Computer Algebra and Material Design

Abstract: This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group theory, have ripened as a new trend, called "computer algebra", outcomes of which now can be available as handy computational packages, and would also be useful to physicists with practical purposes. This article, in the former part, explains how to use the computer algebra for the applications in the solid-state simulation, by means of one of the computer algebra package, the GAP system. The computer algebra enables us to obtain various group theoretical properties with ease, such as the representations, the character tables, the subgroups, etc. Furthermore it would grant us a new perspective of material design, which could be executed in mathematically rigorous and systematic way. Some technical details and some computations which require the knowledge of a little higher mathematics (but computable easily by the computer algebra) are also given. The selected topics will provide the reader with some insights toward the dominating role of the symmetry in crystal, or, the "mathematical first principles" in it. In the latter part of the article, we analyze the relation between the structural symmetry and the electronic structure in C$_{60}$ (as an example to the sysmem without periodicity). The principal object of the study is to illustrate the hierarchical change of the quantum-physical properties of the molecule, in accordance with the reduction of the symmetry (as it descends down in the ladder of subgroups). In order to serve the common interest of the researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained as minutely as possible.