1. (Chapter 1 Title: Introduction to Quantum Hall States.) Historical background of many-body interactions in the context of integral and fractional quantum Hall effect. Review of experimentally observed families of IQL states and their interpretations 2. (Chapter 2 Title: The composite Fermion hierarchy and justification of the CF approach.) Jain's mean field composite Fermion picture, Laughlin-Jastrow type correlations. The work of the University of Tennessee group in condensed matter physics justifying MFCF under certain conditions on the interaction energy. The concept of Effective CF angular momentum associated to the lowest CF Landau level. Partially filled shells and Quasi-electons. Haldane's heirarchy of Laughlin correlated daughter-states and Jain's sequence of IQL states 3. (Chapter 3 Title: Correlation diagrams and the Algebra of correlation functions.) Justification of the CF approach. Establishing (with rigorous proofs) conditions under which Jain's elegant CF approach correctly predicts the angular momentum multiplets. Correlation diagram approach to Moore-Read states provides a new insight and a more general view. Fully symmetric correlation functions for N Fermions in an IQL state for any filing factor less than , obtained from balanced correlation diagrams. Detailed treatment of the algebraic correlation functions and diagrams for small values of N; overlap with the numerical work on diagonalization. 4. (Chapter 4 Title: Invariant-theoretic essentials.) A self-contained brief introduction to the mathematical theory of invariants of binary forms in context of correlation diagrams and their associated correlation functions. Semi-invariants of binary forms and dimension counting theorems. The relationship between the semi-invariants and angular momentum multiplets in presence of quasi-electrons. 5. (Chapter 5 Title: Constructions of correlation diagrams and their correlation functions.) Theorems proving existence of nontrivial fully symmetric correlation functions for a class of correlation diagrams. Symmetry groups of the balanced correlation diagrams and their utility from the computational point of view. 6. (Chapter 6 Title: Trial wave functions for systems of Fermions in IQL state with general filling factor $n / (2 p n + - 1) < 1/2$.) Concrete constructions of fully symmetric correlation functions and detailed examination of their special algebraic features. 7. (Chapter 7 Title: Correlation functions for some Configurations with quasi-electrons and related open questions. ) Concrete constructions of fully symmetric correlation functions for configurations of N interacting Fermions some of which are quasi-electrons, for a family total angular momenta. Various open mathematical questions directly related to such constructions. ukryj opis