Chapter 4 - CHARACTERS OF MATRIX REPRESENTATIONS OF FINITE GROUPS

Abstract

This chapter focuses on the characters of the matrix representations. A number of important theorems concerning the properties of characters can be proved by using the great orthogonality theorem or the considerations that follow from it. The chapter presents and discusses these theorems with examples. The characters of matrices in the same class are identical. When summed over all the symmetry operations R, the character system of irreducible representations is orthogonal and normalized to the order of the group h. A necessary and sufficient condition for two representations to be equivalent is that the characters of the representations must be equal. Two inequivalent irreducible representations cannot have the same characters and irreducible representations with equal characters are equivalent. The number of inequivalent irreducible representations equals the number of classes of the group.