Fermionic Formulas For Unrestricted Kostka Polynomials And Superconformal Characters

The problem of finding fermionic formulas for the many generalizations of Kostka polynomials and for the characters of conformal field theories has been a very exciting research topic for the last few decades. In this dissertation we present new fermionic formulas for the unrestricted Kostka polynomials extending the work of Kirillov and Reshetikhin. We also present new fermionic formulas for the characters of N = 1 and N = 2 superconformal algebras which extend the work of Berkovich, McCoy and Schilling. Fermionic formulas for the unrestricted Kostka polynomials of type A n−1 in the case of symmetric and anti-symmetric crystal paths were given by Hatayama et al. We present fermionic formulas for the unrestricted Kostka polynomials of type A n−1 for all crystal paths based on Kirillov-Reshetihkin modules. Our formulas and method of proof even in the symmetric and anti-symmetric cases are different from the work of Hatayama et al. We interpret the fermionic formulas in terms of a new set of unrestricted rigged configurations. For the proof we give a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths which generalizes a bijection of Kirillov and Reshetikhin. We present fermionic formulas for the characters of N = 1 superconformal models SM(p, 2p + p) and SM(p, 3p − 2p), and the N = 2 superconformal model with central charge c = 3(1 − 2p p′ ). The method used to derive these formulas is known as the Bailey flow. We show Bailey flows from the nonunitary minimal model M(p, p) with p, p coprime positive integers to N = 1 and N = 2 superconformal algebras. We derive a new Ramond sector character formula for the N = 2 superconformal algebra with central charge c = 3(1− 2p p′ ) and calculate its fermionic formula.