We investigate the concept of replicated and $q$-replicated arguments in Schur and Hall-Littlewood symmetric functions. A description of ``dual'' compound symmetric functions is obtained with the help of functions of a replicated argument, while Schur and Hall-Littlewood functions of a $q$-replicated argument are both shown to be related to Macdonald's symmetric functions.
Various tensor product decompositions and winding subalgebra branching rules for the $N=1$ and $N=2$ superconformal algebras are examined by using the triple and quintuple product identities, and various generalizations thereof, concentrating on the particular cases when these decompositions are finite or multiplicity-free.
The boson-fermion correspondence is utilized to develop an algorithm for the calculation of outer products of Schur and $Q$-functions with power sum symmetric functions, and general (outer) multiplication of $S$-functions. A procedure is also developed for the evaluation of (outer) plethysms of Schur functions and power sums. A few examples are given which demonstrate the usefulness of this method for calculating plethysms between Schur functions. By examining the vertex operator realization of Hall-Littlewood functions we are also able to generate an algorithm for expressing Hall-Littlewood functions in terms of Schur functions. The operation of outer plethysm is defined for Hall-Littlewood functions and the algorithm developed for $S$-functions is extended to this case as well.
Kerov's generalized symmetric functions are used to provide a realization for level $k$ Fock space representations of the quantum affine algebra $U_q(\widehat{sl(2)})$. Using these functions, we derive a generalized Macdonald identity which enables the regularized trace of a product of $U_q(\widehat{sl(2)})$ currents to be calculated.
A zipped version of the thesis .dvi file is available
here or a gzipped version is here.
Gauge Theories in Three Dimensions
This thesis begins by considering scalar and spinor QED in 2+1 dimensions, performing perturbation theory to study its behaviour (without allowing the presence or dynamical generation of a parity-violating photon mass). It is found, as first noted by Jackiw and Templeton, that an IR instability prohibits such a perturbative study. The gauge technique is adopted as a non-perturbative alternative, and the photon is allowed to be ``dressed" in a cloud of fermion loops, yielding results which encompass the perturbation results in the UV region, whilst remaining finite at IR momenta.
Chern-Simons theory is then considered, where the photon is allowed to acquire a parity-violating mass. In order to use dimensional regularization to handle the apparently UV divergent integrals which appear, a new formulation of the theory is proposed, allowing the action to be written in arbitrary D dimensions, so that the integrals can be safely evaluated. It is also found that the IR problems which plague the conventional theory are no longer present, as the photon propagator behaviour has been ``softened" by the photon mass, allowing perturbation results to be obtained.
Finally, the idea of mass generation within these theories is considered in more detail, where we see that the presence of a fermion mass will cause a photon mass to be dynamically generated, and vice versa. These ideas are then generalized for arbitrary odd dimensional parity-violating theories.
The thesis postscript file is available in several formats: (Enable load to local disk)