The diagram lattice as structural principle A. New aspects for representations and group algebra of the symmetric group B. Definition of classification character, mixing character, statistical order, statistical disorder; A general principle for the time evolution of irreversible processes

Abstract

It is the aim of this paper to demonstrate the significance of the diagram lattice. This lattice was defined in order to achieve structural insight into the phenomenon of chirality in chemistry. In this context, “Theorie der Chiralitätsfunktionen” [1] may serve as reference. In the introduction of the present paper a summary of the relevant theorems and definitions is given and a few examples of the diagram lattice are graphically illustrated. Parts A and B can be read independently and presuppose knowledge only of the introduction. Part A is of special interest for mathematicians, Part B and [1] for physicists and chemists.

In Part A theorems on the representations of the group$\mathfrak{S}$ _n and certain subgroups of it and on the structure of the group algebra are developed. In Part B the concept “classification character” with the two complementary aspects of “identification” and “distinction” is derived. With the interpretation “mixing character” the partial order relation gains an interpretation through a mixing process, which can be expressed by a bistochastic matrix. This results in another equivalent definition of the diagram lattice. Interpreted as mixing character of a statistical distribution a diagram represents “statistical order” and “statistical disorder” by its row partition and column partition respectively. These concepts and the corresponding lattice structure lead to the hypothesis of growing mixing character as a criterion for the time evolution of isolated systems. The criterion of increasing entropy provides a much weaker condition. A discussion of the master equation leads to a proof of the principle of growing mixing character.[/p]