ABSTRACT

The main objectives of this project are explicit constructions of
representations, related to the fermionic formulas for characters of
representations of Virasoro or other W-algebras and affine Lie
algebras. The project also uses the ideas encountered in fermionic
constructions, to give formulas for graded multiplicities of
irreducible modules in the tensor product of finite-dimensional Lie
algebra modules, or integrable modules in the fusion product of affine
algebra modules (coinvariants or conformal blocks). Some of the
specific goals are: The study of fusion products of
representations of finite-dimensional simple Lie algebras; the
inductive limit of the fusion product as the number of factors becomes
infinite in a stabilized regime, which is expected to give new
realizations of integrable modules of affine Lie algebras;
combinatorial identities for q-series which result from these
constructions; semi-infinite constructions of arbitrary highest
weight representations of affine Lie algebras modules; and
applications of semi-infinite (c.f. Feigin and Styoanovskii's approach)
constructions in solutions of the fractional quantum Hall effect.

This research is at the interface of combinatorial representation
theory and mathematical physics. The constructions are guided by
conformal field theory and exactly solvable models in statistical
mechanics. The results are important in the representation theory of
Lie algebras and affine Lie algebras. Combinatorial questions such as
dimensions of weight spaces in irreducible representations,
multiplicities of irreducible components in tensor products of
Lie-algebra modules are related to the counting of certain matrix
elements or conformal blocks. The physical applications include the
study of wave functions in the quantum Hall effect, and
partition functions in statistical mechanical systems at criticality.

Please report errors in award information by writing to: awardsearch@nsf.gov.