Basic Notions of the Theory of the Symmetric Group

Abstract

The permutational symmetry properties of the spin functions and the spatial functions will play an important role both in the interpretation of the wave function and in the calculation of the matrix elements of the Hamil-tonian and of other operators. In this chapter we shall summarize some of the basic notions of the theory of the symmetric group which will be needed for the further treatment. Special attention will be paid to the Young theory: Young tableaux, Young operators, and the representations given by Young will be used quite frequently. Many of the theorems will be given without proof; for a systematic treatment the reader is referred to the textbooks on the symmetric group (Rutherford,⁽¹⁾ Robinson,⁽²⁾ Hamermesh,⁽³⁾ Boerner⁽⁴⁾). Some of these basic notions are presented in a very nice way in the review of Coleman⁽⁵⁾ (“The symmetric group made easy”).