistical mechanics to describe the combinatorial aspects.

The last EPSRC international subject review has highlighted the need to strengthen cross-disciplinary links between subjects in the mathematical sciences. Our project will make an important contribution as it will involve techniques from 3 different areas. Possible collaborations and impact will be most likely to be of academic nature with presentations of the research at workshops and conferences planned during the final year of the thesis.