Irreducible representations

Abstract

The irreducible representations of the line groups are the starting point for physical applications. Quite general Wigner's theorem [1] singles out unitary representations as the relevant ones in the quantum mechanical framework. Such representations are decomposable to the irreducible components, which are ingredients sufficient for composition of any unitary representations. Hence, in this chapter, we construct and tabulate irreducible unitary representations only, although line groups, since being not compact, have also the non-unitary representations. The construction starts with the first family groups. Then we use simple (induction) procedure to derive the representations of the families 2-8, containing the halving first family subgroup; finally, we use these representations repeating the same procedure in order to get the representations of the largest families (with the first family subgroup of index four). At the end, we make an overview of their properties and physical implications. © 2010 Springer-Verlag Berlin Heidelberg.