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This monograph presents an intuitive theory of trial wave functions
for strongly interacting fermions in fractional quantum Hall
states. The correlation functions for the proposed fermion
interactions follow a novel algebraic approach that harnesses the
classical theory of invariants and semi-invariants of binary forms.
This approach can be viewed as a fitting and far-reaching
generalization of Laughlin's approach to trial wave functions.
Aesthetically viewed, it illustrates an attractive symbiosis
between the theory of invariants and the theory of correlations.
Early research into numerical diagonalization computations for
small numbers of electrons shows strong agreement with the
constructed trial wave functions.The monograph offers researchers
and students of condensed matter physics an accessible discussion
of this interesting area of research.
This monograph presents an intuitive theory of trial wave functions
for strongly interacting fermions in fractional quantum Hall
states. The correlation functions for the proposed fermion
interactions follow a novel algebraic approach that harnesses the
classical theory of invariants and semi-invariants of binary forms.
This approach can be viewed as a fitting and far-reaching
generalization of Laughlin's approach to trial wave functions.
Aesthetically viewed, it illustrates an attractive symbiosis
between the theory of invariants and the theory of correlations.
Early research into numerical diagonalization computations for
small numbers of electrons shows strong agreement with the
constructed trial wave functions.The monograph offers researchers
and students of condensed matter physics an accessible discussion
of this interesting area of research.
Several of the classical sequences in enumerative combinatorics
have q-generalizations arising as generating functions for
statistics defined on finite discrete structures. When q = 1, these
generating functions reduce to the original sequences. When q = -1,
on the other hand, one gets the difference in cardinalities between
those members of a set having an even value for some statistic (on
the set) with those members having an odd value. The current text
provides a systematic study of the case q = -1, giving both
algebraic and combinatorial treatments. For the latter, appropriate
sign-reversing involutions are defined on the associated class of
discrete structures. Among the structures studied are permutations,
binary sequences, Laguerre configurations, derangements, Catalan
words, and finite set partitions. As a consequence of our results,
we obtain bijective proofs of congruences involving Stirling, Bell,
and Catalan numbers. This text studies an interesting problem in
enumerative combinatorics and is suitable for an audience ranging
from motivated undergraduates to researchers in the field.
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