Mathematics – Representation Theory
Scientific paper
1999-03-26
Mathematics
Representation Theory
36 pages, uses plain TeX, added: new Section 6 (First Applications), several figures, and several new statements, made: minor
Scientific paper
The aim of this paper is to present a generalization of Lunn-Senior's mathematical model of isomerism in organic chemistry. The main idea of Lunn and Senior is that if the type of isomerism is fixed, a molecule with a fixed skeleton and d univalent substituents has a symmetry group $W\leq S_d$ which is generally not the molecule's 3-dimensional symmetry group. The unit character of W induces a representation of the symmetric group $S_d$ which governs the combinatorics of the isomers of the given molecule. Lunn-Senior's thesis is that certain non-negative integers established by this representation are upper boundaries of the corresponding numbers, yielded by the experiment (and often coincide with them). Moreover, the authors define (in a particular case) a partial order among the objects of the model, such that some simple substitution reactions correspond to inequalities. These two groups of data determine the group W, and produce so called "type properties" of the molecule (properties which do not depend on the nature of the univalent substituents). Our hypothesis is that if we replace the unit character of $W$ by any one-dimensional character of $W$ (thus we count only a part of the isomers - those having a maximum property), we also get a type property of the molecule. An instance of that is the inventory of the stereoisomers called chiral pairs. The formalism can be generalized naturally and produces some preliminary chemical results. Especially the partial order is defined and studied in the general case and indicates the possible genetic relations among the corresponding molecules. An important result of E. Ruch which connects the dominance order among partitions and the existence of chiral pairs is obtained as a consequence of a more general statement.
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