Fermionic Formulas: 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 book, which is a dissertation we present new fermionic formulas for the unrestricted Kostka polynomials extending the work of Kirillov and Reshetikhin. 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 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. 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. The method used to derive these formulas is known as the Bailey flow.